LLIMIT LINEAR SERIES FOR CURVES NOT OF COMPACTTYPE
BRIAN OSSERMAN
Abstract.
We introduce a notion of limit linear series for nodal curves whichare not of compact type. We give a construction of a moduli space of limitlinear series, which works also in smoothing families, and we prove a corre-sponding specialization result. For a more restricted class of curves whichsimultaneously generalizes two-component curves and curves of compact type,we give an equivalent definition of limit linear series, which is visibly a general-ization of the Eisenbud-Harris definition. Finally, for the same class of curves,we prove a smoothing theorem which constitutes an improvement over knownresults even in the compact-type case. Introduction
The 1980’s saw spectacular progress in the theory of linear series on curvesand their applications, including the proofs of the Brill-Noether (Griffiths-Harris[GH80]) and Gieseker-Petri (Gieseker [Gie82]) theorems, new results on the geom-etry of general linear series (Eisenbud-Harris [EH83]), and the proof that modulispaces of curves of sufficiently high genus are of general type (Harris-Mumford[HM82] and Eisenbud-Harris [EH87]). What these results all had in common wasthat they made central use of degeneration techniques, studying what happens tolinear series as smooth curves degenerate to singular ones. Ultimately, Eisenbudand Harris developed a general theory of “limit linear series” for curves of compacttype, meaning those curves whose dual graphs are trees, or equivalently, whoseJacobians are compact.For more than 25 years, the question of how to extend the Eisenbud-Harris theoryto curves not of compact type has remained open. Aside from the intrinsic appealof the question, there are various reasons one would like to have such a theory: ‚ it would offer the most systematic approach to computing the cohomol-ogy classes of higher-codimension Brill-Noether classes on moduli spaces ofcurves; ‚ it would allow greater flexibility in choosing a degeneration to approachopen questions such as the maximal rank conjecture; ‚ and it likewise offers a more general setting for analyzing generic fibers ofspecific families of curves. For instance, degenerations arising from con-sidering modular curves in positive characteristic are often two-componentnodal curves.The question of limit linear series for curves not of compact type has been exploredby Esteves in various papers, most notably with Medeiros in [EM02], but to date, The author was partially supported by NSA grant H98230-11-1-0159 and Simons Foundationgrant a r X i v : . [ m a t h . AG ] D ec BRIAN OSSERMAN no one has been able to develop a complete theory generalizing that of Eisenbud andHarris. Recently, Amini and Baker [AB] have proposed a notion of limit linear seriesbased on Brill-Noether theory for graphs, which they show generalizes the definitionof Eisenbud and Harris. However, while they prove a specialization theorem, it isnot clear how to prove a smoothing theorem for Amini-Baker limit linear series, orhow to construct moduli spaces.In the present paper, we propose a different notion of limit linear series for curvesnot of compact type, developed independently and motivated in part by work ofthe author in higher rank [Oss14c]. After giving the definition, we construct mod-uli spaces both over individual curves and in smoothing families, and use them toprove a specialization result. We then show that our definition is a generalizationof the Eisenbud-Harris definition. In fact, we do considerably more: for the class ofcurves of “pseudocompact type,” which is a simultaneous generalization of curvesof compact type and curves with two components (see Figure 1 below), we give anequivalent formulation which visibly generalizes the Eisenbud-Harris definition. Inessence, our more general definition is well-suited for abstract theory and construc-tions, while the second definition is more tractable for computations. Finally, forcurves of pseudocompact type we prove a smoothing theorem, which is an improve-ment even for the compact-type case because it does not only apply to refined limitlinear series.To apply our smoothing theorem, it is necessary to produce families of limitlinear series having the expected dimension, and accordingly in [Oss14a] we carryout dimension counts. Using our generalized Eisenbud-Harris definition, we showthat for curves of pseudocompact type the expected dimension of spaces of limitlinear series is always correct, in the sense that if certain gluing conditions imposethe maximal codimension, then the dimension agrees with the Brill-Noether number ρ . We also investigate several families of curves for which we can show the gluingconditions do indeed impose the maximal codimension, giving in particular newcriteria for the generic fiber of a one-parameter family of curves to be Brill-Noethergeneral. One of the families we consider in [Oss14a] is a broad generalization ofthe curves considered by Cools, Draisma, Payne and Robeva in the graph-theoreticcontext in [CDPR12], and we are able to use our theory to shed new light ontheir results, and to suggest further directions of investigation for the Brill-Noethertheory of graphs. The relationship to the Amini-Baker theory will be investigatedmore thoroughly in [Oss14b], but in essence our approach keeps track of more gluingdata, while minimizing the role of graph theory. Although this may in principlemake computations more difficult, in practice this may not be the case, and wehave found that our approach has the desired dimension behavior in some cases(such as binary curves) for which the Amini-Baker theory does not.We now explain the basic ideas that go into our definition of limit linear series.Suppose that B “ Spec R with R a discrete valuation ring, and X Ñ B is a familyof curves over B with smooth generic fiber and reducible nodal special fiber X .Further suppose that the total space X is regular. Then each component Z v of X is a (Cartier) divisor on X , so any given extension of a line bundle L η on thegeneric fiber can be twisted by O X p Z v q to obtain an infinite family of extensions.Given V η an p r ` q -dimensional space of global sections of L η , for any extension of L η there is a unique extension of V η . The idea introduced by Eisenbud and Harriswas to use these twists to concentrate multidegree on each component Z v of X , IMIT LINEAR SERIES FOR CURVES NOT OF COMPACT TYPE 3 and then to restrict the resulting extension of p L η , V η q to Z v , thereby obtaining acollection p L v , V v q v of g rd s on the components of X . The question then becomesto understand which such tuples of g rd s can arise as a limit in this way. Eisenbudand Harris found a compatibility condition in terms of vanishing sequences at thenodes, and used this to define their notion of limit linear series. The power of theirdefinition was that it was fundamentally inductive, describing limit linear seriesalmost independently on each component, and thereby making computations verytractable. However, the drawback of their definition was that it was difficult togeneralize, and also to use for more theoretical purposes, such as moduli spaceconstructions.In [Oss14c], it was shown that one can state an equivalent formulation of theEisenbud-Harris definition as follows: if w is a multidegree of total degree d on X , and L w denotes the extension of L η having multidegree w , then Γ p X , L w | X q contains the extension of V η , and must therefore have dimension at least r ` V η may be obtainedby gluing together sections from the various V v . This leads to a definition of limitlinear series as a generalized determinantal locus (Definition 2.21 below), whichyields new moduli space constructions, and which also lends itself to generalizationto curves not of compact type. The other basic ingredient of our definition isthat we allow for insertion of chain of rational curves at nodes, and keep track ofinformation on these curves as combinatorially as possible, only considering spacesof global sections on the original components.We next discuss our equivalent definition, generalizing the Eisenbud-Harris def-inition to a broader class of curves. We begin by recalling their definition. Given atuple p L v , V v q of g rd s on the components Z v of X , Eisenbud and Harris define thetuple to be a limit linear series if the following condition is satisfied: for everynode of X , given as Z v X Z v , write a v , . . . , a vr and a v , . . . , a v r for the vanishingsequences of p L v , V v q (respectively, p L v , V v q ) at the node in question; then werequire(1.1) a vj ` a v r ´ j ě d for j “ , . . . , r .Our generalized definition builds on this by replacing the vanishing sequencewith a “multivanishing sequence” which keeps track of vanishing at several pointsat a time, and by adding a gluing condition on the spaces V v , which is vacuouslysatisfied in the compact type case. There are additional complications arising fromkeeping track of potential chains of rational curves inserted at each node, but weillustrate the main ideas in the simplest case, where we have two components, anddo not insert any additional rational curves.Some preliminary definitions are as follows. Notation . Let X be a smooth projective curve, D an effective divisor on X ,and p L , V q a g rd on X . Then we denote by V p´ D q the space V X H p X, L p´ D qq . Definition 1.2.
