Voronoi tilings, toric arrangements and degenerations of line bundles III
aa r X i v : . [ m a t h . AG ] F e b VORONOI TILINGS, TORIC ARRANGEMENTS ANDDEGENERATIONS OF LINE BUNDLES III
OMID AMINI AND EDUARDO ESTEVES
Abstract.
We describe limits of line bundles on nodal curves in terms of toric arrange-ments associated to Voronoi tilings of Euclidean spaces. These tilings encode informationon the relationship between the possibly infinitely many limits, and ultimately give riseto a new definition of limit linear series . This article and the first two that preceded itare the first in a series aimed to explore this new approach.In Part I, we set up the combinatorial framework and showed how graphs weightedwith integer lengths associated to the edges provide tilings of Euclidean spaces by certainpolytopes associated to the graph itself and to its subgraphs.In Part II, we described the arrangements of toric varieties associated to the tilingsof Part I in several ways: using normal fans, as unions of orbits, by equations and asdegenerations of tori.In the present Part III, we show how these combinatorial and toric frameworks allowus to describe all stable limits of a family of line bundles along a degenerating family ofcurves. Our main result asserts that the collection of all these limits is parametrized bya connected 0-dimensional closed substack of the Artin stack of all torsion-free rank-onesheaves on the limit curve. Moreover, we thoroughly describe this closed substack and allthe closed substacks that arise in this way as certain torus quotients of the arrangementsof toric varieties of Part II determined by the Voronoi tilings of Euclidean spaces studiedin Part I.
Contents
1. Introduction 22. The space of embedded sheaves 123. Degenerations of line bundles I 244. Degenerations of line bundles II 315. Regeneration 41References 47
Date : February 4, 2021. Introduction
This is a sequel to our previous works [AE20a, AE20b] whose aim is to achieve thedescription of stable limits of line bundles on nodal curves by means of graph theory andtoric geometry. This is motivated by the desire to understand all the possible limits oflinear series g rd over any sequence X , X , . . . of smooth projective curves of genus g whosecorresponding points x , x , . . . in M g converge to a given point x on the Deligne-Mumfordcompactification M g .In this introduction, after providing an overview of the previous works, we focus on thecontribution of this paper and its preceding companions and briefly discuss the content ofour forthcoming work.1.1. Overview.
Nodal curves are curves (one-dimensional, reduced, connected but notnecessarily irreducible projective schemes over an algebraically closed field) that fail tobe smooth in the weakest possible form: the singularities are normal crossings, that is,ordinary nodes. Among them, the most important are (Deligne–Mumford) stable curves,characterized as those having ample canonical bundle. This is in fact the property thatallows for the construction of their moduli space, M g , where g stands for the (arithmetic)genus. One of the key properties of stable curves is the Stable Reduction Theorem, whichsays that a family of stable curves parameterized by the germ of a punctured smooth curvecan be completed, after a finite base change, in a unique way to a family over the wholegerm. It implies that the moduli of stable curves is complete, in fact projective, and isthus a compactification for the moduli of smooth curves of genus at least two. However,no such thing holds, in general, for line bundles and linear series over curves.A linear series over a curve X is simply the data of a line bundle and a linear subspaceof the space of sections of that bundle. If d is the (total) degree of the line bundle and r is the projective dimension of the subspace, we say we have a g rd . If r ≥
0, it correspondsto a rational map X P r , whence the importance of their study for understanding theprojective geometry of curves. Line bundles over a curve are parametrized by the Picardscheme, and the linear series by fibrations by Grassmannians over the Picard scheme. It isall well if X is smooth, as the Picard scheme has projective connected components indexedby the degree d ; the one with degree 0 is even an algebraic group, the Jacobian.However, the Picard scheme of a stable curve has projective components only if the curveis of compact type. Furthermore, whereas for smooth curves, a g rd gives rise to a rationalmap that can even be uniquely extended to the whole curve, the map induced by a g rd ona stable curve may fail to be defined on whole components of the curve. Finally, there isnothing similar to stable reduction; quite to the contrary, the trivial line bundle over thetotal space of a family of smooth curves parameterized by the germ of a punctured smoothcurve can be completed in infinitely many ways over the whole germ if the family of curvesis completed by adding a reducible stable curve. ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 3
A compactification of the Picard scheme of an irreducible curve was suggested by Mum-ford in the sixties, carried out by D’Souza [D’S79] in the seventies, and thoroughly studiedby Altman and Kleiman [AK80, AK79] and others in the years that followed. As it waslater observed by Eisenbud and Harris [EH83], it is possible to complete the proof of thecelebrated Brill–Noether Theorem by considering this compactification over rational irre-ducible cuspidal curves. The compactification parameterizes torsion-free, rank-one sheaves,and the notion of linear series extends simply by just considering spaces of sections of thesesheaves. In fact, in this setup it is even possible to extend the notion of divisors: theseare the pseudo-divisors studied by Hartshorne [Har06]. Finally, there are fine proper mod-uli spaces parameterizing torsion-free, rank-one sheaves of a given degree for families ofirreducible curves.Though even in codimension one (in their moduli), stable curves fail to be irreducible,fortunately, general reducible stable curves are of compact type. Curves of compact typeare those curves for which the connected components of the Picard scheme are compact,indeed projective. Eisenbud and Harris studied degenerations of line bundles along familiesof smooth curves degenerating to curves of compact type, and coined the term limit linearseries [EH86]. Remarkably, they observed that it was actually useful that the family of linebundles could degenerate to many limit line bundles, as just one limit would rarely carryenough information about the degeneration. They found out that for many applicationsit was important, and enough, to consider as many limit line bundles as components ofthe limit stable curve, each focusing on one of the components. Their study allowed fora deep understanding of the moduli of stable curves up to codimension two, and foundmany applications in describing the geometry of a general curve and the theory of divisorsof M g [EH86, EH87a, EH87b]. It is thus no surprise they wished for an extension of theirtheory to more general curves; as they wrote in [EH89]: “The known special cases sufficefor many applications, but there is probably a gold mine awaiting a general insight.”Already in the late seventies, the study of degenerations to more complicated nodalcurves began, in the hands of Oda and Seshadri [OS79]. To do away with the problem ofmany limits line bundles, they introduced the notion of stability, one that had naturallyarisen in the study of vector bundles over smooth curves and the construction of theirmoduli space by Mumford’s Geometric Invariant Theory. Out of the many limits a familyof line bundles could have, at most one would be stable, and a finite nonzero number ofthem would be semistable. Moreover, the semistable limits would be equivalent in the sensethey would all have the same quotients in their Jordan–H¨older filtrations. The convenientnotion of stability has been centerpiece in all the studies that followed, including in thelandmark construction of a compactification of the relative Picard scheme over the wholemoduli of curves by Caporaso in the early 90’s [Cap94].However, as far as limits of linear series are considered, stability appears to be veryrestrictive. The line bundles Eisenbud and Harris considered, one for each component ofthe stable curve, would rarely be semistable; even if one could force one of them to be OMID AMINI AND EDUARDO ESTEVES semistable, by deforming the notion of stability, the others would seldom be. Moreover,generally semistable line bundles in the same equivalence class have almost completelyunrelated spaces of sections.On the other hand, an approach similar to that by Eisenbud and Harris, choosing foreach component of the limit curve a “best” limit line bundle, was tentatively carried out bymany: Ziv Ran had an early draft on this already in the 80’s, whereas the second author,in collaboration with Medeiros and Salehyan [EM02, ES07], used this approach to studylimits of Weierstrass points for a wide class of stable curves in the nineties. However, onecould not carry the approach further.In the middle of the first decade of the present century, Osserman [Oss06] introduced anew idea, that not only extended the theory to include curves in positive characteristic,but also modified substantially the approach by Eisenbud and Harris. Osserman’s ideawas remarkably simple, even though it seemed at first to complicate further the study:One should consider not only the “best” limit line bundle for each component of the limitcurve, but also all of those in between, more precisely, all of those with nonnegative degreeon every component; the intermediate limits carried only partial information over eachcomponent, but perhaps crucial information nonetheless. And, in fact, if Eisenbud’s andHarris’s limit linear series was not refined, Osserman found out that the extra limits didcarry more information. For instance, Osserman and the second author discovered laterthat they described limits of divisors of the family of linear series that cannot be accountedfor only by the “best” line bundles [EO13].The approach by Osserman introduced a new challenge: How to account for, how tokeep track of all the data that show up when considering all effective limit line bundles?Osserman introduced his approach first only for two-component curves of compact type, thesimplest case, and recently extended the approach to more general curves [Oss19b, Oss16],specifically, curves of pseudo-compact type where a parallel theory to that by Eisenbudand Harris can be established.This was not the only challenge: another was to consider one-parameter families ofcurves whose total space was not smooth. For such families, when the limit curve is notof compact type, it might as well happen that a family of line bundles has no line bundlein the limit. One can blow up the total space and replace the stable curve by one of itssemistable models, splitting apart each node to introduce a chain of a variable numberof rational smooth curves. This way one obtains a limit line bundle, but on a differentcurve! Worse, one has to deal with a whole set of new variables, one for each node: thenumber of rational curves in each chain. This procedure, semistable reduction, has alreadyappeared when dealing with curves of compact type; it did not lead to many difficultiesin applications because in them, a class of curves was considered, and the class did notchange by semistable reduction. For more general curves, as it became apparent in thework by Esteves and Medeiros [EM02] on curves with two components, the specific type of
ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 5 reduction that appeared had enormous influence on, for instance, the limits of Weierstrasspoints.The whole study has been combinatorially intensive from the start, already in Oda–Seshadri [OS79]. Farther than obstructing the study of the problem, combinatorics hasbeen one of the main tools used by all those that ventured in the field. It has even spanneda new line of approach, using tropical and non-Archimedean geometry, that has been veryfruitful. In fact, the first author in a joint work with Baker [AB15] has introduced a notionof limit linear series which also extends, in a somewhat more combinatorial way originatingfrom non-Archimedean analysis, the definition by Eisenbud and Harris. He has used thisapproach as an alternative tool in the study of reduction of Weierstrass points and theirdistributions [Ami14]. Moreover, this circle of ideas has been used by Jensen and Payne toprove specific cases of the Maximal Rank Conjecture [JP14, JP16, JP17], as well as in therecent work by Farkas, Jensen and Payne on the Kodaira dimension of the moduli spacesof curves of genus 22 and 23 [FJP20]. A comparison of Osserman’s approach [Oss19b] tothe work of Amini and Baker [AB15] can be found in [Oss19a].The overall approach we take in these series of works features a new interplay between thecombinatorics and the geometry. We do not do away with any data (as it is done, sometimesharmlessly for the applications in mind), but we use the combinatorics to organize all theinformation. In a nutshell we prove in the present paper that the collection of all limittorsion-free, rank-one sheaves of a family of line bundles along a family of smooth curvesdegenerating to a nodal curve X is parameterized by a connected 0-dimensional closedsubstack I of the Artin stack J of all torsion-free, rank-one sheaves on X .More significatively, we characterize combinatorially all the closed substacks I that arisefrom degenerations. The characterization is given by Theorems 4.7 and 5.1, which are themain results in the article. In short, the I are certain torus quotient of a combinatorialarrangement of toric varieties. Thus our theorems provide a far-reaching generalization toall nodal curves of the combinatorial-toric approach carried out by Esteves and Medeirosin [EM02] for the case of stable curves with two irreducible components.In a future work, if a family of linear series is given, in addition to the substack I ,we will show the existence of a closed substack I ′ of G , the Grassmann fibration over J parameterizing vector spaces of sections of torsion-free, rank-one sheaves on X , satisfyingtwo conditions: First, the stack I ′ will lie over I . Second, the induced relative torsion-free rank-one sheaf I on X × I ′ over I ′ and the induced locally free subsheaf V of thepushforward p ∗ I will be such that their restrictions over the points of I ′ parameterize allthe limits of the family of linear series. We will describe I ′ combinatorially as well; it isactually very similar in esprit to the description we give of I .The connection with the work by Osserman will be elaborated in our future study. Inthe case of a two-component curve of compact type, this has been carried out in detailin the doctoral thesis by Rizzo [Riz13], who explains how the collection of linear seriesconsidered by Osserman as a limit linear series for a two-component curve of compact OMID AMINI AND EDUARDO ESTEVES type can actually be viewed as members of a G m -equivariant family of linear series on X parameterized by a 2-punctured chain of rational curves. The quotient of this chain by G m is the stack I ′ truncated in nonnegative degrees.1.2. The present work.
We will now proceed to describe the work we do in this paper,in continuation to what we did in [AE20a] and [AE20b]. Later we will point out what liesahead in the path we are taking.Fix a connected nodal curve X over an algebraically closed field κ . Consider its asso-ciated dual graph G = ( V, E ), which is a pair consisting of a vertex set V in one-to-onecorrespondence with the set of irreducible components of X , and an edge set E in one-to-one correspondence with the set of nodes of X . An edge connects two vertices if thecorresponding node lies on the two corresponding components. For our purposes, we dis-card those edges that form a loop.Let now E be the set of all the oriented edges (also called arcs) obtained out of E : foreach edge, there are two possible arcs, pointing to the two different vertices connected bythe edge. For e ∈ E , we write e = uv to mean that e is an arc connecting u to v , evenif it might not be the only one. Also, we let t e denote the tail and h e the head of e . Inaddition, e denotes the same edge with the reverse orientation.Recall that given a commutative ring A , one associates to the graph G the complex of A -modules d A : C ( G, A ) → C ( G, A ) . Here, C ( G, A ) is the A -module of functions V → A , and C ( G, A ) is the A -module of allfunctions f : E → A satisfying f ( e ) = − f ( e ) for each e ∈ E . And d A ( f )( e ) = f ( v ) − f ( u )for each e = uv ∈ E .The characteristic functions χ v , for v in V , form a basis of the A -module C ( G, A ),whereas the functions χ e − χ e , for a collection of e ∈ E giving an orientation to the whole G , form a basis of C ( G, A ). There are bilinear forms h , i on C ( G, A ) and C ( G, A )satisfying h χ v , χ w i = δ v,w for v, w ∈ V ; h χ e − χ e , χ f − χ f i = δ e,f − δ e,f for e, f ∈ E . Define the homomorphism d ∗ A : C ( G, A ) → C ( G, A ) by putting d ∗ A ( χ e − χ e ) := χ v − χ u for each e = uv ∈ E . Then d ∗ A is the adjoint to d A , that is, h f, d ∗ A ( h ) i = h d A ( f ) , h i foreach f ∈ C ( G, A ) and h ∈ C ( G, A ). In addition, the degree map deg : C ( G, A ) → A ,sending f to P v ∈ V f ( v ), is a cokernel for d ∗ A . The kernel is H ( G, A ).Let H ,A := { f ∈ C ( G, A ) | deg ( f ) = 0 } and F A := Im( d A ). Let ∆ A := d ∗ A d A , theLaplacian of G . The homomorphism d ∗ A induces an injection F A → H ,A . For A = R , itis a bijection, and the bilinear form h , i on C ( G, R ) induces by restriction a norm on F R corresponding via d ∗ R to the quadratic form q on H , R satisfying q ( f ) = h f, ∆ R ( f ) i for each f ∈ C ( G, R ). ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 7
In [AE20a], we described a certain family of tilings of H , R by polytopes. Each tilingconsists of a family of polytopes covering H , R such that • each face of a polytope which is in the tiling belongs itself to the tiling; and • the intersection of a finite number of polytopes in the tiling is a face of each of thepolytopes.By removing from a polytope in the tiling all the faces of positive codimension it contains,we get the corresponding open face. The open faces form then a stratification of the wholespace H , R . We call tiles the polytopes of maximum dimension.For instance, let Λ A := Im( d ∗ A ). Then Λ R = H , R and Λ Z is a sublattice of H , Z of finiteindex equal to the number of spanning trees of G , by the Kirchhoff Matrix Tree Theorem.The standard Voronoi tiling of G is the Voronoi decomposition Vor G of H , R with respectto Λ Z and q . The tiles areVor G ( β ) := (cid:8) η ∈ H , R | q ( η − β ) ≤ q ( η − α ) for every α ∈ Λ Z − { β } (cid:9) for β ∈ Λ Z .This is one of the infinitely many tilings we consider. There are variants of it, thatwe call twisted mixed Voronoi tilings and denote Vor m G,ℓ . Though the standard Voronoitiling is homogeneous, meaning all tiles are translates of the tile centered at the origin,Vor G ( O ), a twisted mixed Voronoi tiling is obtained by putting together translations ofthe tiles Vor H ( O ) associated to connected spanning subgraphs H of G . More precisely,the twisted Voronoi tiling Vor m G,ℓ depends on m ∈ C ( G, Z ) (the “twisting”) and an edgelength function ℓ : E → N ; its tiles are the polytopes d ∗ R ( d m f ) + Vor G m f ( O ) for f ∈ C ( G, Z )with G m f connected, where d m f ∈ C ( G, R ) is a modification of d Z ( f ), namely d m f ( e ) := δ m ,ℓe ( f ) − δ m ,ℓe ( f )2 , where δ m ,ℓe ( f ) := j f ( v ) − f ( u ) + m ( e ) ℓ ( e ) k for each e = uv ∈ E ,and G m f is the spanning subgraph of G obtaining by removing the edges e ∈ E for which d m f ( e ) Z . We refer to [AE20a] for a thorough presentation of these tilings.In the present article we establish a correspondence between the stratifications of H , R associated to the Vor m G,ℓ and the stable limits of line bundles in one-parameter smoothingsof X . Properties of each stratum, and the way they fit together in the stratification of H , R are reflected in properties of the limits and the relationship between them.More explicitly, let π : X → B be a (one-parameter) smoothing of X . Here, B is thespectrum of κ [[ t ]] and π is a projective flat morphism whose generic fiber is smooth andspecial fiber is isomorphic to X . We fix such an isomorphism. Let η and o be the genericand special points of B . The total space X is regular except possibly at the nodes of X .For e ∈ E , the completion of the local ring of X at the corresponding node N e is κ [[ t ]]-isomorphic to κ [[ u, v, t ]] / ( uv − t ℓ e ) for a certain ℓ e >
0, called the singularity degree of π at N e . If ℓ e = 1, then X is regular at N e . If all ℓ e = 1, then π is said to be Cartier ; it is regular if X is regular at all the nodes of X . A finite base change is obtained by sending OMID AMINI AND EDUARDO ESTEVES t to t n for some n . The resulting family is similar to the original one: the special fiber isthe same, the generic fiber is a base field extension of the original one, but the singularitydegrees ℓ e change to nℓ e .Let L η be an invertible sheaf on the generic fiber. If π is Cartier, it extends to an almost invertible sheaf L on X , a sheaf that is invertible at all N e . It is not unique, as L ⊗ O X ( P f ( v ) X v ) is another extension for each f ∈ C ( G, Z ). (Here X v is the componentof X corresponding to v ∈ V , which can and will be viewed as a Cartier divisor of X because π is Cartier.) For general π , the sheaf L η extends to a relatively torsion-free,rank-one sheaf I on X /B , that is, a B -flat coherent sheaf on X whose fibers over B aretorsion-free, rank-one. Again, it is not the unique extension: in [Est01] a procedure similarto the one explicited above shows how to change I into other extensions. Furthermore,one could do a finite base change to π , extend L η to the new generic fiber and consider itsextensions. Of course, they will be extensions on a different total space. But the specialfibers are the same, and thus the restrictions of all these extensions to X are torsion-free,rank-one sheaves that we call the stable limits of L η .Let J denote the Artin stack parameterizing torsion-free, rank-one sheaves on X . Itis the disjoint union of the closed and open substacks J d , each parameterizing sheaves I with degree d , that is, with χ ( I ) − χ ( O X ) = d . Letting d := deg ( L η ), we may considerthe subset I of J d parameterizing all the stable limits of L η . We proceed to describe I thoroughly.First, we give a meaningful structure to J as a quotient stack, as follows. Fixing b ∈ C ( G, Z ), we let J b := Y v ∈ V J b ( v ) v , where J b ( v ) v parameterizes torsion-free, rank-one sheaves of degree b ( v ) on the component X v for each v ∈ V . Over J b we construct a scheme R J b parameterizing gluings along thenodes N e associated to e ∈ E of the sheaves given by points of J b , their modifications andtheir degenerations.More precisely, given torsion-free, rank-one sheaves K v on X v for each v ∈ V , we viewtheir gluings along the nodes as subsheaves of the direct sum ⊕ v K v whose quotients aresupported with length 1 on each and every node N e . This allows for “degenerate” gluings,and thus for torsion-free sheaves that may fail to be invertible at part or all of the N e . Theparameter space for these subsheaves is a product of P , one for each e ∈ E , and describesan irreducible component of the fiber of R J b over the point on J b parameterizing the K v .The other components are obtained by modifying the K v as follows. Fix c ∈ C ( G, Z ).For each c ∈ C ( G, Z ), put K c − cv := K v ⊗ O X v (cid:16) X e ∈ E h e = v ( c ( e ) − c ( e )) N e (cid:17) ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 9 and do the same gluing as above; we obtain another irreducible component (if c = c ) ofthe fiber of R J b over the same point on J b parameterizing the K v , and all components areobtained this way.The fibers of R J b over J b are thus arrangements of an infinite number of simple toricvarieties. Each fiber is the same: what we call the arrangement of toric varieties or simplytoric tiling R associated to the Voronoi decomposition of C ( G, R ) in hypercubes withrespect to C ( G, Z ). More precisely, each Voronoi tile is a rational polytope, to eachrational polytope we may associate its normal fan, and to the normal fan the correspondingtoric variety. The polytopes in the Voronoi decomposition form a complex, in the sensethat each two of them intersect in a common face, when they intersect. We may thus gluethe toric varieties associated to each two polytopes by identifying the orbit closures of thecommon face in each variety with the toric variety associated to the face itself, viewed asa polytope of smaller dimension. This gives us R .There is nothing special about the above Voronoi decomposition with regard to theabove construction. It can be carried out for any tiling of an Euclidean space by rationalpolytopes intersecting each other in faces. As we have already mentioned, we considerother tilings in the present article.The scheme R J b parameterizes torsion-free, rank-one sheaves with a lot of redundancy.Two groups act on it independently. First, there is a natural action of G V m / G m , wherewe view G m , the multiplicative group of κ , embedded diagonally in G V m . The actionis given by observing that G V m is the automorphism group of ⊕ v K c − cv for each point on J b parameterizing the K v and each c ∈ C ( G, Z ), and that a subsheaf is fixed under thediagonal action of G m . The action preserves each fiber of R J b over J b and each componentof that fiber.The second group action moves fibers around. Recall H ( G, Z ), the group of cycles of G , the kernel of d ∗ Z . For each γ ∈ C ( G, Z ) and torsion-free rank-one sheaves K v on X v foreach v ∈ V , put: K − γv := K v ⊗ O X v (cid:16) − X e ∈ E h e = v γ ( e ) N e (cid:17) . If γ ∈ H ( G, Z ) and the K v are parameterized by a point s ∈ J b , so are the K − γv ,parameterized by a point we denote by τ γ ( s ). Then τ γ is an automorphism of J b . Itlifts in a rather trivial way to an automorphism of R J b : Since c − ( c + γ ) = − γ + ( c − c ),we may associate to a point on the component corresponding to c + γ of the fiber of R J b over s the point on the component corresponding to c of the fiber of R J b over τ γ ( s )parameterizing the same subsheaf of the same direct sum, for each c ∈ C ( G, Z ). Theaction of H ( G, Z ) may move fibers and is free, as even if a fiber is fixed, its componentsare not. The two actions are independent and J d can be obtained as the quotient stack:(1.1) J d = " R J b H ( G, Z ) × G V m / G m . We may thus describe the collection I of stable limits of a sheaf L η by describing itsinverse image in R J b . Our Theorem 4.7 claims that this inverse image is the disjointunion of certain connected subschemes of certain fibers of R J b over J b , each subschemeisomorphic to its image in the quotient S d := " R J b H ( G, Z ) , all the images being the same. Under chosen identifications we may view each subschemeas a subscheme of the arrangement of toric varieties R . We prove this subscheme is Y a,bℓ, m for certain choices of ℓ , m , a and b . Each Y a,bℓ, m is itself an arrangement of toric varieties ofdimension | V | − Y a,bℓ, m ⊆ R : characters a : C ( G, Z ) → G m ( κ ) and b : H ( G, Z ) → G m ( κ ), an edge length function ℓ : E → N and an element m ∈ C ( G, Z )we call a twisting. These data arise from the smoothing π : X → B and from L η , asfollows: The length function is simply the collection of singularity degrees ℓ e ; and thecharacter a keeps track of the infinitesimal data of the arc defined by π on the moduli M g or on a versal deformation space of X . The character b describes the gluing data of analmost invertible stable limit, if L η admits one, and then we may set m = 0, no twisting isnecessary. If not, then L η admits an almost invertible limit on X ℓ , the semistable modelof X obtained by splitting the branches of each N e apart, and connecting them by a chainof ℓ e − e ∈ E . It even admits an admissible almostinvertible limit, meaning an almost invertible limit whose restriction to each component ofeach chain has degree zero, but possibly one, where the degree is one. Then b is related tothe gluing data of an admissible almost invertible limit and the twisting m keeps track ofwhere that limit has degree one on each added chain.Different choices of a , b and m may yield the same subscheme Y a,bℓ, m ⊆ R . We have leftthis analysis for a later work. But the structure of Y a,bℓ, m depends on ℓ and m only, itsequations being a deformation of the equations defining Y , ℓ, m , which we denote by Y bt ℓ, m andcall the basic toric tiling, as pointed out in [AE20b], Prop. 4.6.In [AE20b], Section 4, we explained how Y bt ℓ, m is determined from the tiling Vor m G,ℓ of H , R . There we remark we can naturally view Y bt ℓ, m and its deformations Y a,bℓ, m as closedsubschemes of R .Finally, we may consider the subscheme Y a,bℓ, m ⊆ R for arbitrary choices of a , b , ℓ and m ,and under chosen identifications, as a subscheme of a fiber of R J b over J b . If we denoteby I the image of this subscheme in the quotient J d , our Regeneration Theorem 5.1 claims ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 11 that I parameterizes the collection of stable limits of an invertible sheaf under a smoothingof X with singularity degrees ℓ e .Besides the above description, Y a,bℓ, m is also described in [AE20b], Subsection 4.2, by givingthe equations of each of its irreducible components in the corresponding component of R ,and in [AE20b], Thm. 5.3, by means of its orbits under the action of G V m / G m . Thesecharacterizations are the ones we use in the present article.But we have also described Y a,bℓ, m in [AE20b], Thm. 6.3, globally by its (infinitely many)equations in R , and in [AE20b], Thm. 7.2, as an equivariant degeneration of the torus G V m / G m . The first description will be important in understanding the moduli of the Y a,bℓ, m ,to be worked out later. And the second yields that the connected 0-dimensional substack I of J d parameterizing a collection of stable limits is the degeneration of a point!1.3. Future work.
To attain our goal of giving a new definition of limit linear series andconstructing their moduli space, our series of articles will continue. We summarize nowwhat comes ahead.First of all, we have constructed J d as the quotient expressed in (1.1), we have shown itparameterizes torsion-free, rank-one, degree- d sheaves on X , but we have not proved it isactually the moduli stack of these sheaves. That J d is bound to be the moduli stack is anobservation by Margarida Melo and Filippo Viviani, and the proof that it actually is willappear later.Second, given a smoothing π : X → B of X and an invertible sheaf L η on its genericfiber, we may consider the punctured arc in the relative Artin stack J X /B parameterizingtorsion-free rank-one sheaves on the fibers of π . It is natural to think that by adding theArtin substack I of stable limits of L η that the punctured arc will be completed to a B -flatclosed substack A ⊆ J X /B . To show this, we need to construct A , the degeneration ofa point to I , whose existence we proved in [AE20b], Thm. 7.2, and moreover a relativetorsion-free, rank-one sheaf on X × B A whose restriction to the generic fiber is L η . Thiswill also appear later.Third, we will describe the moduli of all collections of stable limits, which is tantamountto describing the various ways the Y a,bℓ, m appear as subschemes of fibers of R J b over J b . Al-ready mentioned above, the global equations of Y a,bℓ, m in R , laid out by [AE20b], Thm. 6.3,and a thorough description of how Y a,bℓ, m actually depends on a , b , ℓ and m are the fundamen-tal pieces in this construction. However, since the Y a,bℓ, m have infinitely many components,the moduli will be formal. To get an actual moduli, we need to truncate R J b . The mostobvious truncation is to restrict to the open subscheme parameterizing sheaves obtainedas gluings of sheaves of nonnegative degrees on the components X v , as for understandinglimits of effective divisors these sheaves are enough.Fourth, the moduli of collections of stable limits can be thought of as a new compactifica-tion of the Picard scheme of the curve X . To show it is natural we will construct a relativeversion of it over the moduli stack of stable curves M g . The construction may require a local analysis and blowups of M g , in the way carried out by Main`o in her constructionof the moduli space of enriched curves [Mai98]. But further blowups will be necessary,infinitely many to get the formal moduli, but finitely many for the truncated moduli.Finally, as mentioned before, our notion of limit linear series, and the construction oftheir moduli space will be similar to what is done for smooth curves: If I is a closed substackof J d parameterizing a collection of stable limits, a “limit linear series” of “sections” of thiscollection is a locally free subsheaf V of the pushforward p ∗ I satisfying certain conditions,where I is the sheaf induced on X × I (or certain base extensions) by the universal sheafover J d .1.4. Organization.
The layout of the paper is as follows. In Section 2 we construct thequotient stack J d describing thoroughly its atlas and the group action. In Section 3 weconsider smoothings of a nodal curve and describe limits of line bundles. Limits of thetrivial bundle are considered in detail, and are explained in terms of the versal deformationspace of the curve, expanding on work done in [Mai98] and [EM02]. In Section 4 we proveone of our main theorems, Theorem 4.7, describing collections of stable limits as quotientsby G V m / G m of the Y a,bℓ, m . Finally, in Section 5 we show that any such a quotient is acollection of stable limits, our Theorem 5.1.1.5. Basic notations.
In addition to the notations already introduced, throughout thepresent article, we will denote by N e the node of X associated to each e ∈ E and by X v the irreducible component of X associated to each v ∈ V .Given a collection A ⊆ E of edges, an orientation of A is simply a section o : A → E over A of the forgetful map E → E . We denote by A o the image of the orientation. To simplifythe presentation, we fix an orientation o : E → E for G . Given e ∈ E , we will write e aswell for the (non-oriented) edge in E . Given e ∈ E , we denote e o := o ( e ).We will drop the subscripts from d A and d ∗ A when appropriate. Given α in C ( G, Z )(resp. C ( G, Z )), we write α x := α ( x ) for each vertex (resp. oriented edge) x of G . Similarly,given a character a of C ( G, Z ) (resp. C ( G, Z )), we write a v := a ( χ v ) for each v ∈ V (resp. a e := a ( χ e − χ e ) for each e ∈ E ). The multiplicative group is denoted G m .2. The space of embedded sheaves
Gluing sheaves.