Let X be a smooth projective curve, r, d ě
0, and D ď D ď¨ ¨ ¨ ď D b ` a sequence of effective divisors on X , with D “ D b ` ą d .Given p L , V q a g rd on X , define the multivanishing sequence of p L , V q along D ‚ to be the sequence a ď ¨ ¨ ¨ ď a r BRIAN OSSERMAN where a value a appears in the sequence m times if for some i we have deg D i “ a ,deg D i ` ą a , and dim p V p´ D i q{ V p´ D i ` qq “ m .Also, given s P V nonzero, define the order of vanishing ord D ‚ s along D ‚ tobe deg D i , where i is maximal so that s P V p´ D i q .Thus, multivanishing sequences generalize usual vanishing sequences and ramifi-cation, incorporating also geometric notions such as secancy conditions (requiringtwo or more points to map to a single a point), bitangency, and so forth. Similarconditions for the case of rational curves were studied by Garc´ıa-Puente et al in[GPHH ` D ‚ does not affect the sequence.Now, suppose that X is obtained by gluing together smooth curves Z and Z at nodes P , . . . , P m . Given d ą
0, let d , d be positive integers such that thereexists b ě d “ d ` d ´ bm , and suppose also that d ´ d i ă m for i “ , d “ d “ b “ d ). For i “ , ď j ď b `
1, set D ij “ j p P ` ¨ ¨ ¨ ` P m q . Now, suppose we are given p L i , V i q a g rd i on Z i for i “ ,
2, and suppose we are also given gluing information ϕ for L and L at the nodes. Then we define the tuple pp L , V q , p L , V q , ϕ q to be a limitlinear series if the following two conditions are satisfied:(I) for i “ ,
2, write a i , . . . , a ir for the multivanishing sequence of p L i , V i q along D i ‚ ; then we require(1.2) a j ` a r ´ j ě bm for j “ , . . . , r ;(II) for i “ ,
2, there exist bases s i , . . . , s ir of the V i such thatord D ‚ s i(cid:96) “ a i(cid:96) for (cid:96) “ , . . . , r, and for all (cid:96) with (1.2) an equality, we have ϕ p s (cid:96) q “ s r ´ (cid:96) . In the above, we have been a bit vague in discussing the gluing; this is made fullyprecise in § §
5, we generalize to the case of curves of pseudocom-pact type, meaning that if we take the dual graph, and collapse all multiple edges,we obtain a tree; see Figure 1. In addition to curves of compact type, this includesinteresting classes of curves such as curves with two components, and chains ofcurves of the sort considered by Cools, Draisma, Payne and Robeva in [CDPR12].One can think of curves of pseudocompact type as being the most general class ofcurves for which one can still analyze gluing conditions by looking at only two com-ponents at a time. Note that there is a close parallel between the above conditions(I) and (II) and the definition of limit linear series for higher-rank vector bundlesgiven by Teixidor i Bigas in [Tei91]. This parallel persists, albeit to a lesser extent,when we allow insertions of chain of rational curves, and consider arbitrary curvesof pseudocompact type. However, this is reflective of the node-by-node aspect ofthe gluing conditions, and for arbitrary nodal curves the behavior is expected to bequite different. More precisely, we obtain a tree from the dual graph by, for each pair of adjacent vertices v, v , replacing all edges connecting v to v with a single edge. IMIT LINEAR SERIES FOR CURVES NOT OF COMPACT TYPE 5
Figure 1.
A dual graph of a curve of pseudocompact type.Finally, in § Theorem 1.3. If X is a curve of pseudocompact type, and the space of limit linearon X has the expected dimension ρ : “ g ` p r ` qp d ´ r ´ g q , then every limit linear series on X can be smoothed to linear series on all nearbysmooth curves. As mentioned above, in comparison to the smoothing theorem of Eisenbud-Harris, our result is stronger because it is not confined to the open subset of refinedlimit linear series. The main tool in the proof of Theorem 1.3 is the theory of linkeddeterminantal loci, which we develop in Appendix A.We conclude with a brief explanation of some of the decisions behind our defi-nitions. First, we originally intended to use torsion-free sheaves to treat specializa-tions, rather than allowing the insertion of rational curves at nodes. However, wediscovered that from this point of view, important gluing conditions are omitted,and as a result, the spaces may no longer have the correct dimension. Next, ofcourse in a general theory of limit linear series, in principle one does not need totreat inserted rational chains differently from other components. However, thereare two compelling reasons for doing so. The first is that it keeps the amount ofdata more manageable; for instance, in the two-component case, we can study limitlinear series in general without having to remember more than two linear series,one for each of the original components. The other reason is that the pseudocom-pact type condition is not preserved under insertion of rational curves at nodes, soour second definition would not be complete (for instance, with respect to special-ization results) if we did not have a system for keeping track of inserted rationalcurves. In addition, our approach is very convenient for working with non-regularsmoothing families. The final comment is that we have not, for the moment, pur-sued the possibility of creating a single proper moduli space of limit linear seriesusing the quasistable curve compactification of the Picard variety. This is a natural
BRIAN OSSERMAN and worthwhile direction to pursue, but because Eisenbud and Harris were able tocarry out all their applications without a compact moduli space (using instead aspecialization result analogous to our Corollary 3.15), it does not seem to be crucialto the basic theory.
Acknowledgements.
I would like to thank Eduardo Esteves for many helpful con-versations, particularly in relation to chain structures and admissible multidegrees.I would also like to thank Frank Sottile for drawing my attention to [GPHH ` Conventions.
All curves we consider are assumed proper, (geometrically) reducedand connected, and at worst nodal. All nodal curves are assumed to be split,meaning that both its nodes and irreducible components are all defined over the basefield. Furthermore, to reduce clutter, we assume that all irreducible componentsare smooth; see Remark 2.23.A graph by default is allowed to have multiple edges, but not, in accordancewith the above, loops. 2.
Fundamental definitions
We begin with some definitions of a combinatorial nature. In the below, Γ willbe obtained by choosing a directed structure on the dual graph of a projective nodalcurve. We assume we have:
Situation 2.1.
Let Γ be a directed graph without loops. For each pair of an edge e and adjacent vertex v of Γ, let σ p e, v q “ e has tail v , and ´ e has head v .The following definitions form the basis for our approach to keeping track ofchains of rational curves inserted at the nodes of the original curve. Definition 2.2. A chain structure on Γ is a function n : E p Γ q Ñ Z ą . A chainstructure is trivial if n p e q “ e P E p Γ q .The chain structure will determine the length of the chain of rational curvesinserted at a given node; for reasons of later convenience, the trivial case (in whichno rational curves are inserted) corresponds to n p e q “ Definition 2.3.
Given n a chain structure on Γ, an admissible multidegree w of total degree d on p Γ , n q consists of a function w Γ : V p Γ q Ñ Z together with atuple p µ p e qq e P E p Γ q , where each µ p e q P Z { n p e q Z , such that d “ t e P E p Γ q : µ p e q ‰ u ` ÿ v P V p Γ q w Γ p v q . The idea behind admissible multidegrees is that in order to extend line bundles,we need only consider multidegrees which have degree 0 or 1 on each rational curveinserted at the node, with degree 1 occurring at most once in each chain. Thus, µ p e q determines where on the chain (if anywhere) positive degree occurs. See Definition2.16 below for details. Definition 2.4.
Given a chain structure n on Γ, let w be an admissible multidegree.Given also v P V p Γ q , the twist of w at v is obtained as follows: for each e adjacentto v , increase µ p e q by σ p e, v q . Now, decrease w Γ p v q by the number of e for which µ p e q had been equal to 0, and for each e , if the new µ p e q is zero, increase w Γ p v q by IMIT LINEAR SERIES FOR CURVES NOT OF COMPACT TYPE 7
1, where v is the other vertex adjacent to v . The negative twist of w at v is theadmissible multidegree w such that the twist of w at v is equal to w .Twists will be the change in multidegrees accomplished by twisting by certainnatural line bundles; see Notation 2.18 below. Example 2.5.