For each v ∈ V and each d ∈ Z , let J dv denote the degree- d compact-ified Jacobian of X v ; it is a projective variety of dimension g v , where g v is the (arithmetic)genus of X v , parameterizing torsion-free, rank-one sheaves on X v of degree d . We refer to[Est01] for basic definitions and results concerning torsion-free rank-one sheaves.Fix an integer d and let b ∈ C ( G, Z ) be of degree d , meaning P v ∈ V b v = d . Define J b := Y v ∈ V J b v v . Note that J b is a projective variety of dimension P g v , and comes with natural projectionmaps to J b v v for each v ∈ V . ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 13
For each vertex v ∈ V , let ι v : X v ֒ → X denote the inclusion map, and L v the pullback to X v × J b of a “Poincar´e sheaf,” or universal sheaf on X v × J b v v . Observe that L v is actuallynot well-defined, in the sense that a Poincar´e sheaf is not unique. However, if we pick apoint P v on the smooth locus of X v , then L v may be defined by imposing the conditionthat it be rigid at P v , that is, L v | P v × J b ∼ = O P v × J b . At any rate, this is not relevant to us,so we just choose one Poincar´e sheaf L v for each vertex v ∈ V . For a point s on J b , wedenote by L v ( s ) the restriction of L v to X v ∼ = X v × s .For each oriented edge e = uv ∈ E , let F e := L u | N e × J b ⊕ L v | N e × J b . It is a rank-two vector bundle over J b , under the natural isomorphism N e × J b → J b . Let P J b ( F e ) be the corresponding P -bundle over J b , and define P J b := Y e ∈ E o P J b ( F e ) , the product fibered over J b . It is a P -bundle over J b , where P := Y e ∈ E o P . Fix now an element c ∈ C ( G, Z ). For each v ∈ V , let e L v denote the pullback of( ι v × J b ) ∗ L v to X × P J b . We will also denote by e F e the pullback of F e to N e × P J b andby q c e : e F e −→ M e the pullback of the universal quotient over P J b ( F e ) to N e × P J b for each e ∈ E o . Noticethat e F e = e L t e | N e × P Jb ⊕ e L h e | N e × P Jb for each e ∈ E o . We use the quotients q c e to construct a natural relative torsion-free, rank-one, degree- d sheaf I c on X × P J b / P J b , the kernel of the composition of surjections: M v ∈ V e L v −→ M v ∈ V M e ∈ E o e ∋ v e L v | N e × P Jb = M e ∈ E o (cid:16) e L t e | N e × P Jb ⊕ e L h e | N e × P Jb (cid:17) −→ M e ∈ E o M e . For a point t on P J b , denote by I c ( t ) the induced torsion-free rank-one sheaf on X ≃ X × t .For each c ∈ C ( G, Z ), we may modify the above construction as follows: Let L cv := L v (cid:16) X e ∈ E h e = v ( c e − c e ) N e × J b (cid:17) , and do the same construction as above but replacing the sheaves L v with the sheaves L cv .More precisely, for each e = uv ∈ E , put F c e e := L cu | N e × J b ⊕ L cv | N e × J b , and set P c J b := Y e ∈ E o P J b ( F c e e ) , the fibered product over J b . Instead of I c , let I c be the relative torsion-free, rank-one,degree- d sheaf on X × P c J b / P c J b obtained from the pullbacks q ce to N e × P c J b of the universalquotients on P J b ( F c e e ), instead of the q c e . As above, for a point t on P c J b , we denote by I c ( t ) the induced torsion-free rank-one sheaf on X . We will also say that a torsion-free,rank-one sheaf I is represented by t ∈ P c J b if I c ( t ) ∼ = I .Observe that for t ∈ P c J b lying over s ∈ J b , and for e = uv ∈ E o , if I c ( t ) is invertible at N e , then it generates both L cu ( s ) and L cv ( s ) in a neighborhood of N e . On the other hand,if I c ( t ) fails to be invertible at N e , then either q ce ( t )( L cu ( s ) | N e ) = 0 or q ce ( t )( L cv ( s ) | N e ) = 0.In the first case, in a neighborhood of N e , the subsheaf of L cv ( s ) generated by I c ( t ) is L cv ( s )( − N e ), whereas that of L cu ( s ) is L cu ( s ) itself. In the second case, the reverse is true,that is, the same statement holds with u and v exchanged. Letting E t denote the set ofedges e ∈ E for which I c ( t ) fails to be invertible at N e , we obtain an orientation o ct : E t → E by assigning to each e ∈ E t the oriented edge whose head v is such that I c ( t ) generates L cv ( s )( − N e ) in a neighborhood of N e . Thus I c ( t ) generates the subsheaf L cv ( s ) (cid:16) − X e ∈ E ct h e = v N e (cid:17) for each v ∈ V , where, for simplicity, we put E ct := E o ct t .2.2. The atlas.
We shall view the sheaves I c as restrictions of a sheaf defined over a largerbase, containing all the schemes P c J b for all c ∈ C ( G, Z ) as closed subschemes. This worksas follows. For each vertex v ∈ V , and each natural number n ∈ N , define the sheaf L ( n ) v as L ( n ) v := L v (cid:16) n X e ∈ E o e ∋ v N e × J b (cid:17) . By means of the natural embeddings O X v ֒ → O X v ( N e ), we may view all the sheaves L cv , forbounded c ∈ C ( G, Z ), more precisely for those c verifying | c e − c e | ≤ n for every e ∈ E ,as subsheaves of the sheaf L ( n ) v . We may thus view the schemes P c J b as closed subschemesof the relative Quot-schemeQuot X × J b / J b (cid:16) M v ∈ V ( ι v × J b ) ∗ L ( n ) v (cid:17) , more precisely, of the piece of the relative Quot-scheme parameterizing subsheaves of rankone and degree d or, equivalently, quotients of finite length equal to (2 n + 1) | E | . ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 15
Given two orientations o and o of a subset A ⊆ E , define ( o , o ) ∈ C ( G, Z ) by( o , o ) e := e ∈ A o − A o , − e ∈ A o − A o ,0 otherwise. Proposition 2.1.
Notations as above, let c, c ′ ∈ C ( G, Z ) with | c e − c e | ≤ n and | c ′ e − c e | ≤ n for all e ∈ E . Viewing P c J b and P c ′ J b in the Quot-scheme, the following statements aretrue:(1) P c J b intersects P c ′ J b if and only if | c ′ e − c e | ≤ for every e ∈ E .(2) More precisely, given t ∈ P c J b , we have that t ∈ P c ′ J b if and only if c − c ′ = ( o ′ , o ct ) where o ′ : E t → E is an orientation of the set E t of edges e for which I c ( t ) fails tobe invertible at N e . In this case, o ′ = o c ′ t .Proof. The first statement follows from the second. Indeed, if t ∈ P c J b ∩ P c ′ J b , then State-ment (2) yields that | c ′ e − c e | = | ( o ′ , o ct ) e | ≤ e ∈ E . Conversely, if | c ′ e − c e | ≤ e ∈ E , we may choose a point t ∈ P c J b such that E t is the subset of edges e ∈ E inthe support of c − c ′ and E ct is the collection of oriented edges e such that c ′ e − c e = 1. If o ′ is the “opposite orientation,” that is, E o ′ t is the collection of oriented edges e such that c e − c ′ e = 1, then c − c ′ = ( o ′ , o t ). Thus, t ∈ P c ′ J b by Statement (2).As for Statement (2), observe first that I c ( t ) is a subsheaf of L ι v ∗ L ( n ) v ( s ) such that I c ( t ) ⊆ M v ∈ V ι v ∗ L cv ( s ) (cid:16) − X e ∈ E ct h e = v N e (cid:17) , where s ∈ J b lies under t . Thus, for each other orientation o ′ of E t , we have I c ( t ) ⊆ M v ∈ V ι v ∗ L cv ( s ) (cid:16) X e ∈ E o ′ t h e = v N e − X e ∈ E ct h e = v N e (cid:17) = M v ∈ V ι v ∗ L c ′ v ( s )if c ′ = c − ( o ′ , o ct ), and thus I c ( t ) = I c ′ ( t ′ ) as subsheaves of L ι v ∗ L ( n ) v ( s ) for a certain t ′ ∈ P c ′ J b such that E t ′ = E t and o c ′ t ′ = o ′ . So t = t ′ in the Quot-scheme, and thus t ∈ P c ′ J b .Conversely, if t = t ′ ∈ P c ′ J b as well, then(2.1) I c ( t ) ⊆ M v ∈ V ι v ∗ L cv ( s ) (cid:16) − X e ∈ A h e = v ( c ′ e − c e ) N e (cid:17) where A := { e ∈ E | c ′ e > c e } . Since I c ( t ) generates L cu ( s ) and L cv ( s )( − N e ) in a neigh-borhood of N e for each e = uv ∈ E ct , it follows that | c ′ e − c e | ≤ e ∈ E . Also, c ′ e − c e = 1 only if e ∈ E ct , that is A ⊆ E ct . Similarly,(2.2) I c ′ ( t ′ ) ⊆ M v ∈ V ι v ∗ L c ′ v ( s ) (cid:16) − X e ∈ A ′ h e = v ( c e − c ′ e ) N e (cid:17) where A ′ := { e ∈ E | c e > c ′ e } . Then A ′ ⊆ E c ′ t ′ .Note that E t = E t ′ , since I c ( t ) = I c ′ ( t ′ ). Thus o c ′ t ′ is another orientation for E t . Also, if e ∈ A then e ∈ A ′ , and hence e ∈ E ct and e ∈ E c ′ t . Thus A ⊆ E ct − E c ′ t ′ .On the other hand, let e ∈ E such that e ∈ E ct − E c ′ t ′ . As subsheaves of L ι v ∗ L ( n ) v ( s ), thetwo sheaves in (2.1) are equal to the corresponding ones in (2.2). Suppose by contradictionthat e A . Then c ′ e ≤ c e . Since e ∈ E ct , we have that I c ( t ) is contained in M v ∈ V −{ h e } ι v ∗ L cv ( s ) (cid:16) − X f ∈ A h f = v ( c ′ f − c f ) N f (cid:17) M ι h e ∗ L c h e ( s ) (cid:16) − X f ∈ A h f =h e ( c ′ f − c f ) N f − N e (cid:17) . But, because of the equality of (2.1) and (2.2), also I c ′ ( t ′ ) is contained in M v ∈ V −{ h e } ι v ∗ L c ′ v ( s ) (cid:16) − X f ∈ A ′ h f = v ( c f − c ′ f ) N f (cid:17) M ι h e ∗ L c ′ h e ( s ) (cid:16) − X f ∈ A ′ h f =h e ( c f − c ′ f ) N f − N e (cid:17) . But then e ∈ E c ′ t ′ , an absurd.It follows that A = E ct − E c ′ t ′ . Then c − c ′ = ( o c ′ t ′ , o ct ). (cid:3) It follows from Proposition 2.1 that we may let n tend to ∞ , and consider the union ofthe P c J b for all c ∈ C ( G, Z ). We will denote this union by R J b := [ c ∈ C ( G, Z ) P c J b . It is a scheme locally of finite type over J b . In fact, there is another way of describing R J b ,which shows that R J b is naturally fibered over J b with fibers equal to R , for the scheme R defined in [AE20b], Subsection 3.2, and recalled below.More precisely, for each e = uv ∈ E o and i ∈ Z , let F ie := L u ( − ( c e − i ) N e × J b ) | N e × J b ⊕ L v (( c e − i ) N e × J b ) | N e × J b . Note that this definition is compatible with that of F c e e , given previously, as the two sheavescoincide if i = c e .As before, we may view the P J b ( F ie ) for integers i with | i − c e | ≤ n as closed subschemesof the relative Quot-schemeQuot X × J b / J b (cid:16) ( ι u × J b ) ∗ (cid:0) L u (cid:0) nN e × J b (cid:1)(cid:1) ⊕ ( ι v × J b ) ∗ (cid:0) L v (cid:0) nN e × J b (cid:1)(cid:1) (cid:17) , more precisely, of the component of the relative Quot-scheme parameterizing quotients offinite length equal to 2 n + 1. Letting n tend to ∞ , we may consider the union of all the ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 17 P J b ( F ie ) for all i ∈ Z . We will denote this union by R e J b := [ i ∈ Z P J b ( F ie ) . It is an R e -bundle over J b , where R e is the doubly infinite chain of smooth rational curvesover κ , as in [AE20b], Subsection 3.2. In addition, R J b can be naturally identified withthe fibered product of the R e J b for all e ∈ E o over J b , that is, R J b = Y e ∈ E o R e J b . It is an R -bundle over J b , where R := Q e ∈ E o R e .Yet more precisely, we identify a fiber of R J b / J b with R in the following way: A point s ∈ J b corresponds to a collection of torsion-free, rank-one sheaves ( L v ; v ∈ V ) with L v ofdegree b v . We fix trivializations L v | N e ∼ = κ for each e ∈ E and each v ∈ e . We fix as welltrivializations O X v ( N e ) | N e ∼ = κ for each e ∈ E and each v ∈ e , and consider the inducedtrivializations O X v ( mN e + D ) | N e ∼ = κ for each m ∈ Z and each Cartier divisor D of X v notcontaining N e in its support. These trivializations induce trivializations L v (cid:16) X e ∈ E h e = v ( c e − c e ) N e (cid:17) | N f ∼ = κ for each f ∈ E , each v ∈ f and each c ∈ C ( G, Z ). These trivializations give rise to anisomorphism between the fiber of R E J b / J b over s and R .The universal subsheaf on X × Quot X × J b / J b (cid:16) M v ∈ V (cid:0) ι v × J b (cid:1) ∗ L ( n ) v (cid:17) for n → ∞ restricts to a relative torsion-free, rank-one, degree- d sheaf on X × R J b / R J b ,which shall be denoted by I . Its restriction to the subscheme X × P c J b is I c for each c ∈ C ( G, Z ).For each c ∈ C ( G, Z ), an open dense subscheme of P c J b parameterizes invertible sheavesof multidegree b + d ∗ Z ( c − c ). Since deg is a cokernel for d ∗ Z , it follows that, up to translation,we could have changed c for any other 1-cochain and b for any other 0-cochain of degree d .Moreover, the orientation o is just a convenient means of ordering the curves in the chain R e for each e ∈ E ; a different orientation would lead to the same fibration R J b / J b .Another way of interpreting Proposition 2.1 is through the following definition andproposition: Given t ∈ R J b , let c ∈ C ( G, Z ) such that t ∈ P c J b . Define c ( t ) ∈ C ( G, Q )by: c ( t ) e := c e + + if e ∈ E ct , − if e ∈ E ct ,0 otherwise. Notice that E t := { e ∈ E | c ( t ) e Z } is precisely the set of oriented edges supported in E t .Define as well, for each c ∈ C ( G, Z ), L cv := L v (cid:16) X e ∈ E h e = v ⌊ c e − c e ⌋ N e × J b (cid:17) and P c J b := \ c ′ ∈ C ( G, Z ) | c ′ e − c e |≤ ∀ e ∈ E P c ′ J b . Proposition 2.2.
Let t ∈ R J b . Then the following statements are true:(1) c ( t ) is well-defined.(2) t ∈ P c J b for c ∈ C ( G, Z ) if and only if | c e − c ( t ) e | ≤ for every e ∈ E . Inparticular, t ∈ P c ( t ) J b .(3) The torsion-free sheaf I ( t ) v generated by I ( t ) on X v for each v ∈ V is isomorphicto L c ( t ) v ( s ) , where s ∈ J b is the point lying under t .Proof. Suppose t lies on P c J b ∩ P c ′ J b . It follows from Proposition 2.1 that c ′ − c = ( o ct , o c ′ t ).Then c ′ e := c e + +1 if e ∈ E ct − E c ′ t − e ∈ E c ′ t − E ct c ( t ) e − − if e ∈ E c ′ t + if e ∈ E c ′ t E ct and E c ′ t are orientations of the same set, E t . Thus, thedefinition of c ( t ) does not change if c ′ were chosen instead of c .Furthermore, from the above argument it follows that | c e − c ( t ) e | ≤ for each e ∈ E and c ∈ C ( G, Z ) such that t ∈ P c J b . Conversely, let c ∈ C ( G, Z ) such that t ∈ P c J b . If c ′ ∈ C ( G, Z ) is such that | c ′ e − c ( t ) e | ≤ for every e ∈ E , then c ′ e = c ( t ) e = c e for each e ∈ E such that c ( t ) e ∈ Z , that is, such that e E t . On the other hand, if e ∈ E t then | c ′ e − c ( t ) e | = . Let o ′ be the orientation of E t such that e ∈ E o ′ t if and only if c ( t ) e − c ′ e = .Then c − c ′ = ( o ′ , o ct ), and it follows from Proposition 2.1 that t ∈ P c ′ J b .Finally, let s ∈ J b be the point lying under t . Let c ∈ C ( G, Z ) such that t ∈ P c J b . Then I c ( t ) generates the subsheaf L cv ( s ) (cid:16) − X e ∈ E ct h e = v N e (cid:17) of L cv ( s ) for each v ∈ V . But this subsheaf is L c ( t ) v ( s ). (cid:3) Group action.
There is a natural action on R J b by H ( G, Z ). Indeed, given γ ∈ H ( G, Z ) and a point s of J b , corresponding to a collection of torsion-free, rank-one sheaves ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 19 ( L v ; v ∈ V ), we associate s ′ ∈ J b , corresponding to the tuple of sheaves ( L − γv ; v ∈ V ),where L − γv := L v ( − X e ∈ E h e = v γ e N e ) . This association gives a map τ γ : J b → J b which sends s to s ′ . Clearly, γ τ γ gives agroup homomorphism H ( G, Z ) → Aut( J b ). Also, L cv ( τ γ ( s )) = L c + γv ( s ) for each s ∈ J b , c ∈ C ( G, Z ) and v ∈ V. By construction, for each point s ∈ J b , the restrictions of I c + γ over points on the fiber of P c + γ J b over s are exactly the same restrictions of I c over points on the fiber of P c J b over τ γ ( s ).Thus, the translation τ γ : J b → J b lifts to an automorphism e τ γ : R J b → R J b sending P c + γ J b to P c J b for each c ∈ C ( G, Z ). More precisely, given t ∈ P c + γ J b over s ∈ J b , correspondingto the subsheaf I c + γ ( t ) ⊆ M v L c + γv ( s ) ,τ γ sends s to the point s ′ corresponding to the tuple ( L v ( s ) − γ ; v ∈ V ) and e τ γ sends t tothe point on P c J b over s ′ which corresponds to the same subsheaf I c + γ ( t ). In other words, I c ( e τ γ ( t )) = I c + γ ( t ) as subsheaves of M v L cv ( τ γ ( s )) = M v L c + γv ( s ) . Proposition 2.3.