In the case of trivial chain structure, a twist at v simply reduces w Γ p v q by the valence of v while increasing w Γ p v q by the number of edges connecting v to v , for each v ‰ v . This is the same as the chip firing considered by Baker andNorine in [BN07]. Remark . Given Γ and n , let r Γ be the (directed) graph obtained from Γ bysubdividing each edge e into n p e q edges. Thus, we have a natural inclusion V p Γ q Ď V p r Γ q . Then if w is an admissible multidegree for p Γ , n q , we obtain a weight function r w : V p r Γ q Ñ Z on r Γ (which we think of as being a multidegree for the trivial chainstructure) by setting r w p v q “ w Γ p v q for all v P V p Γ q , and setting r w p v q “ v R V p Γ q , unless v lies over an edge e of Γ, and is the µ p e q th new vertex lying over e . In the latter case, we set r w p v q “ p Γ , n q are imbedded into the set of multide-grees on r Γ, and this imbedding is compatible with twists as follows: twisting w at v P V p Γ q is the same as twisting r w by v , and then also by all new vertices between v and the σ p e, v q µ p e q th new vertex lying over e , for each e P E p Γ q adjacent to v . Inthe above, we take the representative of σ p e, v q µ p e q between 0 and n p e q ´
1. Seealso Notation 2.18 below for the geometric version of this statement.
Example 2.7.
In the two-component case, with components v and v , and edgesoriented from v to v , we describe twists in terms of multidegrees on r Γ as in Remark2.6. The idea is that twisting by v moves the positive-degree new vertices awayfrom v and towards v . Specifically, when twisting w at v , for each e P E p Γ q , thedegree-1 new vertex over e shifts by one away from v . If the vertex with degree 1is already adjacent to v , then the degree on v is increased, and no new verticesover e will have positive degree. If no new vertices over e have degree 1, then thedegree on v is decreased, and the first new vertex over e is given degree 1.Note that twists are invertible, since twisting at every vertex of Γ returns tothe initial multidegree. Thus, the negative twist at v can be expressed also as thecomposition of the twists at all v ‰ v . We will primarily be interested in (positive)twists, but the utility for us of negative twists is in the following definition. Definition 2.8.
An admissible multidegree w is concentrated at a vertex v P V p Γ q if there is an ordering on V p Γ q starting with v , and such that for each sub-sequent vertex v , we have that w becomes negative in index v after taking thecomposition of the negative twists at all the previous vertices.A more canonical condition which implies concentration (but is in general strictlystronger) is that for all v ‰ v , and all v adjacent to v , the negative twist of w at v is negative in index v . We have elected to use the above definition as the mostgeneral for which one can make the argument of Proposition 3.3 below. Example 2.9.
The concentration condition is the generalization of the multide-grees considered by Eisenbud and Harris in the compact type case, where they haddegree d on one component, and degree 0 on all the others. BRIAN OSSERMAN
In our generalized setting, w will be concentrated at v if it is negative on all v ‰ v . If the chain structure is trivial, it is enough to have degree at most 0 at all v ‰ v , but in general this is not the case, since with nontrivial chain structures, anegative twist at v adjacent to v need not reduce the degree on v .However, at the opposite extreme, even with nontrivial chain structures we canhave a multidegree simultaneously concentrated at two adjacent vertices. For in-stance, if Γ has only two vertices, connected by n edges, then a multidegree whichis strictly less than n on each vertex, and with µ p‚q identically zero, will be con-centrated on both vertices. This corresponds to usual linear series (of restrictedmultidegrees) on the relevant two-component curves.See also Remark 2.24 below for further comments on the role of the concentrationcondition. Proposition 2.10.
Given any admissible multidegree w , and any v P V p Γ q , thereexists an admissible multidegree w , concentrated at v , and obtained from w byrepeated twisting at vertices v other than v .Proof. First note that the composition of negative twists over a collection S of ver-tices of Γ is equivalent to the composition of (positive) twists over the complementof S . For each n ě
0, let Γ v,n denote the subset of V p Γ q consisting of all vertices v such that there is a path (undirected) in Γ of length less than or equal to n from v to v . Let N be maximal such that Γ v,N Ĺ V p Γ q . Taking sufficiently many negativetwists of w at all vertices of Γ v,N , we can achieve negative degrees at all verticesof V p Γ q (cid:114) Γ v,N . Repeating this process for Γ v,N ´ achieves negative degree onΓ v,N (cid:114) Γ v,N ´ without affecting the degree on V p Γ q (cid:114) Γ v,N , and continuing in thisway down to Γ v, , we achieve negative degree at all vertices other than v , which inparticular implies concentration at v . (cid:3) The following directed graph keeps track of all the multidegrees we will want toconsider starting from any one admissible multidegree.
Notation . Let G p w q be the directed graph with vertex set V p G p w qq Ď Z V p Γ q ˆ ź e P E p Γ q Z { n p e q Z consisting of all admissible multidegrees obtained from w by sequences of twists,and with an edge from w to w if w is obtained from w by twisting at some vertex v of Γ.Given w P V p G p w qq and v , . . . , v m P V p Γ q (not necessarily distinct), let P p w, v , . . . , v m q denote the path in V p G p w qq obtained by starting at w , and twist-ing successively at each v i .By the invertibility of twists, G p w q “ G p w q if and only if w P G p w q . Whileour directed structure on Γ is just a convenience, the directedness of G p w q iscrucial. Although it is not important for our present purposes, we also mentionthat G p w q can be expressed as the collection of admissible multidegrees which arelinearly equivalent to w on r Γ , using the theory of linear equivalence on graphs asdeveloped by Baker and Norine in [BN07].Also, note that P p w, v , . . . , v m q is independent of the ordering of the v i . Proposition 2.12. If P p w, v , . . . , v m q is a minimal path in G p w q from w to some w , then m and the v i are uniquely determined up to reordering. IMIT LINEAR SERIES FOR CURVES NOT OF COMPACT TYPE 9
More generally, paths P p w, v , . . . , v m q and P p w, v , . . . , v m q have the same end-point if and only if the multisets of the v i and the v i differ by a multiple of V p Γ q .Proof. We have already observed the “if” direction. For the converse, in light ofRemark 2.6 the desired statement for Γ and n follows from the same statement forthe graph r Γ constructed by subdividing every edge e of Γ into n p e q edges, withthe trivial chain structure. We thus consider the matrix M indexed by V p r Γ q , with p v, v q entry given by the negative of the valence of v , and for v ‰ v , with p v, v q entry given by the number of edges of r Γ connecting v to v . We wish to see thatthe vector p , . . . , q generates the kernel of M . If we consider (cid:15)M ` I , with 1 { (cid:15) at least the maximal valence in r Γ, we have a symmetric doubly stochastic matrixwith nonnegative entries, which is irreducible because Γ is connected. The Perron-Frobenius theorem then implies that the maximal eigenvalue is 1, and is simple,which implies that the eigenvalue 0 of M is likewise simple, as desired. (cid:3) We now move on to definitions which involve geometry more directly.
Situation 2.13.
Let X be a projective nodal curve, with dual graph Γ, andchoose an orientation on Γ. For v P V p Γ q , let Z v be the corresponding irreduciblecomponent of X , and Z cv the closure of the complement of Z v in X .A preliminary definition (see also Maino [Mai98]) is the following. Definition 2.14. If X is a nodal curve with dual graph Γ, an enriched structure on X consists of the data, for each v P V p Γ q of a line bundle O v on X , satisfyingthe following conditions:(I) for any v P V p Γ q , we have O v | Z v – O Z v p´p Z cv X Z v qq , and O v | Z cv – O Z cv p Z cv X Z v q ;(II) we have â v P V p Γ q O v – O X . Note that it follows from the definitions that each O v has degree 0. Enrichedstructures always exist; they amount to suitable gluing choices at the nodes, andthey are unique when X is of compact type. However, an enriched structure isalways induced by any regular smoothing of X ; see Proposition 3.10.We now explicitly introduce the chains of rational curves induced by a chainstructure on X . Definition 2.15.
Given X and a chain structure n , let r X denote the nodal curveobtained from X by, for each e P E p Γ q , inserting a chain of n p e q ´ r Γ be the dual graph of r X , with a naturalinclusion V p Γ q Ď V p r Γ q . We refer to the new components of r X as the exceptionalcomponents .Note that the above r Γ is compatible with that of Remark 2.6.
Definition 2.16.
Using our orientation of E p Γ q , an admissible multidegree w oftotal degree d on p X , n q gives a multidegree of total degree d on r X by assigning,for each e P E p Γ q , degree 0 on each component of the corresponding chain ofprojective curves, except for degree 1 on the µ p e q th component when µ p e q ‰ The reason for restricting to such multidegrees is that extensions of line bundlesmay always be chosen to have such degrees; see Corollary 3.15.From now on, we will assume we have fixed an enriched structure together withsuitable global sections, as follows.
Situation 2.17.
In Situation 2.13, suppose we have also a chain structure n onΓ, and an enriched structure p O v q v on the resulting r X , and for each v P V p r Γ q , fix s v P Γ p r X , O v q vanishing precisely on Z v .The sections s v will be convenient in describing maps between different twists ofline bundles; they will not be unique even for curves of compact type, but in ourcase they are just a formal convenience, and do not ultimately affect our definitionof limit linear series. See Remark 2.22 for further discussion.We next describe how, given an enriched structure on r X , and a line bundle L of multidegree w , we get a collection of line bundles indexed by V p G p w qq , withmorphisms between them indexed by E p G p w qq . Notation . In Situation 2.17 assume we are given also an admissible multidegree w on p Γ , n q . Then for any edge ε P E p G p w qq , starting at w “ p w Γ , p µ p e qq e P E p Γ q q and determined by twisting at v P V p Γ q , we have the corresponding twisting linebundle O ε on r X defined as O ε “ O v b â e P E p Γ q σ p e,v q µ p e q â i “ O v e,i , where the first product is over edges e adjacent to v , and for any such pair, v e,i denotes the i th rational curve in r X from Z v on the chain corresponding to e .In addition, we have the section s ε of O ε obtained from the tensor product ofthe relevant sections s v and s v e,i .Similarly, given w, w P V p G p w qq , let P “ p ε , . . . , ε m q be a minimal path from w to w in G p w q , and set O w,w “ m â i “ O ε i . In Notation 2.18, if µ p e q “
0, the product over i is empty for the given e , and wetake the representative of σ p e, v q µ p e q between 0 and n p e q ´
1. Note that it followsfrom Proposition 2.12 that the constructions of Notation 2.18 are independent ofchoices of (minimal) paths. The reason for the notation O w,w is that, as one caneasily verify, tensoring by O w,w take a line bundle of multidegree w to one ofmultidegree w . Notation . In Situation 2.17, suppose L is a line bundle on r X of multidegree w . Then for any w P V p G p w qq , set L w : “ L b O w ,w . Given also w v P V p G p w qq concentrated at v , set L v : “ L w v | Z v . Given an edge ε from w to w in G p w q , corresponding to twisting at v , theneither L w “ L w b O ε , or L w “ L w b O w ,w . In the former case, we get a morphism IMIT LINEAR SERIES FOR CURVES NOT OF COMPACT TYPE 11 L w Ñ L w induced by s ε . In the latter case, we observe that O w ,w b O ε – O Ă X ,and fixing such an isomorphism and again using s ε gives an induced morphism L w Ñ L w b O ε “ L w b O w ,w b O ε – L w . In either case, pushing forward gives an induced morphism f ε : Γ p r X , L w q Ñ Γ p r X , L w q . Finally, if P “ p ε , . . . , ε m q is any path in G p w q , set f P : “ f ε m ˝ ¨ ¨ ¨ ˝ f ε . If P is a minimal path from w to w , write f w,w : “ f P . We have the following simple consequence of Proposition 2.12:
Corollary 2.20.
For any w, w P V p G p w qq , the morphism f w,w is independent ofthe choice of minimal path. We can now give the definition of a limit linear series. As mentioned previously,the idea is simply that a collection of g rd v s on the components Z v of X should con-stitute a limit linear series precisely when it is possible to use them to glue togetheran p r ` q -dimensional space of sections on all of X in any desired multidegree. Definition 2.21.
Let X be a projective nodal curve, n a chain structure, w anadmissible multidegree of total degree d on p X , n q , and p O v q v P V p Γ q an enrichedstructure on r X . Choose also a tuple p w v q v P V p Γ q of vertices of G p w q , with each w v concentrated at v , and sections p s v q v as in Situation 2.17. Then a limit linearseries on p X , n q consists of a line bundle L of multidegree w on r X , togetherwith subspaces V v of Γ p Z v , L v q for each v P V p Γ q , satisfying the condition thatfor all w P V p G p w qq , the natural morphism(2.1) Γ p r X , L w q Ñ à v P V p Γ q Γ p Z v , L v q{ V v has kernel of dimension at least r `
1, where (2.1) is obtained as the compositionΓ p r X , L w q ‘ f w,wv Ñ à v P V p Γ q Γ p r X , L w v qÑ à v P V p Γ q Γ p Z v , L v q Ñ à v P V p Γ q Γ p Z v , L v q{ V v . Clearly, the choices of concentrated multidegrees are necessary to even definethe data underlying a limit linear series. However, we will show in Proposition 3.5below that the resulting moduli space of limit linear series does not depend on thischoice.
Remark . Even in the compact type case, the sections s v of Situation 2.17are not typically unique, even up to scaling: indeed, if v disconnects Γ, then s v can be scaled independently on (the subcurves corresponding to) each resultingconnected component. Thus, a priori our definition of limit linear series dependson extra data even in the compact type case. However, the choice of s v is uniqueup to scaling on each component, and because the maps (2.1) are obtained byrestricting to individual components, their kernels do not depend on the choice of s v . Consequently, we see that the notion of limit linear series is in fact independentof the choices of the s v . This is different from the notion of linked linear seriesintroduced in [Oss14c] (generalizing [Oss06]), where even for curves of compacttype, the choice of s v does have an effect. Remark . We have chosen not to allow self-nodes (i.e., nodes on single irre-ducible components) not because they are harder to handle, but because they arealready better understood, and our techniques don’t add anything new for them.It is not difficult to combine our techniques with those developed for self-nodes,but we have chosen to present our definitions and results without any self-nodesbecause we would have to systematically treat the two types of nodes differently.If one wants to treat limit linear series on a reducible curve with some self-nodes,there are several options: the first, which is simplest to state but probably leasteffective for computation is to simply introduce new rational components at eachself-node, thereby removing all self-nodes; another option is to work with sheaveswhich are allowed to be torsion-free (but not invertible) at the self-nodes. In thelatter case, one can study the resulting linear series by partially normalizing atthe self-nodes and studying linear series on the resulting smooth component(s),imposing a secancy condition at each pair of points lying above self-nodes at whichthe sheaf was invertible (above nodes at which the sheaf was not invertible, one doesnot have a gluing condition, but the degree on the relevant component is decreased).This approach was developed already by Kleiman [Kle76] nearly 40 years ago.
Remark . Obviously, concentrated multidegrees are not unique, so a choiceof these is a necessary input to our definition of limit linear series. AlthoughProposition 3.5 asserts that in fact the resulting limit linear series moduli spaceswill not depend on the choice of the w v , it is still natural to wonder to what extentone can make canonical choices of the tuples p w v q v of concentrated multidegrees.The answer likely comes from the theory of v -reduced divisors, which plays animportant role in Brill-Noether theory for graphs. However, since our theory goesthrough fully as long as the w v are concentrated, it seems potentially advantageousnot to place any further restrictions on them. Thus, the question of canonicalchoices is rather orthogonal to the purpose of the present paper, and for the sakeof simplicity we do not pursue it here.3. Families and moduli schemes
In this section, we construct a moduli scheme of limit linear series, show that itis independent of the choice of tuple of concentrated multidegree, and finally givean alternate description which generalizes to the case of smoothing families. Themain technical tool is the generalized determinantal loci introduced in Appendix Bof [Oss14c], and the main issue that needs to be addressed for smoothing familiesis that the limit linear series are defined in terms of (sections of) line bundles L v on individual components of the reducible curve, which no longer makes sense ina smoothing family. This difficulty is resolved by working instead with the linebundles L w v on the whole curve, with multidegree concentrated on the relevantcomponent.This section is of a foundational nature, and later sections are largely inde-pendent from it, with the exception of Theorem 6.1, our smoothing result. Note,however, that our specialization result, Corollary 3.15, is proved in this section. IMIT LINEAR SERIES FOR CURVES NOT OF COMPACT TYPE 13
First, we set the following notation.