Notations as above, the assignment γ e τ γ defines an injective grouphomomorphism H ( G, Z ) → Aut( R J b ) . Moreover, e τ γ has fixed points only if γ = 0 .Proof. By construction, the assignment e τ • : H ( G, Z ) → Aut( R J b ) is clearly a grouphomomorphism. We prove the second statement, from which the injectivity of e τ • follows.Let γ ∈ H ( G, Z ), and suppose e τ γ has a fixed point t ∈ P c J b for a certain c ∈ C ( G, Z ).Since e τ γ sends P c ′ + γ J b to P c ′ J b for each c ′ ∈ C ( G, Z ), it follows that t lies on P c − nγ J b for everyinteger n ≥
0. This is possible, by Proposition 2.1, only if γ = 0. (cid:3) The action of H ( G, Z ) on R J b is thus free. We may consider the quotient stack: S d := " R J b H ( G, Z ) . Since I ( t ) = I ( e τ γ ( t )) for each t ∈ R E J b and γ ∈ H ( G, Z ), the sheaf I descends to atorsion-free, rank-one, degree- d sheaf on X × S d / S d , which we shall also denote by I .We call S d the moduli of embedded sheaves . We have the following moduli description. Proposition 2.4.
The stack S d parameterizes the data of torsion-free, rank-one sheaves M v on X v for each v ∈ V , and subsheaves I ⊆ ⊕ M v of degree d such that all the inducedmaps h v : I → M v are surjective. Proof.
For each such data, let E I be the collection of edges e ∈ E for which I fails to beinvertible at N e . Let u : E I → E be an orientation. For each v ∈ V , let f M v := M v (cid:16) X e ∈ E u I h e = v N e (cid:17) . Then P v ∈ V deg ( f M v ) = d , whence there are c ∈ C ( G, Z ) and s ∈ J b such that f M v ∼ = L cv ( s )for each v ∈ V . (Given c , the point s is unique.) Now, since I ⊆ M v ∈ V M v ⊆ M v ∈ V L cv ( s ) , it follows that the data corresponds to a unique point t on P c J b over s . Since the h v aresurjective, o ct = u .A choice u ′ : E I → E different from u would correspond to a point t ′ on P c ′ J b over thesame s , for c ′ := c − ( u ′ , u ). Note however that t ′ = t on R J b by Proposition 2.1.Furthermore, for a fixed u , any other choice of c differs from the one above by an elementof H ( G, Z ). More precisely, if c ′ ∈ C ( G, Z ) and s ′ ∈ J b are such that f M v ∼ = L c ′ v ( s ′ ) forevery v ∈ V , then γ := c − c ′ ∈ H ( G, Z ) and s ′ = τ γ ( s ). As the construction yields thesame subsheaf I , we have that the corresponding point t ′ on P c ′ J b over s ′ satisfies t ′ = e τ γ ( t ).Then the images of t and t ′ on S d are the same.Conversely, given a point on S d , let t ∈ R J b be a lifting, and s ∈ J b its image. Let c bean element of C ( G, Z ) such that t ∈ P c J b . Then s corresponds to the sheaves L v ( s ) for v ∈ V and t to the subsheaf(2.3) I c ( t ) ⊆ M v ∈ V L cv ( s ) . For each v ∈ V , the sheaf I c ( t ) generates the subsheaf M v := L cv ( s )( − X e ∈ E ct h e = v N e ) , Then I := I c ( t ) ⊆ M v ∈ V M v with surjective induced maps I → M v .Picking a different c will not change the data of the M v and I ⊆ ⊕ M v , since replacing ⊕L cv ( s ) by the larger sheaf ⊕ v L ( n ) v ( s ) for large n does not change the sheaves M v obtained.If t ′ ∈ R J b is another lifting, lying over s ′ ∈ J b , then s ′ = τ γ ( s ) and t ′ = e τ γ ( t ) for acertain γ ∈ H ( G, Z ). Since t ∈ P c J b we have t ′ ∈ P c − γ J b . Also, the inclusion in (2.3) is thesame as I c − γ ( t ′ ) ⊆ M v ∈ V L c − γv ( s ′ ) . Thus we obtain the same data of sheaves M v on X v and subsheaf of ⊕ M v . ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 21
We omit the simple verification that the two procedures described above are inverse toeach other. (cid:3)
Observe that a surjection h v : I → M v from a torsion-free, rank-one sheaf on X to oneon X v induces an isomorphism e h v : I v → M v , where I v is the restriction of I to X v modulotorsion. So we may view the data defining a point on S d as the equivalence class of thedata of an abstract torsion-free, rank-one degree- d sheaf I on X together with elements z v ∈ G m ( κ ) for each v ∈ V ; these data give rise to a map I ֒ −→ ⊕ I v ( z v ; v ∈ V ) −−−−−−−→ ⊕ I v , and the equivalence identifies data with the same image.Another way to put this is as follows. Observe that there is a natural action of thecharacter group of C ( G, Z ), henceforth denoted G V m , on R J b . In fact, the action of each z ∈ G V m ( κ ) is the restriction of the one on Quot X × J b / J b (cid:16) L v ∈ V ( ι v × J b ) ∗ L ( n ) v (cid:17) inducedby the isomorphisms L ( n ) v z v −→ L ( n ) v for v ∈ V .
If all the z v are equal, the action is the identity. We have thus defined an action of G V m / G m , where G m is embedded diagonally in G V m , on R J b that sends a fiber over J b toitself. Moreover, the isomorphism between a fiber of R E J b over J b and R is equivariant withrespect to the action of G V m / G m on R defined in [AE20b], Subsection 5.1, and recalledbelow in Section 4. Also, the action on R J b restricts to an action on P c J b for each c ∈ C ( G, Z ). Finally, this action commutes with the action of H ( G, Z ) and thus induces anaction on S d .The (abstract) torsion-free, rank-one sheaves I of degree d on X correspond to orbits of S d under G V m / G m . The specific point on the orbit tells us how I is viewed inside ⊕ I v . So,we may view the space of orbits, the quotient stack J d := " S d G V m / G m , as a parameter space for torsion-free, rank-one sheaves of degree d on X . The sheaf I on X × S d is G V m / G m -invariant, and thus descends to a relative torsion-free, rank-one, degree- d sheaf on X × J d / J d , a universal torsion-free, rank-one, degree- d sheaf.However, the quotient J d is not well-behaved because the orbits of the action havevariable dimension. Proposition 2.5.
The orbits of maximum dimension of S d under the action of G V m / G m have dimension | V | − and correspond to simple torsion-free, rank-one sheaves of degree d .Proof. Simple, torsion-free, rank-one sheaves on X are direct images of invertible sheaveson connected partial desingularizations of X . So the proof reduces to considering invertible sheaves on X , and to showing that the subscheme of the Picard scheme of X parameterizinginvertible sheaves with given restrictions to the components X v is isomorphic to G | E |−| V | +1 m .This has been shown by a number of authors — see e.g. [OS79, 10.2] — and follows formallyfrom the exact sequence1 → H ( G, G m ) → Pic ( X ) → Y v ∈ V Pic ( X v ) → . (cid:3) Thus, restricting to orbits of dimension | V | −
1, and considering the orbit space, we getthe moduli space of simple, torsion-free, rank-one sheaves of degree d on X . This space hasbeen constructed by Altman and Kleiman as an algebraic space [AK80, AK79]. (Actually,they prove the representability by an algebraic space of a functor parameterizing far moregeneral objects.) The second author of the present article has shown later that this spaceis actually a scheme locally of finite type and universally closed over the field, though notseparated; see [Est01].The second author has also considered certain open subspaces of the moduli space ofsimple sheaves which are actually proper over the field, actually projective [Est09], thusproducing various compactifications of the Jacobian, even for general families of curvessatisfying certain mild conditions [Est01]. Instead of doing this, however, we will considerclusters of orbits. The following proposition, which collects statements proved along thesection and states new ones, will serve us in describing these clusters. Proposition 2.6.
Let I be a torsion-free, rank-one, degree- d sheaf on X . Let E I be thecollection of edges e ∈ E for which I fails to be invertible at N e . For each v ∈ V , let I v denote the maximum torsion-free quotient of I | X v . Then the following statements hold:(1) For each orientation u : E I → E , there are c ∈ C ( G, Z ) and s ∈ J b such that (2.4) L cv ( s ) ∼ = I v (cid:16) X e ∈ E u I h e = v N e (cid:17) for each v ∈ V. (2) For each u , c and s satisfying Equations (2.4) , there is a point t ∈ P c J b over s representing I with o ct = u .(3) For each point s ∈ J b and each element c ∈ C ( G, Z ) satisfying e ∈ E I if and onlyif c e Z , and such that (2.5) L c v ( s ) ∼ = I v for each v ∈ V, there is a point t ∈ P c J b over s representing I with c ( t ) = c .(4) Conversely, given c ∈ C ( G, Z ) and s ∈ J b , and a point t on the fiber of P c J b over s representing I , Equations (2.4) hold for u := o ct and Equations (2.5) hold for c := c ( t ) . ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 23 (5) For each two sets of data ( u , c , s ) and ( u , c , s ) satisfying (2.4) , and each twopoints t and t representing I on the fibers of P c J b and P c J b over s and s , respec-tively, with o c t = u and o c t = u , we have that γ := c − c + ( u , u ) = c ( t ) − c ( t ) is in H ( G, Z ) , satisfies τ γ ( s ) = s and is such that t differs from e τ γ ( t ) by theaction of an element of G V m / G m ( κ ) .Proof. For each orientation u : E I → E , let f u ∈ C ( G, Z ) be defined by f u ( v ) := deg ( I v ) + | E u I ( v ) | for each v ∈ V , where E u I ( v ) := { e ∈ E u I | h e = v } . Then (2.4) is satisfied for u , c and s only if(2.6) f u = d ∗ ( c − c ) + b . Furthermore, if the latter holds for given u and c ∈ C ( G, Z ), then there is a unique s ∈ J b such that (2.4) holds for u , c and s . Indeed, as the points s ∈ J b parameterize all tuplesof sheaves with degrees given by b , the tuples ( L cv ( s ) | v ∈ V ) run through all tuples ofsheaves with degrees given by d ∗ ( c − c ) + b .Now, given u , since the union of all E u I ( v ) for v ∈ V is E u I , it follows that f u has degree d , the same as b . So there is c ∈ C ( G, Z ) such that f u = d ∗ ( c − c ) + b . This finishes theproof of Statement (1).Statement (2) and the first part of Statement (4) have already been addressed before, inthe proof of Proposition 2.4. The remaining of Statement (4) follows from the definitionof c ( t ) for t ∈ P c J b .As for Statement (3), let u : E I → E be any orientation. And let c ∈ C ( G, Z ) such that c e = c e for each e such that c e ∈ Z and c e = ⌊ c e ⌋ for each e ∈ E u I . Then L cv ( s ) ∼ = L c v ( s ) (cid:16) X e ∈ E u I h e = v N e (cid:17) for each v ∈ V ,from which it follows that u , c and s satisfy Equations (2.4). By Statement (2), there is apoint t ∈ P c J b over s representing I with o ct = u . From the definition of c ( t ), it follows that c ( t ) = c .Finally, we prove Statement (5). First observe that f u ( v ) − f u ( v ) = | E u I ( v ) | − | E u I ( v ) | = d ∗ (( u , u ))( v ) for each v ∈ V. Since (2.4) holds for the two sets of data, ( u , c , s ) and ( u , c , s ), instead of ( u , c, s ),Equations (2.6) hold, that is, f u = d ∗ ( c − c ) + b and f u = d ∗ ( c − c ) + b . Thus d ∗ ( c − c ) = f u − f u = d ∗ (( u , u )) , and hence γ , as defined, is in H ( G, Z ). The equality c − c + ( u , u ) = c ( t ) − c ( t )follows from the definition of c ( t ) and c ( t ), using that o c t = u and o c t = u .Now, since (2.4) holds for ( u , c , s ) and for ( u , c , s ), instead of ( u , c, s ), we have L c v ( τ γ ( s )) ∼ = L c + γv ( s ) = L c − ( u , u ) v ( s ) ∼ = L c v ( s ) (cid:16) X e ∈ E h e = v ( u , u ) e N e (cid:17) ∼ = I v (cid:16) X e ∈ E u I ( v ) N e + X e ∈ E u I ( v ) N e − X e ∈ E u I ( v ) N e (cid:17) ∼ = I v (cid:16) X e ∈ E u I ( v ) N e (cid:17) ∼ = L c v ( s )for each v ∈ V , whence τ γ ( s ) = s .Also, e τ γ ( t ) represents the same sheaf as t , and e τ γ ( t ) lies on P c − γ J b , on the same fiberof R J b over J b as t , that over s . We may thus assume s = s and γ = 0.Now, o c t = u and o c t = u . Since c = c + ( u , u ), it follows from Proposition 2.1 that t ∈ P c J b as well. We may thus assume c = c , whence u = u . Set c := c .Since t and t represent I , there is an isomorphism I c ( t ) → I c ( t ). This isomorphisminduces isomorphisms I c ( t ) v → I c ( t ) v for each v ∈ V making the diagram commute: I c ( t ) −−−→ L I c ( t ) v y y I c ( t ) −−−→ L I c ( t ) v . Since o ct = u = u = o ct , it follows from (2.4) that the isomorphisms I c ( t ) v → I c ( t ) v extend to isomorphisms L cv ( s ) → L cv ( s ) making the extended diagram commute: I c ( t ) −−−→ L v ∈ V I c ( t ) v −−−→ L v ∈ V L cv ( s ) y y y I c ( t ) −−−→ L v ∈ V I c ( t ) v −−−→ L v ∈ V L cv ( s ) . But an automorphism of L cv ( s ) is multiplication by a certain z v ∈ G m ( κ ). It follows that I c ( t ) = ( z v | v ∈ V ) I c ( t ) as subsheaves of ⊕L cv ( s ), and hence that t and t differ by theaction of a κ -point on G V m / G m . (cid:3) Degenerations of line bundles I
Admissible extensions.
Let B be the spectrum of the power series ring κ [[ t ]]. Let π : X → B be a flat, projective map with smooth generic fiber and special fiber isomorphicto X . We say π is a smoothing of X . We will identify the special fiber with X , through ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 25 a fixed isomorphism. For each e ∈ E , let ℓ e be the singularity degree of X at N e . Moreprecisely, ℓ e is the unique positive integer such that the completion of the local ring of X at N e is κ [[ t ]]-isomorphic to κ [[ x, y, t ]] / ( xy − t ℓ e ). Let ℓ : E → N be the function assigning ℓ e to e for each e ∈ E . For each v ∈ V , the irreducible component X v of X fails to be aCartier divisor of X at the nodes N e for which ℓ e > N e , we obtain a map σ : e X → X forwhich the strict transform of X v is Cartier in e X for each v ∈ V . Then all irreducible com-ponents of σ − ( X ) are Cartier in e X . We choose e X minimal with this property. (Observethat σ is an isomorphism over any node of X other than the N e , so e X is not regular if thesingularity degree of any such node is greater than 1.) We call σ the Cartier reduction of π . It is the semistable reduction if e X fails to be regular only at the N e .Put e π := πσ and let X ℓ be the special fiber of e π . It is obtained by replacing each N e by a chain Z e := σ − ( N e ) of rational smooth curves of length ℓ e −
1. For each e = uv in E we will order the components of Z e as Z e , . . . , Z eℓ e − , where Z e intersects X u and Z eℓ e − intersects X v . Sometimes it will be convenient to set Z e := X u and Z eℓ e := X v . We willidentify the remaining components of X ℓ with their corresponding images on X . The curve X ℓ and the restriction σ ℓ := σ | X ℓ : X ℓ → X depend only on ℓ .Let H be the graph obtained as a subdivision of G by inserting ℓ e − e = uv in E by a path P e = uz e z e . . . z eℓ e − v ,for new vertices z e , . . . , z eℓ e − . Sometimes it will be convenient to set z e := u and z eℓ e := v .The graph H coincides precisely with the graph obtained from the dual graph of X ℓ byremoving all loop edges: the new components Z ei correspond to the vertices z ei and thenodes Z ei ∩ Z ei +1 to the edges { z ei , z ei +1 } on the path P e . Abusing the notation, given avertex v (resp. edge e ) of H , we will also denote by X v (resp. N e ) the correspondingcomponent (resp. node) of X ℓ . To avoid confusion, we will sometimes denote by V ( G ) and E ( G ) the vertex and edge sets of G , likewise for H . We will also view V ( G ) as a subset of V ( H ) in the natural way.To an almost invertible sheaf L on X , that is, a torsion-free, rank-one sheaf on X that isinvertible at all the nodes N e for e ∈ E ( G ), we may associate a divisor D ∈ Div( G ), namely, D := P deg ( L | X v ) v . Likewise for sheaves on X ℓ and divisors on H . Recall from [AE20a,Subsection 2.3] that a divisor D on H is called G -admissible if D contains in its supportat most one vertex among the new vertices z e , . . . , z eℓ e − for each edge e ∈ E ( G ), and if so,with coefficient equal to 1. We set t De := ℓ e X j =1 ( ℓ e − j ) D ( z ej ) for each D ∈ Div( H ) and e ∈ E ( G )to identify the vertex. Similarly, we say an almost invertible sheaf L on X ℓ is σ ℓ -admissible if its associated divisor is G -admissible. If L is a σ ℓ -admissible almost invertible sheaf ofdegree d , then the push-forward σ ℓ ∗ L is a torsion-free, rank-one sheaf on X of degree d ; see e.g. [EP16], Thm. 3.1, p. 63 (where a more general notion of admissibility, for which thestatement still holds, is considered).Let now L η be an invertible sheaf on the generic fiber of π , which is the same as thatof e π . Let d := deg L η . Since σ is the Cartier reduction of π , the sheaf L η extends to an almost invertible sheaf on e X , that is, a relative torsion-free, rank-one sheaf on e X /B whoserestriction to the special fiber X ℓ is almost invertible. The extension is not unique though:Given an almost invertible sheaf L on e X , all the sheaves of the form L ( f ) := L ⊗ O e X (cid:16) X v ∈ V ( H ) f ( v ) X v (cid:17) for f ∈ C ( H, Z ) have the same restriction to X η as L , and these are the only almostinvertible sheaves with this property. The invertible sheaves T f := O e X (cid:16) X v ∈ V ( H ) f ( v ) X v (cid:17) are called twisters . Given f ∈ C ( H, Z ), the divisor on H associated to T f | X ℓ does notdepend on the choice of the smoothing π . In addition, two twisters T f and T h yield thesame divisor if and only if f − h is constant.The divisor on H associated to T f | X ℓ is the principal divisor defined bydiv( f ) := X e ∈ E ( H ) ( f (t e ) − f (h e ))h e for each f ∈ C ( H, Z ). The assignment defines a group homomorphismdiv : C ( H, Z ) → Div( H )whose kernel is the subgroup of constant functions and whose image defines an equivalencerelation on Div( H ), called linear equivalence [BLN97, BN07]. It follows that the divisorsin Div( H ) associated to the L| X ℓ for all almost invertible extensions L of L η to e X are thosein a certain linear equivalence class.We are interested in special extensions of L η to e X , those that yield meaningful extensionsof L η to X as well, so we make the following definition. Definition 3.1.
An almost invertible sheaf L on e X is called σ - admissible if L| X ℓ is σ ℓ -admissible.The admissible sheaves we consider are special in the sense that the last statement ofthe following proposition holds. Proposition 3.2.
Let L be a σ -admissible almost invertible sheaf on e X . Then R σ ∗ L = 0 and I := σ ∗ L is a (relative) torsion-free, rank-one sheaf on X /B , with formation commut-ing with base change. In particular, the restriction of I to X η is the same as that of L .Furthermore, I v ∼ = L| X v for each v ∈ V ( G ) , where I v is the restriction of I to X v modulotorsion. ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 27
Proof.
The first statement follows from [EP16], Thm. 3.1, p. 63. As for the last statement,set L := L| X ℓ and I := I| X . By the first statement, I = σ ℓ ∗ L . Let Z ⊆ X ℓ be the unionof the chains Z e of exceptional components over which L has total degree 1, and W ⊆ X ℓ the union of the remaining components. Let µ : X ′ → X be the partial normalization of X along the nodes N e at which I fails to be invertible. Since I = σ ℓ ∗ L , it follows fromThm. 3.1 in loc. cit. that σ ℓ induces upon restriction a map ν : W → X ′ . It follows fromLemma 2.1 in loc. cit. that applying σ ℓ ∗ to the short exact sequence0 −→ L | Z (cid:16) − X P ∈ Z ∩ W P (cid:17) −→ L −→ L | W −→ I ∼ = µ ∗ ν ∗ ( L | W ). As W is a semistable model of X ′ , and L | W has de-gree 0 on every exceptional component of W , it follows again from Thm. 3.1 in loc. cit. that ν ∗ ( L | W ) is almost invertible and ν ∗ ν ∗ ( L | W ) = L | W . In particular, ν ∗ ( L | W ) | X v ∼ = L | X v foreach v ∈ V . Finally, since µ is finite, the natural map µ ∗ µ ∗ ν ∗ ( L | W ) → ν ∗ ( L | W ) is surjec-tive. Thus, restricting it to X v we get a surjection µ ∗ ν ∗ ( L | W ) | X v → L | X v . Since L is almostinvertible, whence L | X v is torsion-free, and I ∼ = µ ∗ ν ∗ ( L | W ), it follows that I v ∼ = L | X v ,finishing the proof. (cid:3) Admissible extensions exist, according to the following proposition.
Proposition 3.3.
There is a σ -admissible almost invertible extension L of L η to e X . Fur-thermore, if L η admits a nonzero section, there is such an extension whose restriction to X ℓ has an effective associated divisor in Div( H ) .Proof. The first statement is essentially a restatement of the results proved in [AE20a,Section 2]. Indeed, it is enough to see that any divisor on H is linearly equivalent to a G -admissible divisor. This is proved in [AE20a, Thm. 2.10]. As for the second statement,first observe that for any almost invertible extension L of L η , any nonzero section of L η extends to a section of L with a nonzero restriction to L| X ℓ . For each component X v of X ℓ , this section vanishes to a certain finite order, say − f ( v ), on X v . So it induces asection of M := L ⊗ T f whose zero scheme is finite over B . In particular, the divisor D ∈ Div( H ) associated to M| X ℓ is effective. Hence t De ≥ δ e (0; t D ) := ⌊ t De ℓ e ⌋ ≥ e ∈ E ( G ). The statement follows now from [AE20a, Prop. 2.9], which yields thatthe G -admissible divisor D ′ = D + div( h ) is effective, where h ∈ C ( H, Z ) is the canonicalextension of the zero function in C ( G, Z ) with respect to D . (cid:3) Special twisters allow us to pass from one σ -admissible almost invertible sheaf on e X toany other with the same restriction to X η .First, we introduce notation. By [AE20a], Prop. 2.7, for each divisor D ∈ Div( H ) andeach f ∈ C ( G, Z ), there is a unique extension e f ∈ C ( H, Z ) of f such that D + div( e f )is G -admissible. The function e f is called the canonical extension of f with respect to D .Thus, for each almost invertible sheaf L on e X with D as the associated divisor to L| X ℓ , the sheaf L ( e f ) is σ -admissible. As e f depends in this case on L and f , we abuse the notationby setting L ( f ) := L ( e f ).We will often deal with the special case where D ∈ Div( H ) is G -admissible and f = χ v ∈ C ( G, Z ) for a vertex v ∈ V . In this case we denote e f by f D,v . We have that f D,v ( z ei ) = 1for each e ∈ E with h e = v and each i = ℓ e − t De , . . . , ℓ e , whereas f D,v ( w ) = 0 for all othervertices w ∈ V ( H ). Also, for a σ -admissible almost invertible sheaf L on e X with D asthe associated divisor to L| X ℓ , we denote L ( f ) by M v ( L ). The divisor on H associatedto M v ( L ) | X ℓ is the G -admissible chip firing move of D at v , denoted M v ( D ) in [AE20a,Subsection 2.6]. So the notations are coherent. Proposition 3.4.
Let L be a σ -admissible almost invertible sheaf on e X . Then:(1) For each pair of vertices v, w ∈ V ( G ) , we have M v ( M w ( L )) = M w ( M v ( L )) .(2) Enumerating the vertices of G as v , . . . , v n , we have M v ( M v ( · · · M v n ( L ) · · · )) ∼ = L . (3) For each σ -admissible almost invertible M on e X with the same restriction to X η as L , there exists a sequence v , . . . , v m of vertices of G such that M ∼ = M v ( M v ( · · · M v m ( L ) · · · )) . Furthermore, the sequence is unique up to reordering the vertices and adding orsubtracting all the vertices of G .Proof. Statements (1) and (2) are easy to check, and follow from [AE20a], Prop. 2.11. Theexistence part in Statement (3) follows from Prop. 2.14 in loc. cit.As for the uniqueness part in Statement (3), the first two statements yield that the men-tioned operations to a sequence v , . . . , v m do not change the resulting sheaf. In addition,the proof of Prop. 2.14 in loc. cit. shows that D ′ = D + div( h ), where D (resp. D ′ ) is thedivisor in H associated to L| X ℓ (resp. M| X ℓ ), and h ∈ C ( H, Z ) is the canonical extensionof f := χ v + · · · + χ v m with respect to D . Thus, for any other sequence v ′ , . . . , v ′ p ofvertices of G with the same property as v , . . . , v m , we have div( h ) = div( h ′ ), where h ′ isthe canonical extension of f ′ := χ v ′ + · · · + χ v ′ p with respect to D . Then h ′ − h is constant.Since h ′ − h extends f ′ − f , it follows that f ′ − f is constant, finishing the proof. (cid:3) Let L be a σ -admissible almost invertible sheaf on e X . Set I ( f ) := σ ∗ L ( f ) for each f ∈ C ( G, Z ). It will be useful to have a direct interpretation of the I ( f ). It is given byProposition 3.5 below, from which we see how to obtain the I ( f ) recursively from I := σ ∗ L . Proposition 3.5.
Notations as above, for each v ∈ V ( G ) and f ∈ C ( G, Z ) , the sheaf I ( f ) is the kernel of the surjection (3.1) I ( f + χ v ) −→ I ( f + χ v ) v , where the sheaf to the right is the torsion-free sheaf generated on X v by I ( f + χ v ) . ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 29
Proof.
The kernel of the surjection (3.1) is a relatively torsion-free, rank-1 sheaf on X /B byan argument analogous to the one found in [Lan75], Prop. 6, p. 100; see [Est01], Section 3.On the other hand, there is a natural exact sequence on e X ,(3.2) 0 −−−→ L ( f ) −−−→ L ( f + χ v ) −−−→ L ( f + χ v ) | Y −−−→ , where Y is the subcurve of X ℓ which is the union of X v and the components Z ej for all e ∈ E with h e = v and j = ℓ e − t De + 1 , . . . , ℓ e −
1, where D ∈ Div( H ) is the divisorassociated to L ( f + χ v ) | X ℓ . Since L ( f + χ v ) has degree 0 on these Z ej , we have σ ∗ ( L ( f + χ v ) | Y ) = L ( f + χ v ) | X v = I ( f + χ v ) v , and the surjection in (3.2) restricts to the surjection (3.1). It follows that σ ∗ L ( f ) is thekernel of (3.1), as claimed. (cid:3) Generalized enriched structures.
We apply the above recursive interpretation todescribe the O X ( f ). (Here O X ( f ) := σ ∗ O e X ( f ) for each f ∈ C ( G, Z ).) That interpretationallows us to view O X ( f ) as a sheaf of fractional ideals of X , which we will describe below.In her thesis [Mai98], Main`o defined the notion of enriched structures over X , andconstructed the moduli space of enriched curves, that is, stable curves with enriched struc-tures. Under the interpretation given by the second author and Medeiros [EM02], anenriched structure on X arises from a smoothing X → B with regular total space X , thus ℓ e = 1 for each e ∈ E ( G ), as the group homomorphism L : C ( G, Z ) → Pic ( X ) given by L ( f ) := O X ( f ) | X for each f ∈ C ( G, Z ).In the general case of higher singularity degrees, we may thus see the O X ( f ) | X as partof a generalized enriched structure over X ; the precise definition is given in Definition 4.1.Main`o constructed in loc. cit. a quasiprojective variety parameterizing enriched curves overthe moduli space of stable curves. One of our goals, to be pursued in a subsequent workwith the contributions given here, is to compactify this variety in a meaningful way. (Werefer to recent work by Biesel and Holmes [BH16] for a compactification of Main`o’s modulispace following a different approach.)Let V /M be the versal deformation of X . As explained in [DM69], pp. 79–81 andreviewed in [EM02], pp. 288–9, we have M = Spec( R ), where R is the power series ringover κ in the variables t e , for e ∈ E o , and s , . . . , s p , for a certain integer p . Furthermore,the variables can be chosen so that for each e ∈ E o we have an isomorphism of R -algebras, ψ e : b O V,N e → R [[ z e , w e ]] / ( z e w e − t e ) . The versal deformation comes with an identification of the special fiber of
V /M with X ,thus we may view X ⊆ V . We may assume that z e = 0 corresponds to the component X v and w e = 0 to X u , where v := h e and u := t e . Letting ˆ z e and ˆ w e denote the elementsof b O V,N e corresponding to z e and w e , we have that ˆ z e restricts to a local parameter z e of b O X u ,N e , whereas ˆ w e restricts to a local parameter w e of b O X v ,N e . As X /B is a deformation of X , there is a natural Cartesian diagram factoring theinclusion of X in V : X −−−→ X −−−→ V y π y y Spec( κ ) −−−→ B −−−→ M The map B → M sends via pullback t e to a e t ℓ e ξ e , for certain a e ∈ κ and ξ e ∈ κ [[ t ]] with ξ e (0) = 1. If we denote by ˜ z e and ˜ w e the pullbacks under the map X → V of ˆ z e and ˆ w e ,respectively, we have ˜ z e ˜ w e = a e t ℓ e ξ e in b O X ,N e for each e ∈ E o .Now, for each f ∈ C ( G, Z ) define δ e ( f ) := j f ( v ) − f ( u ) ℓ e k for each e = uv ∈ E . We claim that, for each e = uv ∈ E o , the sheaf of fractional ideals O X ( f ) is generatedlocally analytically at N e by(3.3) ( t − f ( u ) ˜ z − δ e ( f ) e , t − f ( v ) ˜ w − δ e ( f ) e ) . Indeed, suppose f ( v ) ≥ f ( u ). As t = 0 gives X , we have that O X ( f ) = t − f ( v ) O X (cid:0) f − f ( v ) χ V (cid:1) , where χ V := X χ v . And, as shown in [CEG08], p. 14, locally analytically at N e the sheaf O X ( f − f ( v ) χ V ) is theideal generated by ( ˜ w q +1 e , ˜ w qe t r ), where q and r are the quotient and the remainder of theEuclidean division of f ( v ) − f ( u ) by ℓ e . In other words, q = δ e ( f ) and r = f ( v ) − f ( u ) − ℓ e q .It is now easy to check, using ˜ z e ˜ w e = a e t ℓ e ξ e , that t − f ( v ) ( ˜ w q +1 e , ˜ w qe t r ) is the fractional ideal(3.3). An analogous argument works when f ( v ) ≤ f ( u ). Notice that t − f ( v ) ˜ w − δ e ( f ) e = ( a e ξ e ) δ e ( f ) t − f ( u ) ˜ z − δ e ( f ) e if ℓ e divides f ( v ) − f ( u ); in particular, O X ( f ) is principal at N e in this case.For each f ∈ C ( G, Z ), the sheaf of fractional ideals t f ( v ) O X ( f ) of X generates a sheafof fractional ideals J v ( f ) of X v for each v ∈ V ( G ). Given the above description, for each e = uv ∈ E o we have that, locally analytically at N e , the element ˜ z − δ e ( f ) e is mapped to z − δ e ( f ) e in J u ( f ), whereas ˜ w − δ e ( f ) e is mapped to w − δ e ( f ) e in J v ( f ). On the other hand, if ℓ e does not divide f ( v ) − f ( u ), then t f ( u ) − f ( v ) ˜ w − δ e ( f ) e and t f ( v ) − f ( u ) ˜ z − δ e ( f ) e are mapped to 0 in J u ( f ) and J v ( f ), respectively. It follows that J v ( f ) = O X v (cid:16) X e ∈ E h e = v δ e ( f ) N e (cid:17) for each v ∈ V ( G ).As in [EM02], p. 292, we may consider the image J ( f ) of the composition O X ( f ) −−−→ L v ∈ V t f ( v ) O X ( f ) −−−→ L v ∈ V J v ( f ) . ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 31
Then J ( f ) is a sheaf of fractional ideals of X isomorphic to O X ( f ) | X . Clearly, J ( f ) gener-ates J v ( f ) for each v ∈ V ( G ). As the J v ( f ) are described above, to describe the subsheaf J ( f ) we need only describe it locally analytically at the N e where J ( f ) is invertible, thatis, for e = uv ∈ E o such that ℓ e divides f ( v ) − f ( u ).Let K := Q v ∈ V κ ( X v ), the product of the fields of rational functions of the irreduciblecomponents of X . And put K := Q v ∈ V K v , the product of the constant sheaves of rationalfunctions of the irreducible components of X . For each e ∈ E and v ∈ e , let b K v,e denote thefield of fractions of b O X v ,N e ; it contains the field of fractions of O X v ,N e , which is κ ( X v ). Wemay thus view any local description of a sheaf of fractional ideals of X at N e in b K u,e × b K v,e ,for each e = uv ∈ E o .In particular, for each f ∈ C ( G, Z ) and each e = uv ∈ E o , the sheaf of fractional ideals J ( f ) is generated at N e locally analytically in b K u,e × b K v,e by ( a δ e ( f ) e z − δ e ( f ) e , w − δ e ( f ) e ) if ℓ e divides f ( v ) − f ( u ) and by (( z − δ e ( f ) e , , (0 , w − δ e ( f ) e )) otherwise.For each e = uv ∈ E o , we may use z e to establish isomorphisms α e,m : O X u ( mN e ) | N e ∼ = κ for each m ∈ Z , by analytically identifying O X u ( mN e ) at N e with the fractional idealgenerated by z − me , and taking z − me to 1. Doing the same for w e , we get isomorphisms β e,m : O X v ( mN e ) | N e ∼ = κ . Under these isomorphisms the sheaf J ( f ) is the subsheaf of L v ∈ V J ( f ) v given locally at N e , for each e = uv ∈ E o such that ℓ e divides f ( v ) − f ( u ), asthe kernel of the surjection O X u ( δ e ( f ) N e ) | N e ⊕ O X v ( δ e ( f ) N e ) | N e ( α e,δe ( f ) ,β e,δe ( f ) ) −−−−−−−−−−−→ ∼ = κ ⊕ κ (1 , − a δe ( f ) e ) −−−−−−−→ κ. Notice that as X /B varies among smoothings of X with singularity degree function ℓ ,the a e on which the O X ( f ) ultimately depends vary freely. The O X ( f ) depends thus on thefree choice of a homomorphism a : C ( G, Z ) → G m . Also, by Nakayama Lemma, givingan isomorphism α e : O X v ( N e ) | N e ∼ = κ for each e ∈ E and v ∈ e is the same as choosing ananalytic local parameter for X v at N e . We have chosen above, as a result of consideringa versal deformation of X , certain analytic local parameters z e and w e for X u and X v at N e for each e = uv ∈ E o . Different choices z ′ e and w ′ e can be expressed as power series z ′ e = τ e z e + · · · and w ′ e = σ e w e + · · · for τ e , σ e ∈ κ , and we would obtain the same subsheaf J ( f ) of L v ∈ V J ( f ) v by replacing the a e by τ e a e σ e for each e ∈ E o .4. Degenerations of line bundles II
Let B be the spectrum of the power series ring κ [[ t ]]. Let π : X → B be a smoothing of X . Let ℓ : E → N be the function assigning to e the singularity degree ℓ e of X at N e . Let σ : e X → X be the Cartier reduction of X . Put e π := πσ . Let X ℓ be the special fiber of e π .Keep the remaining notation of Section 3. As f runs in C ( G, Z ), the sheaf I ( f ) | X runsthrough what we call limits of L η . These are not all of what we call stable limits though.To obtain all of them, we consider base changes t t n for positive integers n , as explainedbelow. For each n ∈ N , let µ n : B → B be the base change map given by t t n , and let π n : X n → B be the base extension of π . Let σ n : e X n → X n be the Cartier reduction. If m ∈ N divides n , then we have the following commutative diagram of maps: e X n σ n −−−→ X n π n −−−→ B y y µ n/m ye X m σ m −−−→ X m π m −−−→ B. The square to the right is Cartesian, so the geometric fibers of X n /B are the same as thoseof X /B but the one to the left is not. The singularity degree of X n at N e is now nℓ e foreach e ∈ E ( G ), so the special fiber of e X n /B is X nℓ . We have the same configuration asbefore. The only difference is that ℓ is replaced by nℓ .Given an invertible sheaf L η on the generic fiber of π m , for given m ∈ N , and given n ∈ N divisible by m , we may pull L η back to an invertible sheaf L nη on the generic fiber of π n ,and consider its extensions to e X n , as we did before for the case n = 1. Given an extension L of L η to e X m we may pull it back to an extension L n of L nη to e X n . If L is σ m -admissible,then L n is σ n -admissible. We will also denote by I n the pullback of a relative torsion-freerank-one sheaf I on X m /B ; it is one on X n /B . If I = σ m ∗ L then I n = σ n ∗ L n .As before, to each f ∈ C ( G, Z ) and each σ m -admissible almost invertible extension L of L η to e X m , we associate a σ n -admissible extension L n ( f ) of L nη . Again by [EP16], Thm. 3.1,p. 63, the pushforward I n ( f ) := σ n ∗ L n ( f )is a relative torsion-free rank-one sheaf on X n /B . The notation is consistent, as I n (0) = I n . Definition 4.1.