Notation . In the situation of Definition 2.21, let P rw ‚ p X , n , p O v q v q be thescheme parametrizing tuples p L , p V v q v P V p Γ q q , where L is a line bundle on r X of multidegree w , and each V v is an p r ` q -dimensional space of global sectionsof the induced line bundle L v on Z v .Thus, if we write d v “ deg L v , then P rw ‚ p X , n , p O v q v q can naturally be con-structed as a fibered product of Pic w p r X q with the spaces G rd v p Z v q , fibered overthe spaces Pic d v p Z v q .We then construct a moduli scheme of limit linear series as follows. Definition 3.2.
In the situation of Definition 2.21, write M for the universal linebundle on P rw ‚ p X , n , p O v q v q ˆ X , and V v for the universal subbundles of the in-duced p ˚ M v . Then let G r ¯ w p X , n , p O v q v q be the closed subscheme of P rw ‚ p X , n , p O v q v q defined by the intersection over w P V p G p w qq of the p r ` q st vanishing loci of themaps(3.1) p ˚ M w Ñ à v P V p Γ q p p ˚ M v q{ V v . In the above, the p r ` q st vanishing locus is a canonical scheme structure on theset of points on which the kernel has dimension at least r `
1, defined in AppendixB of [Oss14c]. Thus, G r ¯ w p X , n , p O v q v q is a canonical scheme structure on theset of limit linear series described in Definition 2.21. The notation ¯ w representsthe collection of admissible multidegrees obtained from w by twisting (that is, V p G p w qq ); we use it because we will prove shortly, in Proposition 3.5, that thechoice of the w v does not affect the resulting moduli scheme.The T -valued points of P rw ‚ p X , n , p O v q v q are tuples p L , p V v qq , where L is aline bundle on T ˆ r X of multidegree w , and each V v is a rank- p r ` q subbundleof p ˚ L v (in the sense of Definition 4.2 of [Oss06]). Such a tuple is a T -valuedpoint of G r ¯ w p X , n , p O v q v q if for all w P V p G p w qq , the map(3.2) p ˚ L w Ñ à v P V p Γ q p p ˚ L v q{ V v has p r ` q st vanishing locus equal to all of T .Our next task is to show that in fact, for a fixed w , the spaces of limit linearseries for different choices of the w v are canonically identified with one another.A preliminary fact is the following. Proposition 3.3.
Let L be a line bundle of multidegree w P V p G p w qq on r X ,and suppose that w is concentrated at v . Then the restriction map H p r X , L q Ñ H p Z v , L | Z v q is injective.Proof. The main point is that for any vertices v , v , and any section s P H p r X , L q which vanishes on Z v , then the number of zeroes consequently imposed on Z v isequal to the change in index v when we take the negative twist of w at v . Indeed,if there are m nodes of X connecting Z v to Z v for which L is trivial on theassociated exceptional chain (equivalently, for which µ p e q “ s | Z v mustvanish at these m nodes, but m is also the amount by which the negative twist of w at v reduces the degree at v . Given this, if s vanishes on v , then we simplytraverse Γ in the ordering provided by the definition of concentration, and vanishingon Z v for all the previous vertices v implies vanishing at the next component aswell. (cid:3) Corollary 3.4.
Suppose that w is concentrated at v , that w v can be obtained from w by twisting at vertices other than v , and that we have a T -valued tuple p L , p V v q v q such that the p r ` q st vanishing locus of (3.2) is all of T . Then the kernel of (3.2) is a subbundle of p ˚ L w of rank r ` , and is equal to the preimage of V v under themap p ˚ L w Ñ p ˚ L v . Moreover, both statements hold after arbitrary base change.Proof. First observe that the hypotheses on w , together with Proposition 3.3, implythat the map p ˚ L w Ñ p ˚ L w | Z v Ñ p ˚ L v is injective on points, and hence universally injective. Now, by hypothesis the p r ` q st vanishing locus of (3.2) is all of T . On the other hand, at any point, thekernel is contained in the preimage of (the corresponding fiber of) V v , which hasdimension r ` p r ` q nd vanishing locus of(3.2) is empty. The statement of the corollary then follows from Proposition B.3.4and Lemma B.2.3 (iv) of [Oss14c]. (cid:3) Proposition 3.5.
In the situation of Definition 2.21, let p w v q v be another choice ofadmissible multidegrees concentrated at the v P V p Γ q . Then the schemes G r ¯ w p X , n , p O v q v q obtained from p w v q v and p w v q v are canonically isomorphic.Proof. It is clearly enough to treat the case that w v “ w v for all v other thansome fixed choice of v . We first observe that it follows from Proposition 2.12 thatgiven any w, w P V p G p w qq , there is some w such that the minimal paths from w to w and from w to w do not require twisting at v . Indeed, if we take a minimalpath P from w to w , and let w be obtained by all twists in P except those at v , then the minimal path from w to w involves only twists at v , so the minimalpath from w to w does not require twisting at v . Moreover, if w is concentratedat v , then we see from the construction that w is also concentrated at v . Thus,to prove the proposition we may further assume that w v is obtained from w v bytwisting at vertices other than v . In particular, if, for a given L of multidegree w ,we let L v be as usual, and L v the corresponding line bundle obtained from w v ,then the map L v Ñ L v is (universally) injective. It follows that a subbundle V v of L v induces a subbundle V v of L v , so we obtain a morphism P rw ‚ p X , n , p O v q v q Ñ P rw p X , n , p O v q v q which we wish to show is an isomorphism on the closed subschemes of limit linearseries.First, if p L , p V v q v q is a T -valued point of G r ¯ w p X , n , p O v q v q Ď P rw ‚ p X , n , p O v q v q ,and w P V p G p w qq , we need to check that the p r ` q st vanishing locus of(3.3) p ˚ L w Ñ p ˚ L v { V v ‘ à v ‰ v p p ˚ L v q{ V v is all of T . But by construction, p ˚ L v { V v injects into p ˚ L v { V v (universally),so if there is a minimal path from w to w v factoring through w v , then the kernelof (3.3) is identified with that of (3.2), so the hypothesis that p L , p V v q v q is in G r ¯ w p X , n , p O v q v q together with Proposition B.3.2 of [Oss14c] implies that the p r ` IMIT LINEAR SERIES FOR CURVES NOT OF COMPACT TYPE 15 q st vanishing locus of (3.3) is all of T . Otherwise, we have that the compositionof minimal paths from w to w v and from w v to w v is not minimal, meaning that itincludes a twist at v ; since the latter does not have such a twist, we conclude thatthe minimal path from w to w v includes a twist at v . In this case, the minimal pathfrom w v to w does not include a twist at v . Let V v denote the kernel of (3.2) inmultidegree w v ; by Corollary 3.4 this is a subbundle which is equal to the preimageof V v . Because the minimal path from w v to w does not include a twist at v , and w v is concentrated at v , we see that the map p ˚ L w v Ñ p ˚ L w is universally injective,so the image of V v is a subbundle of rank r `
1, which is easily verified to be inthe kernel of (3.3), since V v is in the kernel of (3.2). We conclude from PropositionB.3.4 of [Oss14c] that the p r ` q st vanishing locus is all of T , as desired.Now, suppose that p L , p V v q v q is a T -valued point of G r ¯ w p X , n , p O v q v q Ď P rw p X , n , p O v q v q . In order to lift to P rw ‚ p X , n , p O v q v q , we will set V v “ V v for all v ‰ v . At v , we consider (3.3) for w “ w v , and apply Corollary 3.4 again toconclude that the kernel of (3.3) is a subbundle of rank r ` V v under the universal injection p ˚ L w v ã Ñ p ˚ L v ã Ñ p ˚ L v . Put differently, V v must be contained in (the image of) p ˚ L w v . Then set V v tobe the preimage of V v in p ˚ L v , or equivalently, the image of the kernel of (3.3).This gives a ( T -valued) point of P rw ‚ p X , n , p O v q v q mapping to p L , p V v q v q , and itis clear from the above injectivities that such a point is unique. It thus remains tocheck that the point we have constructed lies in G r ¯ w p X , n , p O v q v q .Given any w , we know that the kernel of (3.3) has p r ` q st vanishing locus equalto T , and we wish to verify the same for the kernel of (3.2). If there is a minimalpath from w to w v factoring through w v , then we are in the same situation as above,and we get the desired statement. On the other hand, if the minimal path from w to w v includes a twist at v , then in (3.2) the map to the summand p p ˚ L v q{ V v iszero, so we conclude that (3.2) factors through (3.3), and then by Corollary B.3.5of [Oss14c] it follows that the p r ` q st vanishing locus of (3.2) is all of T . Theproposition follows. (cid:3) We now describe a second version of the moduli space construction, which isless immediately related to our definition of limit linear series, but which workstransparently in families of curves; we will then show in Proposition 3.8 that on thespecial fiber, the two constructions are canonically isomorphic.
Notation . In the situation of Definition 2.21, let r P rw ‚ p X , n , p O v q v q be thescheme parametrizing tuples p L , p V v q v P V p Γ q q , where L is a line bundle on r X ofmultidegree w , and each V v is an p r ` q -dimensional space of global sections ofthe induced line bundle L w v on r X .Denote by G rw p r X q the moduli scheme of pairs p L , V q , where L has multidegree w on r X , and V is an p r ` q -dimensional space of global sections of L . We thus havethat r P rw ‚ p X , n , p O v q v q can naturally be constructed as the product over v P V p Γ q ofthe spaces G rw v p r X q , fibered over Pic w p r X q via twisting by O w v ,w . In particular, r P rw ‚ p X , n , p O v q v q is proper over Pic w p r X q . Definition 3.7.
In the situation of Notation 3.6, let Ă M be the universal linebundle on r P rw ‚ p X , n , p O v q v q ˆ X , and for each w P V p G p w qq , let Ă M w be in-duced by twisting as before. Then for each v P V p Γ q , let r V v be the universalsubbundles of p ˚ Ă M w v , and let r G r ¯ w p X { B, n , p O v q v q be the closed subscheme of r P rw ‚ p X { B, n , p O v q v q defined by the intersection over w P V p G p w qq of the p r ` q stvanishing loci of the maps(3.4) p ˚ Ă M w Ñ à v P V p Γ q p p ˚ Ă M w v q{ r V v . Thus, a T -valued point of r P rw ‚ p X , n , p O v q v q is a tuple p L , p V v qq , where L is aline bundle on T ˆ r X of multidegree w , and each V v is a rank- p r ` q subbundleof p ˚ L w v . Such a tuple is a T -valued point of r G r ¯ w p X , n , p O v q v q if for all w P V p G p w qq , the map(3.5) p ˚ L w Ñ à v P V p Γ q p p ˚ L w v q{ V v has p r ` q st vanishing locus equal to all of T .We now check that our two constructions are equivalent. Note that it follows inparticular that r G r ¯ w p X , n , p O v q v q is also independent of the choice of p w v q v . Proposition 3.8.
In the situation of Definition 2.21, restriction to the components Z v induces an isomorphism r G r ¯ w p X , n , p O v q v q „ Ñ G r ¯ w p X , n , p O v q v q . Proof.
We first verify that restriction to the Z v induces a morphism(3.6) r P rw ‚ p X , n , p O v q v q Ñ P rw ‚ p X , n , p O v q v q , which amounts to the assertion that if V v is a subbundle of p ˚ L w v on some scheme T over Spec k , then restricting V v to Z v induces a subbundle of the same rank of p ˚ L v . By Lemma B.2.3 (iii) of [Oss14c], this follows from injectivity of restrictionon points, which is Proposition 3.3. Next, that (3.6) induces a morphism r G r ¯ w p X , n , p O v q v q Ñ G r ¯ w p X , n , p O v q v q is immediate from the fact that (3.2) factors through (3.5), using Corollary B.3.5of [Oss14c].It thus remains to prove that this morphism is an isomorphism, or equiva-lently that every T -valued point of G r ¯ w p X , n , p O v q v q lifts to a unique point of r G r ¯ w p X , n , p O v q v q . Accordingly, suppose that p L , p V v q v P V p Γ q q is a T -valued pointof G r ¯ w p X , n , p O v q v q ; by the injectivity of the maps p ˚ L w v Ñ p ˚ L v , a lift p L , p V v q v P V p Γ q q is unique, if it exists. Next, for any v P V p Γ q , if we considerthe multidegree w v , Corollary 3.4 implies that the kernel of (3.2) is a subbundleof p ˚ L w v of rank r `
1, which is the preimage of V v . We thus set this kernelas our V v . Thus, it is enough to see that with this choice of the bundles V v , wehave that for every multidegree w , the p r ` q st vanishing locus of (3.5) is all of T .But by construction, for each v the natural map p p ˚ L w v q { V v Ñ p p ˚ L v q { V v isinjective, even after arbitrary base change, so it follows that for any w , the kernelsof (3.2) and (3.5) are identified, likewise after arbitrary base change. Then the p r ` q st vanishing loci agree by Proposition B.3.4 of [Oss14c], giving the desiredstatement. (cid:3) IMIT LINEAR SERIES FOR CURVES NOT OF COMPACT TYPE 17
We conclude this section by explaining how the construction of Definition 3.7works in families, and applying it to prove a specialization statement.First, the families of curves we will consider are as follows:
Definition 3.9.
We say that π : X Ñ B is a smoothing family if B “ Spec R for R a DVR, and further:(I) π is flat and proper;(II) the special fiber X of π is a (split) nodal curve;(III) the generic fiber X η of π is smooth;(IV) π admits sections through every component of X .If further X is regular, we say that π is a regular smoothing family.See Remark 3.17 below for discussion of our choice of level of generality. Con-dition (IV) is always satisfied after etale base change, and is used to ensure theexistence of a Picard scheme with universal line bundle.Associated to a smoothing family we still have a dual graph Γ: namely, the dualgraph of the special fiber X . We then continue to use the notation Z v to denotethe component of X corresponding to a vertex v of Γ. In this situation, one maydefine an enriched structure as before, with the additional condition that thereshould exist sections s v as in Situation 2.17. We then see: Proposition 3.10. If π : X Ñ B is a regular smoothing family, then an enrichedstructure is uniquely determined by setting O v “ O X p Z v q , and s v as in Situation2.17 are then induced by the canonical inclusions O X Ñ O X p Z v q . Moreover, thischoice induces an enriched structure together with suitable sections on X via re-striction. Now, we introduce the following terminology to take chain structures into ac-count.
Definition 3.11.
Given p X , n q and a regular smoothing family r π : r X Ñ r B with r B the spectrum of a DVR, we say that r π is of fiber type p X , n q if the specialfiber of r π is isomorphic to (a base extension of) the curve r X obtained from p X , n q .Given also a smoothing family π : X Ñ B , with special fiber X , we say that r π isan extension of π if it is obtained from π via base extension followed by iteratedblowups at the nodes of the special fiber.Whenever we say π is of fiber type p X , n q , we implicitly assume that we havefixed an isomorphism between the special fiber of π and the appropriate base ex-tension of r X .(Regular) smoothing families of type p X , n q arise naturally in two differentways: the first is as extensions of a given regular smoothing family π , taken forinstance in order to extend the generic point to a field of definition of a line bundleon the geometric generic fiber, in which case the line bundle will extend over theextended family. The second is as regularizations of irregular families, in whichcase no base change is involved. The former will be more immediately importantto us, but our theory is general enough to handle both situations at once. Situation 3.12.