Let m ∈ N and L η be an invertible sheaf on the generic fiber of X m /B .Let L be a σ m -admissible almost invertible extension of L η to e X m . For each n ∈ N divisibleby m and f ∈ C ( G, Z ), let L n ( f ) := L n ( f ) | X nℓ and I n ( f ) := I n ( f ) | X . We call I := { I n ( f ) | n ∈ m N , f ∈ C ( G, Z ) } the collection of stable limits of L η . In case L η is the structure sheaf of the generic fiber of X /B , we call I a generalized enriched structure .If ℓ is the constant function 1, and L η is the structure sheaf of the generic fiber of X /B ,Main`o called an enriched structure the subset { I ( χ v ) | v ∈ V ( G ) } . In this case, the fullset I is obtained from this subset by tensor products and degeneration.In general, I is a set of torsion-free, rank-one sheaves of degree equal to that of L η . If κ has characteristic 0, then the field of Puiseux series is the algebraic closure of the field ofLaurent series, the field of fractions of κ [[ t ]], and we may thus think of I as the collectionof all the limits of the pullback of L η to the geometric generic fiber of X /B . ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 33
For each d ∈ Z , denote by J d the stack parameterizing torsion-free, rank-one sheaves ofdegree d on X ; see Section 2. If L η has degree d , we may view I as a subset of J d . Themain theorem of this section, Theorem 4.7, asserts that the collection I is the support ofa closed substack of J d , and describes thoroughly the structure of this substack.We need a few preliminary results though. Recall the notation introduced in [AE20a]and [AE20b]: For each n ∈ N , each m ∈ C ( G, Q ) and f ∈ C ( G, Z ), we set δ m ,nℓ,e ( f ) := j f (h e ) − f (t e ) + n m e nℓ e k for each e ∈ E ,and let d m ,nℓ,f ∈ C ( G, Z ) be defined by putting d m ,nℓ,f ( e ) = 12 (cid:16) δ m ,nℓ,e ( f ) − δ m ,nℓ,e ( f ) (cid:17) for each e ∈ E . If ℓ is fixed we drop the subscript ℓ .In addition, let H n denote the graph obtained from the dual graph of X nℓ by removing theself loops. For each e ∈ E ( G ), we let Z e,n ⊆ X nℓ denote the chain of rational smooth curvesof length nℓ e − N e of X . For each e = uv ∈ E , order the components Z e,n , . . . , Z e,nnℓ e − of Z e,n by assuming that Z e,ni intersects Z e,ni +1 for i = 1 , . . . , nℓ e − Z e,n intersects X u at N e (whence Z e,nnℓ e − intersects X v at N e ). For convenience, we set Z e,n := X u and Z e,nnℓ e := X v . We denote by z e,nj the vertex of H n associated to Z e,nj for each j = 0 , . . . , nℓ e . As with H we may view V ( G ) as a subset of V ( H n ), and more generally V ( H m ) as a subset of V ( H n ) for each m ∈ N with m | n .If n = 1, the overscript n is dropped throughout. Proposition 4.2.
Let n ∈ N and D be a G -admissible divisor of H n . For each e ∈ E o ,put m e := 1 n nℓ e − X i =1 iD ( z e,ni ) , and let m be the element of C ( G, Q ) satisfying this. Then, for each f ∈ C ( G, Z ) and v ∈ V , we have O e X n (cid:0) e f (cid:1) | X v ∼ = O X v (cid:16) X e ∈ E t e = v ⌊ d m ,nf ( e ) ⌋ N e + X e ∈ E t e = v m e < N e (cid:17) , where e f ∈ C ( H n , Z ) is the canonical extension of f with respect to D . (Recall that e f is the unique extension of f to V ( H n ) such that D +div( e f ) is G -admissible;see [AE20a, Subsection 2.5].) Proof.
Since D is G -admissible, 0 is the canonical extension of 0 with respect to D . Clearly, O e X n (0) is trivial. On the other hand, if e ∈ E is such that t e = v , then d m ,n ( e ) = 0 unless m e = 0. In this case, since | m e | < ℓ e , if m e > d m ,n ( e ) = 1 /
2. And if m e < d m ,n ( e ) = − /
2. In any case, the proposition holds for f = 0. Observe that f can be replaced by f + bχ V for any b ∈ Z , where χ V := P χ v . We maythus assume f ≥
0. We argue by induction on deg ( f ), the initial case, deg ( f ) = 0, havingjust been considered.Suppose first that f = χ w for a certain w ∈ V ( G ). Then O e X n (cid:0) e f (cid:1) = O e X n (cid:16) X w + X e ∈ E t e = w r e X i =1 Z e,ni (cid:17) , where r e = 0 if m e = 0; otherwise r e is the only integer i such that D ( z e,ni ) = 1.If v = w then O e X n (cid:0) e f (cid:1) | X v ∼ = O X v (cid:16) X N e (cid:17) , where the sum is over the e ∈ E such that e = wv and either nℓ e = 1 or D ( z ¯ e,n ) = 1.For such e we have that d m ,nf (¯ e ) = 1 and m ¯ e = 0 if nℓ e = 1; and if nℓ e > m ¯ e = − /n and d m ,nf (¯ e ) = 0 if e ∈ E o , whereas m ¯ e = ( nℓ e − /n and d m ,nf (¯ e ) = 1 if ¯ e ∈ E o .In all cases, N e is in the support of U := X e ∈ E t e = v ⌊ d m ,nf ( e ) ⌋ N e + X e ∈ E t e = v m e < N e with multiplicity 1. Furthermore, suppose N e is in the support of U for a certain e ∈ E .Then v ∈ e and we may suppose h e = v . If t e = w , then, as before, ⌊ d m ,nf (¯ e ) ⌋ = 0 if m ¯ e ≥ ⌊ d m ,nf (¯ e ) ⌋ = − N e is in the support of U we must then have e = wv .Suppose nℓ e >
1. If m ¯ e ≥
0, since | m ¯ e | < ℓ e , we have ⌊ d m ,nf (¯ e ) ⌋ = 0 unless n m ¯ e = nℓ e − ⌊ d m ,nf (¯ e ) ⌋ = 1. But this can only happen if ¯ e ∈ E o and D ( z e,n ) = 1. On theother hand, if m ¯ e < ⌊ d m ,nf (¯ e ) ⌋ = − n m ¯ e = −
1, in which case ⌊ d m ,nf (¯ e ) ⌋ = 0.But this can only happen if e ∈ E o and D ( z e,n ) = 1. At any rate, the statement of theproposition holds.If v = w then O e X n (cid:0) e f (cid:1) | X v ∼ = O X v (cid:16) − X N e (cid:17) , where the sum is over the e ∈ E such that t e = v and m e = 0. For such e we have that d m ,nf ( e ) = −
1, thus N e is in the support of U with multiplicity -1. Furthermore, suppose N e is in the support of U for a certain e ∈ E . As before, we may suppose t e = v . If m e > | m ¯ e | < ℓ e , we have ⌊ d m ,nf (¯ e ) ⌋ = 0. Since N e is in the support of U we must thenhave m e ≤
0. But if m e < ⌊ d m ,nf (¯ e ) ⌋ = −
1. Thus, since N e is in the support of U wemust have m e = 0. It follows that the statement of the proposition holds.In the general case, we may assume that deg ( f ) > w ∈ V ( G ) such that f ( w ) >
0. Let g := f − χ w . Then O e X n (cid:0) e f (cid:1) ∼ = O e X n (cid:0) e g (cid:1) ⊗ O e X n (cid:0) e h (cid:1) ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 35 where e g ∈ C ( H n , Z ) is the canonical extension of g with respect to D , where D ′ = D + div( e g ), and where e h ∈ C ( H n , Z ) is the canonical extension of χ w with respect to D ′ .The isomorphism holds by [AE20a, Prop. 2.11]. Now, by induction, O e X n (cid:0) e g (cid:1) | X v ∼ = O X v (cid:16) X e ∈ E t e = v ⌊ d m ,ng ( e ) ⌋ N e + X e ∈ E t e = v m e < N e (cid:17) . Thus we need only prove that O e X n (cid:0) e h (cid:1) | X v ∼ = O X v (cid:16) X e ∈ E t e = v (cid:0) ⌊ d m ,nf ( e ) ⌋ N e − ⌊ d m ,ng ( e ) ⌋ (cid:1) N e (cid:17) , or, using what we have proved above, that for each e ∈ E with t e = v we have that(4.1) ⌊ d m ,nf ( e ) ⌋ − ⌊ d m ,ng ( e ) ⌋ = e = w and either nℓ e = 1 or D ′ ( z e,n ) = 1; − v = w and m ′ e = 0;0 otherwise,where m ′ ∈ C ( G, Q ) satisfies m ′ e := 1 n nℓ e − X i =1 iD ′ ( z e,ni )for each e ∈ E o .Indeed, if v = w , then both sides in (4.1) are zero unless h e = w . Suppose e = vw . Thenthe left-hand side in (4.1) is 1 if d m ,nf ( e ) ∈ Z and 0 otherwise. But d m ,nf ( e ) ∈ Z if and onlyif either nℓ e = 1 or D ′ ( z e,n ) = 1.If v = w then the left-hand side in (4.1) is − d m ,ng ( e ) ∈ Z and 0 otherwise. But d m ,ng ( e ) ∈ Z if and only if m ′ e = 0.In any case, Equation (4.1) follows. (cid:3) Recall the definitions of R and the subschemes Y a,bℓ, m ⊆ R from [AE20b], which we reviewnow.First, as seen in Section 2, R := Y e ∈ E o R e , where R e is the doubly infinite chain of smooth rational curves. We may order the rationalcurves in the chain, and denote them by P e,i for i ∈ Z . We may give them coordinates( x e,i : x e,i ) such that the point 0 e,i , given by x e,i = 0, is the point of intersection of P e,i with P e,i +1 , and the point ∞ e,i , given by x e,i = 0, is the point of intersection of P e,i with P e,i − . We can also define P e,i for each i ∈ R − Z as the point of intersection of P e, ⌊ i ⌋ with P e, ⌈ i ⌉ . Then R = [ α ∈ C ( G, Z ) P α , where, more generally, P α := Y e ∈ E o P e,α e for each α ∈ C (cid:0) G, Z (cid:1) .We see that P α ⊇ P β if and only if | β e − α e | ≤ for each e ∈ E , with α e ∈ Z whenever β e ∈ Z . Removing from each P α all those P β contained in it, we obtain the interior of P α ,the open subscheme denoted P ∗ α . The P ∗ α for α ∈ C ( G, Z ) form a stratification of R .Let a : C ( G, Z ) → G m ( κ ) and b : C ( G, Z ) → G m ( κ ) be characters. Let ℓ : E → N be an edge length function and m ∈ C ( G, Z ). To each f ∈ C ( G, Z ) we associate thesubvariety P a,bℓ, m ,f of P d m f given by the equations ∀ oriented cycle γ in G m f , Y e ∈ ¯ γ ∩ E o b e a d m f ( e ) e Y e ∈ γ ∩ E o x e, d m f ( e ) Y e ∈ ¯ γ ∩ E o x e, d m f ( e ) = Y e ∈ γ ∩ E o b e a d m f ( e ) e Y e ∈ ¯ γ ∩ E o x e, d m f ( e ) Y e ∈ γ ∩ E o x e, d m f ( e ) , where G m f is the spanning subgraph of G whose edges are those of G for which d m f is aninteger. The equation corresponding to γ may also be written in the format:(4.2) Y e ∈ γ x e, d m f ( e o ) = Y e ∈ γ b e a d m f ( e o ) e Y e ∈ ¯ γ x e, d m f ( e o ) , where e o := e if e ∈ E o and e o := e otherwise.It is clear from (4.2) that P a,bℓ, m ,f depends rather on the restriction of b to H ( G, Z ). Asany character of H ( G, Z ) extends to one of C ( G, Z ), we will assume later that b is rathera character of H ( G, Z ).We denote by Y a,bℓ, m the union of the P a,bℓ, m ,f for all f ∈ C ( G, Z ).There is a natural action of the character group of C ( G, Z ), which we denote by G E m , on R : To a character c : C ( G, Z ) → G m ( κ ) and a point p on P α with coordinates ( x e,α e , x e,α e ),for each e ∈ E o , we associate the point on the same P α with coordinates ( c e x e,α e , x e,α e ).The homomorphism d ∗ induces a homomorphism from the character group of C ( G, Z ),which we denote G V m , to G E m . The induced action on R of c ∈ G V m takes the point p tothat on the same P α with coordinates ( c v x e,α e , c u x e,α e ) for each e = uv ∈ E o . Finally, thedegree map induces a “diagonal” injective homomorphism G m ֒ → G V m , and the inducedaction of G m on R is trivial. We may thus speak of the action of the quotient G V m / G m on R .It is clear from Equations (4.2) that the action of G V m / G m on R leaves each Y a,bℓ, m in-variant. We may thus describe each Y a,bℓ, m in terms of its orbits. This was done in [AE20b],Thm. 5.3, and we reproduce it and its Cor. 5.4 here for later use. Theorem 4.3.