Suppose r π : r X Ñ r B is a regular smoothing family of fiber type p X , n q , and we also fix an admissible multidegree w on r X , as well as a tuple p w v q v P V p r Γ q of vertices of G p w q , with each w v concentrated at v . Let p O v , s v q v P V p Γ q be the enriched structure and associated sections on r X given by Proposition 3.10. In Situation 3.12, given w P V p G p w qq , denote by Pic w p r X { r B q the moduli schemesof line bundles of degree d which have multidegree w on fibers lying over the closedpoint of r B . Then denote by G rw p r X { r B q the moduli scheme of pairs p L , V q , where L is in Pic w p r X { r B q , and V is an p r ` q -dimensional space of global sections of L .The representability of these spaces is standard; one can argue just as in the proofof Theorem 5.3 of [Oss06], for instance. The maps (3.4) generalize to this situation,and we can then generalize the previous constructions to the case of families. Notation . Construct r P rw ‚ p r X { r B, X , n , p O v q v q as the product over v P V p Γ q ofthe spaces G rw v p r X q , fibered over Pic w p r X q , and let r G r ¯ w p r X { r B, X , n , p O v q v q be theclosed subscheme defined by the intersection of the p r ` q st vanishing loci of themaps (3.4), as w varies over V p G p w qq .We then have the following basic fact. Proposition 3.14.
The moduli scheme r G r ¯ w p r X { r B, X , n , p O v q v q is proper over Pic w p r X { r B q . Its generic fiber is naturally identified with G rd p X η q , and its specialfiber with (the appropriate base extension of ) r G r ¯ w p X , n , p O v q v q .Proof. The first statement is immediate from the construction, as is the statementon the special fiber. The description of the generic fiber follows from the observationthat the maps f w,w are all isomorphisms over the generic fiber; in fact, we claimthat if we fix any v , then an arbitrary choice of V v uniquely determines V v as theimage of V v for all v ‰ v . Indeed, using Proposition B.3.4 and Lemma B.2.3 (iv)of [Oss14c], we see that for a given choice of V v , if we consider w “ w v we will havethe desired condition on the p r ` q st vanishing locus of (3.5) if and only if V v mapsinto each of the V v , which is the same as saying that V v is the image of V v . Onthe other hand, if V v is the image of V v for all v , we see that the kernel of (3.5) forany w is simply the image of V v , so we have the desired behavior of the p r ` q stvanishing locus. (cid:3) Corollary 3.15.
Let π : X Ñ B be a smoothing family, with special fiber X .Let p L , V q be a g rd on the geometric generic fiber X ¯ η . Then there exists a chainstructure n on X , an extension r π : r X Ñ r B of π having fiber type p X , n q , and anadmissible multidegree w on the resulting r X such that L extends to a line bundleof multidegree w on r X .For any such n , r π , and w , and any collection of w v P V p G p w qq concentratedat each v P V p Γ q , we have that p L , V q extends to a limit linear series on r X .Proof. This is mostly standard, but also brief, so we include it for the convenienceof the reader. The last assertion is immediate from Propositions 3.14 and 3.8.For the first assertion, we necessarily have p L , V q defined over some finite exten-sion η of η ; let B be the corresponding integral closure of B , localized at a closedpoint. If X “ X ˆ B B , then if we repeatedly blow up the non-smooth locus of X over B to obtain a regular total space, we obtain our n and r X . Since L is nowdefined over the new generic fiber, and r X is still regular, we can extend L to allof r X . It remains to see that the extension can be chosen to have admissible multi-degree, but this is easily achieved by twisting first at non-exceptional componentsto achieve sufficiently positive degree on each chain of exceptional components,and then twisting at exceptional components first to achieve nonnegativity on eachcomponent, and then admissibility. (cid:3) IMIT LINEAR SERIES FOR CURVES NOT OF COMPACT TYPE 19
Remark . In fact, we see from the proof of Corollary 3.15 that we have thefollowing refined statement: let n be the chain structure on X obtained by setting n p e q to be one greater than the number of blowups required to make make X regular at the point corresponding to e . Then the n produced in the proof is of theform n p e q “ m n p e q for all e , where m is the ramification index of B over B .Thus, the collection of chain structures we need to consider in order to extendline bundles on the initial family are not arbitrary, but are restricted to multiplesof the “base” chain structure n . Remark . The base B in Definition 3.9 may be generalized considerably, butthis makes the conditions more complicated; compare Definitions 2.1.1 and 2.2.2 of[Oss14c]. Moreover, imposing the existence of an enriched structure will imply thateven if B is higher-dimensional, the geometry of the family all occurs in codimension1, so there seems to be little reason to introduce additional technical complications.4. The two-component case
In order to give the equivalent definition which will ultimately generalize that ofEisenbud and Harris, the two-component case is the simplest situation to consider.Conveniently, it is also the base case of an induction argument for the more generalsituation, so we will first develop the key comparison result for curves with twocomponents. In this case, we simplify our notation as follows.
Situation 4.1.
Let X consist of two smooth curves Z , Z glued to one another atnodes P , . . . , P m . Fix the orientation on Γ with all edges going from Z to Z . Let n be a chain structure, and for i “ , . . . , m , write n i : “ n p P i q . For i “ , . . . , m ,and j “ , . . . , n i ´
1, let E i,j denote the j th exceptional component of r X lyingover P i on X . Fix an admissible multidegree w on p X , n q , and multidegrees w , w P V p G p w qq concentrated at Z , Z respectively. Write µ i : “ µ p P i q , where w “ pp w q Γ , µ p‚qq . Let b be the number of twists at Z required to get from w to w . Identify V p G p w qq with Z by sending w to the number of twists at Z requiredto get from w to w .We will assume throughout this section that we are in the above situation. Inthis case, G p w q is an unbounded chain, with edges going in each direction. Wehave identified w with 0, and w with b . Accordingly, for any line bundle L ofmultidegree w on r X , for i P Z we will write L i for the line bundle L w , where w is obtained from w by twisting i times at Z . As we have already done above,when convenient we will write nodes or components in place of the correspondingedges or vertices of the dual graph.We then introduce the following notation as well. Notation . For any line bundle L of multidegree w on r X , write L : “ L | Z ,and L : “ L b | Z .We now define sequences of effective divisors supported on the P i which will beused to give multivanishing sequences. Definition 4.3.
Let D , . . . , D b ` be the sequence of effective divisors on Z de-fined by D “
0, and for i ě D i ` ´ D i “ ÿ j : µ j ` i ” p mod n j q P j , and similarly define D , . . . , D b ` on Z by D “
0, and for i ě D i ` ´ D i “ ÿ j : µ j ` b ´ i ” p mod n j q P j . The relationship between the twisting divisors and line bundles is given by thefollowing basic proposition, whose proof is left to the reader.
Proposition 4.4.
For i “ , . . . , b ` , we have L i | Z “ L p´ D i q and L i ´ | Z “ L p´ D b ` ´ i q , where we use equality to denote canonical isomorphism.We also have for all i “ , . . . , b that P j is in the support of D i ` ´ D i if and onlyif L i has degree on E j,(cid:96) for all (cid:96) , and P j is in the support of D i ` ´ D i if and onlyif L b ´ i has degree on E j,(cid:96) for all (cid:96) . In particular, if E i denotes the union over j such that P j in the support of D i ` ´ D i of the chains of exceptional componentslying over the P j , L i | E i – O E i , and we thus get an induced isomorphism ϕ i : L p´ D i q{ L p´ D i ` q „ Ñ L p´ D b ´ i q{ L p´ D b ` ´ i q for each i . We think of the ϕ i as being gluing maps; in the case of trivial chain structure,the ϕ i are each defined on all nodes at once, but in general they are only definedon subsets of the nodes, which depend on i . Definition 4.5.
In the situation of Definition 1.2, we say that j is critical for D ‚ if D j ` ‰ D j .Our main comparison result in the two-component case is as then follows: Lemma 4.6.