Let ℓ : E → N be an edge length function, m ∈ C ( G, Z ) , and let a : C ( G, Z ) → G m ( κ ) , b : C ( G, Z ) → G m ( κ ) ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 37 be characters. For each n ∈ N and f ∈ C ( G, Z ) , let p nf be the point on P d m ,nf given by thecoordinates ( b e a d m ,nf ( e ) e : 1) for each e ∈ E o with d m ,nf ( e ) ∈ Z . Then Y a,bℓ, m is the union of the orbits of the p nf under the action of G V m / G m . Furthermore,given α ∈ C ( G, Z ) , we have that P ∗ α ∩ Y a,bℓ, m = ∅ if and only if α = d m ,nf for certain n ∈ N and f ∈ C ( G, Z ) , and in this case P ∗ α ∩ Y a,bℓ, m is the orbit of p nf and is dense in P α ∩ Y a,bℓ, m . Recall from Section 2 the definition of the schemes J b and R J b and the stacks S d and J d . Recall that given a point s ∈ J b corresponding to a collection of torsion-free,rank-one sheaves ( L v ; v ∈ V ) with L v of degree b v , fixing trivializations L v | N e ∼ = κ and O X v ( N e ) | N e ∼ = κ for each e ∈ E and each v ∈ e , we obtain an isomorphism between thefiber of R E J b / J b over s and R . The isomorphism is uniquely defined once we specify that P c + c J b is taken to P c for each c ∈ C ( G, Z ). For simplification, we assume from now on that c = 0.The importance of the Y a,bℓ, m stems from the following proposition. Proposition 4.4.
Let m ∈ N and L η be an invertible sheaf of degree d on the generic fiberof X m /B . Let L be a σ m -admissible almost invertible extension of L η to e X m . For each n ∈ N divisible by m , let L n be the pullback of L to e X n , and for each f ∈ C ( G, Z ) let L n ( f ) := L n ( f ) | X nℓ and I n ( f ) := σ nℓ ∗ L n ( f ) . Let D m ∈ Div( H m ) be the divisor associated to L m (0) . For each e ∈ E o , let m e := 1 m mℓ e − X i =1 iD m ( z e,mi ) , and let m be the unique element of C ( G, Q ) satisfying this. Let E ⊆ E be the support of m and put K v := I m (0) v (cid:16) X e ∈ E o h e = v N e (cid:17) for each v ∈ V ( G ) , where I m (0) v is the torsion-free, rank-one sheaf generated by I m (0) on X v . Let b ∈ C ( G, Z ) satisfying b v = deg ( K v ) for each v ∈ V ( G ) . Fix trivializations K v | N e ∼ = κ and O X v ( N e ) | N e ∼ = κ for each e ∈ E and v ∈ e . Then the following three statements hold:(1) The degree of b is d and the K v are represented by a unique point s ∈ J b .(2) For each n ∈ N with m | n and f ∈ C ( G, Z ) , there is a unique (modulo action of G V m / G m ) point t nf on the fiber of R J b over s representing I n ( f ) with c ( t nf ) = d m ,nf .(3) There are characters a : C ( G, Z ) → G m ( κ ) and b : C ( G, Z ) → G m ( κ ) such that, under the chosen trivializations, the equivariant isomorphism of the fiberof R J b over s with R , taking P c J b to P c for each c ∈ C ( G, Z ) , takes a point on the orbit of t nf for each n ∈ N divisible by m and f ∈ C ( G, Z ) to the point on P d m ,nf given by the coordinates ( b e a d m ,nf ( e ) e : 1) for each e ∈ E o with d m ,nf ( e ) ∈ Z . (4) In particular, the union of the orbits of all the t nf is a closed subset of the fiber of R J b over s isomorphic to Y a,bmℓ,m m .Proof. The first statement is immediate from the fact that deg b = deg I m (0) = deg L η = d .As for the second statement, let n ∈ N divisible by m and f ∈ C ( G, Z ). Let D n denote the pullback of D m to H n : we have D n ( v ) = D m ( v ) for each v ∈ V ( G ) and D n ( z e,nj ) = 0 for each e ∈ E and j = 1 , . . . , nℓ e −
1, unless n | jm and D m ( z e,njm/n ) = 1, inwhich case D n ( z e,nj ) = 1. Also, D n is the associated divisor to L n (0). In particular, D n is G -admissible. Notice that m e = 1 n mℓ e − X i =1 nm iD n ( z e,nni/m ) = 1 n nℓ e − X i =1 iD n ( z e,ni )for each e ∈ E o . Then, by Propositions 3.2 and 4.2, I n ( f ) v ∼ = L n ( f ) | X v ∼ = L m (0) | X v ⊗ O X v (cid:16) X e ∈ E t e = v ⌊ d m ,nf ( e ) ⌋ N e + X e ∈ E t e = v m e < N e (cid:17) for each vertex v ∈ V ( G ), where I n ( f ) v is the restriction modulo torsion of I n ( f ) to X v .Furthermore, since m e > e ∈ E o , we have I n ( f ) v ∼ = I m (0) v (cid:16) X e ∈ E o h e = v N e (cid:17) ⊗ O X v (cid:16) X e ∈ E h e = v ⌊ d m ,nf (¯ e ) ⌋ N e (cid:17) ∼ = K v (cid:16) X e ∈ E h e = v ⌊− d m ,nf ( e ) ⌋ N e (cid:17) ∼ = L d m ,nf v ( s ) , the latter isomorphism following from the definition in Subsection 2.2. In addition, since I n ( f ) = σ nℓ ∗ L n ( f ), it follows that I nf fails to be invertible at a node N e if and only if d m ,nf ( e ) Z . The statement follows now from Proposition 2.6.We prove now Statement (3). First, given the trivializations, the O X n ( f ) | X depend on a : C ( G, Z ) → G m ( κ ) arising from the deformation X /B , as explained in Subsection 3.2.More explicitly, O X n ( f ) | X is isomorphic to the subsheaf of M v ∈ V ( G ) O X v (cid:16) X e ∈ E h e = v d ,nf (¯ e ) N e (cid:17) whose cokernel is supported at the nodes N e for e = uv ∈ E o such that d ,nf ( e ) ∈ Z and isequal in a neighborhood of such a N e to the quotient of the vector space O X u (cid:0) d ,nf ( e ) N e (cid:1) | N e ⊕ O X v (cid:0) − d ,nf ( e ) N e (cid:1) | N e ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 39 by a certain one-dimensional vector subspace. The space is identified with κ ⊕ κ , underour choices of trivializations, and the subspace is that generated by ( a d ,nf ( e ) , e = uv ∈ E o , consider the sheaf I m ( m m e χ u ). Then d m ,mm m e χ u ( e ) ∈ Z ,whence I m ( m m e χ u ) is invertible at N e . Furthermore, it is naturally in a neighborhood of N e a subsheaf of K u ⊕ K v whose quotient is the same as the quotient of a one-dimensionalvector subspace of K u | N e ⊕ K v | N e . The latter is identified with κ ⊕ κ , under our choices oftrivializations. Define b e ∈ κ such that the one-dimensional vector subspace is generatedby ( b e , b : C ( G, Z ) → G m ( κ ). Its restriction to H ( G, Z ) isdenoted by b by abusing the notation.It follows that, under the above trivializations, given n ∈ N , f ∈ C ( G, Z ) and c ∈ C ( G, Z ) such that | c e − d m ,nf ( e ) | ≤ / e ∈ E , the fiber of P c J b over s is identifiedwith P c , and the point t nf on the fiber representing I n ( f ) corresponds to the point on P d m ,nf ⊆ P c with coordinates( b e a c e e : 1) for each e ∈ E o with d m ,nf ( e ) ∈ Z . Indeed, for each e = uv ∈ E o with d m ,nf ( e ) ∈ Z we have that d m ,nf ( e ) = d ,ng ( e ), where g := f − n m e χ u . Furthermore, I n ( f ) is equal to the tensor product of I n ( n m e χ u ) and O X n ( g ) | X in a neighborhood of N e , thus equal in that neighborhood to the subsheaf of K u (cid:16) d m ,nf ( e ) N e (cid:17) ⊕ K v (cid:16) − d m ,nf ( e ) N e (cid:17) whose quotient is, under the identification with κ ⊕ κ of the restriction of the latter sheafto N e , that of κ ⊕ κ by the subspace generated by ( b e a d m ,nf ( e ) e , d m ,nℓ,f = d m m ,n/mmℓ,f for each f ∈ C ( G, Z ) and each n ∈ N divisible by m . (cid:3) Corollary 4.5.
Notations as in Proposition 4.4, the sheaf I n ( f ) is a flat degeneration ofthe sheaf I q ( h ) if P d m ,nf ⊆ P d m ,qh , or equivalently, for each e ∈ E the following two conditionsare satisfied:(1) | d m ,nf ( e ) − d m ,qh ( e ) | ≤ / .(2) If d m ,nf ( e ) ∈ Z then d m ,qh ( e ) ∈ Z .Proof. By Proposition 4.4, there is a closed subscheme Y of the fiber of R J b over s , withthe induced reduced structure, which is the union of all the orbits of the t nf . Consider thequotient stack I := [ Y / ( G V m / G m )]. Let s nf denote the image of t nf in I for each n and f .As observed in Section 2, there is a relative torsion-free rank-one sheaf I on X × I / I suchthat I| X ×{ s nf } ∼ = I n ( f ) for each n and f .By Proposition 4.4, there is an equivariant isomorphism from Y to Y a,bmℓ,m m for certain m , a and b , taking each t nf to the point p nf on P ∗ d m ,nf given in Theorem 4.3. The isomorphism is equivariant, whence we may view I as a quotient of Y a,bmℓ,m m where each s nf is the imageof the point p nf . Pulling back I we obtain a relative torsion-free rank-one sheaf L on X × Y a,bmℓ,m m /Y a,bmℓ,m m such that L| X ×{ y } ∼ = I n ( f ) for each point y on the orbit of p nf .Let n, q ∈ N divisible by m and f, h ∈ C ( G, Z ). Suppose P d m ,qh ⊇ P d m ,nf . It is nowenough to observe that P ∗ d m ,qh ∩ Y a,bmℓ,m m is the orbit of p qh and is dense in P d m ,qh ∩ Y a,bnℓ,m m byTheorem 4.3. (cid:3) Now we need only one technical statement before stating the main result of the section.
Proposition 4.6.
Let f , f ∈ C ( G, Z ) and n , n ∈ N . If d m ,n f − d m ,n f ∈ H ( G, Z ) , then d m ,n f = d m ,n f .Proof. Since d m ,n i f i = d m ,pn i pf i for each p ∈ N , we may assume n = n . Then the propositionfollows from [AE20b], Prop. 5.1. (cid:3) Theorem 4.7 (Degeneration) . Let I be the collection of stable limits of an invertible sheaf L η of degree d on the generic fiber of X m /B for given integer m ∈ N and smoothing X /B with singularity degrees ℓ : E → N of the nodal curve X . Then I is the supportof a reduced closed substack of J d , also denoted I . The inverse image of I in R J b isan infinite disjoint union of connected closed subschemes of certain fibers of R J b over J b . Each connected component is the image of each other, under the action of a uniqueelement of H ( G, Z ) , and each maps isomorphically to the same closed subscheme Y ⊆ S d .Furthermore, each connected component, under an identification of the fiber containing itwith R , is equal to the subscheme Y a,bℓ, m for certain choices of m ∈ C ( G, Z ) and characters a : C ( G, Z ) → G m ( κ ) and b : H ( G, Z ) → G m ( κ ) . In particular, Y has pure dimension | V | − and I has pure dimension .Proof. Using the notation in Proposition 4.4, we have that I is the collection of the I nf for n ∈ N divisible by m and f ∈ C ( G, Z ). Let the K v and b be as in Proposition 4.4, as wellas s ∈ J b and the t nf ∈ R J b in Statements (1) and (2) therein.Let Y ⊆ R J b be the union of the orbits of the t nf for all n ∈ N and f ∈ C ( G, Z ). ByProposition 4.4, it is a closed subset of the fiber of R J b over s . We give it the reducedstructure, so that Y becomes isomorphic to Y a,bℓ, m for certain choices of m ∈ C ( G, Z ) andcharacters a : C ( G, Z ) → G m ( κ ) and b : H ( G, Z ) → G m ( κ ), according to Proposition 4.4.Then Y is a connected closed subscheme of R J b of pure dimension | V | − Y → S d . Since S d is the quotientof R J b by the free action of the discrete group H ( G, Z ), to prove our claim it is enoughto prove that the translate of Y by any nonzero element γ ∈ H ( G, Z ) does not intersect Y .So, let γ ∈ H ( G, Z ). Suppose that there are two points t and t on Y that differ bythe action of γ . Let c , c ∈ C ( G, Z ) such that t lies on the fiber of P c J b over s and t onthe fiber of P c J b over s . We may assume that c = c + γ , and thus c ( t ) = c ( t ) + γ . But, ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 41 by construction of Y , there are n , n ∈ N and f , f ∈ C ( G, Z ) such that c ( t i ) = d m ,n i f i for i = 1 ,
2. Then Proposition 4.6 implies that γ = 0.Any two points on R J b representing I n ( f ) for each n ∈ N and f ∈ C ( G, Z ) differby the action of H ( G, Z ) × G V m / G m . Since the action of H ( G, Z ) commutes with thatof G V m / G m , it follows that all the points on R J b representing I n ( f ) are in e τ γ ( Y ) for γ ∈ H ( G, Z ). So the inverse image of I in R J b is the union S e τ γ ( Y ) for γ runningin H ( G, Z ). Now, it follows from what we proved above that the intersection of e τ γ ( Y )with e τ γ ′ ( Y ) for distinct γ, γ ′ ∈ H ( G, Z ) is empty. Thus the e τ γ ( Y ) are the connectedcomponents of the inverse image.It follows that each of the e τ γ ( Y ) is mapped isomorphically to the same closed subscheme Y ⊆ S d , and that Y has pure dimension | V | −
1. Then I has the structure of a closedsubstack of J d of pure dimension 0. All the statements have been proved. (cid:3) Remark 4.8.
Each I should be parameterized by a point on Hilb J d , the Hilbert stackof J d , and each Y should be parameterized by a point on Hilb G V m / G m S d , the Hilbert stackparameterizing G V m / G m -invariant closed substacks of S d , if this Hilbert stack exists! More-over, the analysis of a few examples leads us to ask whether these points, as π and L η vary,form a closed substack of Hilb J d or Hilb G V m / G m S d . This substack should be regarded as a newcompactified Jacobian for X .The situation is simpler when X is of compact type. Then there are no cycles, so R J b = S d . Furthermore, Y is a fiber of R J b over J b . Thus the “new compactifiedJacobian” is J b , which is of course nothing new in this case. The general situation, wherethe “new compactified Jacobian” is indeed a new construction, will be addressed in a futurework. Remark 4.9.
There is a natural G V m / G m -invariant relative torsion-free, rank-one, degree- d sheaf on X × Y / Y which is given by the restriction of the sheaf on X × S d we called I in Section 2. Being invariant, it is the pullback of a relative torsion-free, rank-one sheaf on X × I / I . It is this sheaf that gives the structure of I as a closed substack of J d .5. Regeneration
Recall from Section 4 the definition of R and of the Y a,bℓ, m . Recall from Section 2 thedefinition of the schemes J b and R J b and the stacks S d and J d . Recall that given a point s ∈ J b corresponding to a collection of torsion-free, rank-one sheaves ( L v ; v ∈ V ) with L v of degree b v , fixing trivializations L v | N e ∼ = κ and O X v ( N e ) | N e ∼ = κ for each e ∈ E and each v ∈ e , we obtain an isomorphism between the fiber of R E J b / J b over s and R .The isomorphism is uniquely defined once we specify that P c + c J b is taken to P c for each c ∈ C ( G, Z ). For simplification, we assume from now on that c = 0.In this section, we give a converse to Theorem 4.7. Theorem 5.1 (Regeneration) . Let ℓ : E → N be a length function, m ∈ C ( G, Z ) and let a : C ( G, Z ) → G m ( κ ) and b : H ( G, Z ) → G m ( κ ) be characters. Let Y a,bℓ, m be the corresponding subscheme of R . Let b ∈ C ( G, Z ) . For each v ∈ V , let K v be a torsion-free, rank-one sheaf of degree b v on X v . Let s ∈ J b be thecorresponding point. Fix trivializations K v | N e ∼ = κ and O X v ( N e ) | N e ∼ = κ for each e ∈ E and v ∈ e . Let Y ⊆ R J b be the closed subscheme of the fiber over s corresponding to Y a,bℓ, m under the isomorphism of the fiber with R induced by the trivializations. Let I ⊆ J d be theimage of Y . Then there exist a smoothing π : X → B of X and an invertible sheaf L η onthe generic fiber of π such that the following two statements hold:(1) The total space X is regular everywhere but possibly at the nodes N e of X , where ithas singularity degree ℓ e for each e ∈ E ( G ) , and at the remaining nodes of X wherethe sheaf ⊕ K v is not invertible, where it has singularity degree .(2) I is the collection of stable limits of L η . The whole section is devoted to the proof of the above theorem.Let G ′ denote the full dual graph of X , containing self loops. Consider a versal defor-mation V /M of X , as explained in Subsection 3.2. More precisely, M is the spectrum ofthe power series ring R over κ in the variables t e for e ∈ E ( G ′ ), and variables s , . . . , s p fora certain integer p , and we have an isomorphism of R -algebras ψ e : b O V,N e → R [[ z e , w e ]] / ( z e w e − t e )for each e ∈ E ( G ′ ), with z e = 0 corresponding to the component X v of X and w e = 0 to X u if e ∈ E ( G ) and e o = uv . The pullbacks of z e and w e under ψ e yield under restrictionanalytic local parameters of X u and X v at N e , respectively. We may assume that theisomorphisms O X v ( N e ) | N e ∼ = κ are given by them, for all e ∈ E ( G ) and v ∈ V ( G ) with v ∈ e .We may then consider the smoothing π : X → B of X induced by the map B → M pulling back the s i to t , and pulling back t e to a e o t ℓ e for each e ∈ E ( G ), to t for each e ∈ E ( G ′ ) − E ( G ) if K v is invertible at the corresponding node for v ∈ e , and to t otherwise.Then π is as in the statement of the theorem. Also, as explained in Subsection 3.2, thesheaves O X n ( f ) | X are determined by the homomorphism a .Let σ : e X → X be the Cartier reduction of π . The composition e π := πσ : e X → B is asmoothing of X ℓ . As in Section 4, we will let e π n : e X n → B denote the Cartier reduction ofthe extension of π by the base change map µ n : B → B for each n ∈ N ; it is a smoothingof X nℓ .Recall from Theorem 4.3 that Y a,bℓ, m is a union of orbits in R under the action of G V m / G m ,the orbits of the p nf for f ∈ C ( G, Z ) and n ∈ N , where p nf is a point on P d m ,nf whosecoordinates are given in terms of a and b , as made precise in that theorem. Now, for each f ∈ C ( G, Z ) and n ∈ N , let I nf be the torsion-free, rank-one sheaf on X parameterized bythe point t nf ∈ Y corresponding to p nf . For each e = uv ∈ E , write(5.1) f ( v ) − f ( u ) + n m e = nℓ e δ m ,ne ( f ) + nℓ e − i ne ( f ) , ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 43 where i ne ( f ) is an integer satisfying 0 < i ne ( f ) ≤ nℓ e . We can write I nf = σ nℓ ∗ L nf for a certainalmost invertible sheaf L nf on X nℓ , admissible for σ nℓ , such that L nf has degree zero oneach rational smooth curve Z e,ni contracted by σ nℓ onto N e for each e = uv ∈ E , unless d m ,nf ( e ) Z and i = i ne ( f ).The sheaf L nf is not unique. Its associated divisor in H n , denoted henceforth by D nf , is.At any rate, with the choices we have already made, we have the following claim. Claim 5.2.
Notation as above, for each f, h ∈ C ( G, Z ) and n ∈ N , the sheaves (5.2) L nh and L nf ⊗ O e X n (cid:0) e g (cid:1) | X nℓ have the same degrees on the exceptional components of X nℓ and the same restrictions to X v for each v ∈ V ( G ) , where e g is the canonical extension of g := h − f with respect to D nf .In particular, D nh = D nf + div( e g ) . Proof.
Indeed, both sheaves in (5.2) are σ nℓ -admissible. Furthermore, for each e = uv ∈ E ,the sheaf on the right in (5.2) has degree 0 on Z e,ni for each i = 1 , . . . , nℓ e −
1, unless g ( u ) − g ( v ) + i ne ( f ) = nℓ e j g ( u ) − g ( v ) + i ne ( f ) nℓ e k + i. Substituting for i ne ( f ) from Equation (5.1) we get g ( u ) − g ( v ) + f ( u ) − f ( v ) + nℓ e δ m ,ne ( f ) + nℓ e − n m e = nℓ e j g ( u ) − g ( v ) + i ne ( f ) nℓ e k + i, whence h ( v ) − h ( u ) + n m e = nℓ e (cid:16) δ m ,ne ( f ) − j g ( u ) − g ( v ) + i ne ( f ) nℓ e k(cid:17) + nℓ e − i. Comparing with Equation (5.1), with f replaced by h , it follows that both sheaves in (5.2)have the same degree on each exceptional component Z e,ni .Now, given v ∈ V ( G ) and q ∈ C ( G, Z ), it follows from Propositions 2.2 and 3.2 that(5.3) L nq | X v ∼ = ( I nq ) v ∼ = K v ⊗ O X v (cid:16) X e ∈ E t e = v ⌊ d m ,nq ( e ) ⌋ N e (cid:17) , where ( I nq ) v is the restriction of I nq to X v modulo torsion. In addition, by Proposition 4.2,(5.4) O e X n (cid:0) e g (cid:1) | X v ∼ = O X v (cid:16) X e ∈ E t e = v ⌊ d p ,ng ( e ) ⌋ N e + X e ∈ E t e = v p e < N e (cid:17) , where p ∈ C ( G, Z ) satisfies(5.5) p e = 1 n nℓ e − X i =1 iD nf ( z e,ni ) for each e ∈ E o .Since D nf is the divisor associated to L nf , we have that n p e = i ne ( f ) for each e ∈ E o with i n ¯ e ( f ) = nℓ e , whence n p e = nℓ e − i ne ( f ). The latter holds even if i ne ( f ) = nℓ e , because then p e = 0 and i ne ( f ) = nℓ e . Now, if e ∈ E − E o , then n p e = i ne ( f ), whence n p e = − i ne ( f ),unless i ne ( f ) = nℓ e , in which case n p e = 0. To summarize,(5.6) n p e = ( nℓ e − i ne ( f ) if p e ≥ , − i ne ( f ) if p e < . Finally, for each e ∈ E , nℓ e ⌊ d m ,nh ( e ) ⌋ = h (h e ) − h (t e ) + n m e − nℓ e + i ne ( h )= h (h e ) − h (t e ) − f (h e ) + f (t e ) − i ne ( f ) + nℓ e ⌊ d m ,nf ( e ) ⌋ + i ne ( h )= g (h e ) − g (t e ) + n p e − n p e − i ne ( f ) + nℓ e ⌊ d m ,nf ( e ) ⌋ + i ne ( h )= nℓ e ⌊ d p ,ng ( e ) ⌋ + ρ − n p e − i ne ( f ) + nℓ e ⌊ d m ,nf ( e ) ⌋ + i ne ( h )for a certain integer ρ satisfying 0 ≤ ρ < nℓ e . Since 0 < ρ + i ne ( h ) < nℓ e , and Equation (5.6)yields that n p e + i ne ( f ) is a multiple of nℓ e , it follows that ρ + i ne ( h ) = nℓ e and(5.7) ⌊ d m ,nh ( e ) ⌋ = ( ⌊ d p ,ng ( e ) ⌋ + ⌊ d m ,nf ( e ) ⌋ if p e ≥ , ⌊ d p ,ng ( e ) ⌋ + ⌊ d m ,nf ( e ) ⌋ + 1 if p e < . It follows now from Isomorphisms (5.3) for q = h and q = f , and from Equation (5.4), thatthe restrictions to X v of both sheaves in (5.2) are equal, for each v ∈ V ( G ), finishing theproof of our claim. (cid:3) Notice as well that, since ρ + i ne ( h ) = nℓ e , if d m ,nh ( e ) ∈ Z , then i ne ( h ) = nℓ e and thus ρ = 0, implying that also d p ,ng ( e ) ∈ Z . And if d m ,nf ( e ) ∈ Z then p e = 0. Now, if both I nf and I nh are invertible at N e , then all of d m ,nf ( e ) , d m ,nh ( e ) , d p ,ng ( e ) are integers, p e = 0 and itfollows from (5.7) that(5.8) d m ,nh ( e ) = d p ,ng ( e ) + d m ,nh ( e ) . Furthermore, since we have already shown that the sheaves in (5.2) have the same degreeson the exceptional components of X nℓ , it follows that also the pushforward σ nℓ ∗ O e X n ( e g ) | X nℓ is invertible at N e . Finally, the gluing data of I nf and I nh at N e are b e a d m ,nf ( e ) e and b e a d m ,nh ( e ) e , re-spectively, as given by the coordinates of p nf and p nh . As for the gluing data of σ nℓ ∗ O e X n ( e g ) | X nℓ at N e , since p e = 0, it is the same as if we assumed that e g is the canonical extension of g with respect to the divisor 0; it follows from Subsection 3.2 that the gluing data is a d ,ng ( e ) e .Clearly, d ,ng ( e ) = d p ,ng ( e ), and hence it follows from (5.8) that the sheaves in (5.2) have thesame gluing data in a neighborhood of Z e,n .Since the chains Z e,n are curves of compact type, an invertible sheaf on each of them isdetermined by its degrees on the components. It follows that the sheaves in (5.2) coincide ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 45 up to the chosen gluing along the intersections of the chains Z e,n with the rest of X nℓ forthe e ∈ E such that either d m ,nh ( e ) or d m ,nf ( e ) is not an integer.In fact, we have not yet specified the gluing giving L nf along Z e,n if I nf is not invertibleat N e . We do it now: For each oriented edge e = uv ∈ E o such that d m ,nf ( e ) Z , give L nf the gluing along Z e,n such that the pushforwards under σ nℓ of(5.9) L nf e and L nf ⊗ O e X n (cid:0) e r (cid:1) | X nℓ have the same gluing data at N e , where e r is the canonical extension of r := ( nℓ e − i ne ( f )) χ u with respect to the divisor D nf , and f e := f + r . Notice that I nf e is invertible at N e , andthus the gluing data for L nf e along Z e,n is determined from that of I nf e at N e . From theabove prescription, the gluing data for L nf is now fixed everywhere for each f ∈ C ( G, Z )and n ∈ N .Finally, with the above choices, we can claim more than Claim 5.2: Claim 5.3.
Notation as above, for each f, h ∈ C ( G, Z ) and n ∈ N , we have L nh ∼ = L nf ⊗ O e X n (cid:0) e g (cid:1) | X nℓ , where e g is the canonical extension of g := h − f with respect to the divisor D nf in H n associated to L nf .Proof. Indeed, given e ∈ E o , notice first that, if q ∈ C ( G, Z ) is such that I nq is invertibleat N e , we have proved that the gluing data along Z e,n of L nq and L nf e ⊗ O e X n (cid:0) e g (cid:1) | X nℓ coincide, where g := q − f e and e g is the canonical extension of g with respect to the divisor D nf e in H n associated to L nf e . It follows that the gluing data along Z e,n of L nq and L nf ⊗ O e X n (cid:0) e g + e r (cid:1) | X nℓ coincide, where e r is the canonical extension of r := ( nℓ e − i ne ( f )) χ u with respect to thedivisor D nf . But D nf e = D nf + div( e r ) by Claim 5.2. It thus follows from [AE20a], Prop. 2.11,that e r + e g is the canonical extension of r + g with respect to the divisor D nf . Notice that r + g = q − f .Thus, given q ∈ C ( G, Z ) such that d m ,nq ( e ) ∈ Z , the gluing data along Z e,n of thesethree sheaves,(5.10) L nq , L nh ⊗ O e X n (cid:0) e p (cid:1) | X nℓ and L nf ⊗ O e X n (cid:0) e g (cid:1) | X nℓ , coincide, where e p (resp. e g ) is the canonical extension of p := q − h (resp. g := q − f ) withrespect to the divisor D nh (resp. D nf ). But then so do the gluing data of L nh ⊗ O e X n (cid:0) e p + e r (cid:1) | X nℓ and L nf ⊗ O e X n (cid:0) e g + e r (cid:1) | X nℓ , where e r is the canonical extension of r := h − q with respect to the divisor D nq . Sinceall three sheaves in (5.10) have the same associated divisor in H n , it follows again from[AE20a], Prop 2.11, that e p + e r is the canonical extension of p + r = 0 and that e g + e r is the canonical extension of g + r = h − f with respect to D nf . Since D nf is G -admissible, e p + e r = 0. So the sheaves above are exactly those in (5.2), and we have concluded thatthey have the same gluing data along Z e,n for every e ∈ E o , whence our claim. (cid:3) We need two more claims to finish the proof of the theorem.
Claim 5.4.
Notation as above, L n is the pullback of L under the map X nℓ → X ℓ .Proof. Notice that p n = p for every n ∈ Z , since d m ,n = d m , , and thus I n = I . Thus therestrictions of L n and L to the components X v for v ∈ V ( G ) coincide. Also, i ne (0) = ni e (0)for each e ∈ E , and thus D n is the pullback of D . Thus the restriction of L n to Z e,n coincides with that of the pullback of L for each e ∈ E .Also, the gluing data of L n along Z e,n for each e ∈ E o such that d m , ( e ) ∈ Z is the gluingdata of the pullback of L . On the other hand, by construction, for each e = uv ∈ E o suchthat d m , ( e ) Z , the gluing data of L n along Z e,n is such that the gluing data at N e of thepushforwards of L ng n and L n ⊗ O e X n (cid:0) e g n (cid:1) | X nℓ coincide, where g n := ( nℓ e − i ne (0)) χ u and e g n is the canonical extension of g n with respectto D n . Same for L , for n replaced by 1. Clearly, since i ne (0) = ni e (0), we have that g n = ng , whence O e X n (cid:0) e g n (cid:1) is the pullback of O e X (cid:0) e g (cid:1) to e X n . Finally, the gluing dataof the pushforward I ng n at N e is given by p ng n , whereas that of I g is given by p g . Since g n = ng , we have that d m ,ng n = d m , g , and thus p ng n = p g . It follows that the gluing data of L n and of the pullback of L coincide everywhere, proving the claim. (cid:3) Claim 5.5.
Notation as above, there is an almost invertible sheaf L on e X whose restrictionto X ℓ is L .Proof. The sheaf L fails to be invertible only at the nodes of X ℓ lying over nodes of X other than the N e , at which the corresponding K v is not invertible. Let b X denote theblowup of e X at these nodes and by b X the special fiber of b X /B . The curve b X is obtainedby separating the branches of every node in the center of the blowup and connecting themby a smooth rational curve. Then L is the pushforward of an invertible sheaf on b X , havingdegree one on the added rational curves, by [EP16], Prop. 5.5, p. 77. Since this sheaf isinvertible, since the relative Picard scheme of b X /B is smooth over B , and since B is thespectrum of κ [[ t ]], it follows that it extends to an invertible sheaf on b X . The pushforwardof this extension under the blowup map b X → e X is the required sheaf L . (cid:3) Proof of Theorem 5.1.
We claim that L η := L| X η is the required sheaf. Indeed, we needonly prove that L nh = L n ( h ) | X nℓ for each h ∈ C ( G, Z ) and n ∈ N , where L n is the pullbackof L to e X n . But notice that L n | X nℓ = L n by Claims 5.4 and 5.5. And L n ( h ) = L n ⊗ O e X n (cid:0) e h (cid:1) for each h ∈ C ( G, Z ), where e h is the canonical extension of h with respect to the divisor D n on H n associated to L n . Applying Claim 5.3 with f = 0 finishes the proof. (cid:3) ORONOI TILINGS, TORIC ARRANGEMENTS AND DEGENERATIONS OF LINE BUNDLES III 47
Acknowledgements.
This project benefited very much from the hospitality of the Math-ematics Department at the ´Ecole Normale Sup´erieure (ENS) in Paris and the Instituto deMatem´atica Pura e Aplicada (IMPA) in Rio de Janeiro during mutual visits of both au-thors. We thank the two institutes and their members for providing for those visits. Weare also specially grateful to the Brazilian-French Network in Mathematics for providingsupport for a visit of E.E. to ENS Paris and a visit of O.A. to IMPA.
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CNRS - CMLS, ´Ecole Polytechnique, Palaiseau, France
Email address : [email protected] Instituto Nacional de Matem´atica Pura e Aplicada, Estrada Dona Castorina 110,22460-320 Rio de Janeiro RJ, Brazil
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