In Situation 4.1, fix also an enriched structure on r X , and sections s v as in Situation 2.17. For a given p L , p V , V qq , and i “ , , denote by a i , . . . , a ir the multivanishing sequence of V i along the D i ‚ . Then p L , p V , V qq is a limitlinear series if and only if (I) for (cid:96) “ , . . . , r , if a (cid:96) “ deg D j with j critical for D ‚ , then (4.1) a r ´ (cid:96) ě deg D b ´ j ;(II) for i “ , , there exist bases s i , . . . , s ir of the V i such that ord D ‚ s i(cid:96) “ a i(cid:96) for (cid:96) “ , . . . , r, and for all (cid:96) with (4.1) an equality, we have ϕ j p s (cid:96) q “ s r ´ (cid:96) when we consider s (cid:96) P V p´ D j q and s r ´ (cid:96) P V p´ D b ´ j q , with j as in (I).Remark . Although condition (I) appears asymmetric, in fact this is not thecase; indeed, Proposition 4.4 says that the construction of the D i ‚ implies that j iscritical for D ‚ if and only if b ´ j is critical for D ‚ , so (I) is equivalent to requiringthat if a r ´ (cid:96) “ deg D b ´ j with b ´ j critical for D ‚ , then a (cid:96) ě deg D j .As an intermediate step, it is convenient to consider a bounded version of G p w q as follows. IMIT LINEAR SERIES FOR CURVES NOT OF COMPACT TYPE 21
Notation . Let ¯ G p w q denote the directed subgraph of G p w q consisting of allvertices between w and w (inclusive), and with all edges of G p w q connectingvertices in V p ¯ G p w qq .It turns out that in the definition of limit linear series, considering multidegreesin ¯ G p w q suffices. Proposition 4.9.
In the situation of Lemma 4.6, p L , p V , V qq is a limit linearseries if and only if (2.1) has kernel of dimension at least r ` for all w P V p ¯ G p w qq .Proof. Since V p ¯ G p w qq Ď V p G p w qq , one direction is trivial. Conversely, supposethat (2.1) has kernel of dimension at least r ` w P V p ¯ G p w qq , and let w P V p G p w qq be arbitrary; we need to show that (2.1) also has kernel of dimensionat least r ` w . Considering w “ i for some i P Z , there are threecases to consider: either 0 ď i ď b , or i ă
0, or i ą b . The first case is the sameas having w P V p ¯ G p w qq , so there is nothing to show. The other two cases beingsymmetric, we only treat the case that i ă
0. In this case, we claim that the kernel W of (2.1) in multidegree w injects into the kernel of (2.1) in multidegree w under f w ,w . Indeed, it is clear that the entire image of f w ,w is contained in the kernelof (2.1), so it suffices to see that f w ,w is injective on W . But f w ,w is inducedby a map which is an inclusion on Z , so the desired injectivity is an immediateconsequence of Proposition 3.3. (cid:3) Next, in ¯ G p w q , we can reinterpret the kernel of (2.1) as follows. Proposition 4.10.
In the situation of Lemma 4.6, for i “ , . . . , b , consider themap (4.2) V p´ D i q ‘ V p´ D b ´ i q Ñ L p´ D b ´ i q{ L p´ D b ´ i ` q induced by taking quotients, and applying ´ ϕ i on the first factor. Then our mor-phisms H p r X , L i q Ñ H p Z j , L j q for j “ , induce an isomorphism between thekernel of (2.1) and the kernel of (4.2) .Proof. The image of H p r X , L i q in H p Z , L q (respectively, H p Z , L q ) is con-tained in H p Z , L p´ D i qq (respectively, H p Z , L p´ D b ´ i qq ) by construction,so a section of H p X , L i q which lies in the kernel of (2.1) necessarily restricts to V p´ D i q on Z and V p´ D b ´ i q on Z . That it in fact yields an element in the ker-nel of (4.2) is essentially the definition of ϕ i . To see that the constructed map is bi-jective, the main point is that given a pair p s , s q P H p Z , L i | Z q‘ H p Z , L i | Z q ,an extension of p s , s q to a global section s P H p r X , L i q is unique if it exists, andit exists if and only if ϕ i p s q “ s , using the identifications of Proposition 4.4.Indeed, the assertion is clear on the union of exceptional chains E i from the con-struction of ϕ i , so it is enough to check that there is always a unique extension overthe exceptional chains not contained in the E i . But if E is such a chain, then L i | E has degree 1 on exactly one irreducible component, and degree 0 on the others, andit follows that L i | E has a unique global section with arbitrary prescribed values ateither end of E , giving the desired assertion. The desired bijectivity follows. (cid:3) We can now finish our examination of the two-component case.
Proof of Lemma 4.6.
First, by Propositions 4.9 and 4.10, we have reduced to show-ing that p L , p V , V qq satisfies (I) and (II) if and only if (4.2) has kernel of dimen-sion at least r ` i “ , . . . , b . Now, observe thatdim V p´ D i q “ t (cid:96) : a (cid:96) ě deg D i u , and dim V p´ D b ´ i q “ t (cid:96) : a (cid:96) ě deg D b ´ i u . For each i , let r i be the rank of (4.2); note that r i “ i is critical for D ‚ .Choose (cid:96) and (cid:96) minimal with a (cid:96) ě deg D i and a (cid:96) ě deg D b ´ i , and (cid:96) and (cid:96) maximal with with a (cid:96) ď deg D i and a (cid:96) ď deg D b ´ i . Here, if a (cid:96) ă deg D i for all (cid:96) , set (cid:96) “ r `
1, if a (cid:96) ą deg D i for all (cid:96) , set (cid:96) “ ´
1, and similarly for (cid:96) and (cid:96) .Then the kernel of (4.2) has dimension equal todim V p´ D i q ` dim V p´ D b ´ i q ´ r i “ r ` ´ (cid:96) ` r ` ´ (cid:96) ´ r i , so for the kernel of (4.2) to have dimension at least r ` (cid:96) ` (cid:96) ` r i ď r ` . In addition, we see that condition (II) of the lemma is equivalent to having that, foreach critical i for D ‚ , the images of V p´ D i q and V p´ D b ´ i q under (4.2) overlapwith dimension at least equal to(4.4) t (cid:96) : (cid:96) ď (cid:96) ď (cid:96) , and (cid:96) ď r ´ (cid:96) ď (cid:96) u . Now, if we assume condition (I), we claim that for all i , we have (cid:96) ` (cid:96) ď r ` i is critical for D ‚ , we also have (cid:96) ď r ´ (cid:96) and (cid:96) ď r ´ (cid:96) . For thefirst claim, note that by definition a (cid:96) ´ ă deg D i ; if we let i be critical for D ‚ with a (cid:96) ´ “ deg D i , then (I) implies that a r ` ´ (cid:96) ě deg D b ´ i ě deg D b ´ i , so (cid:96) ď r ` ´ (cid:96) , giving the first claimed inequality. Next, if i is critical for D ‚ ,then there are two cases to consider: if deg D i does not occur in a , we will have (cid:96) “ (cid:96) ´
1, and in this case the inequality (cid:96) ď r ´ (cid:96) is the same as (cid:96) ď r ` ´ (cid:96) ,which we have just proved. On the other hand, if deg D i does occur in a , then(I) gives a r ´ (cid:96) ě D b ´ i , which means (cid:96) ď r ´ (cid:96) , as desired. The proof of the lastclaimed inequality (cid:96) ď r ´ (cid:96) is similar, taking into account Remark 4.7.Still assuming (I), we next claim that (4.2) having kernel of dimension at least r ` i “ , . . . , b is equivalent to condition (II). If i is not critical for D ‚ ,then r i “
0, so we see from (4.3) that the desired kernel size follows from theinequality (cid:96) ` (cid:96) ď r `
1, which we have already proved. If i is critical for D ‚ ,using (cid:96) ď r ´ (cid:96) and (cid:96) ď r ´ (cid:96) . the inequalities in (4.4) simplify to r ´ (cid:96) ď (cid:96) ď (cid:96) .Thus, the existence of the desired basis is equivalent to requiring that the imagesof V p´ D i q and V p´ D b ´ i q under (4.2) overlap with dimension at least equal to (cid:96) ` ´ p r ´ (cid:96) q . On the other hand, the dimension of this overlap is equal to thesum of the dimensions of the images of V p´ D i q and V p´ D b ´ i q , minus r i , whichis to say, (cid:96) ` ´ (cid:96) ` (cid:96) ` ´ (cid:96) ´ r i , so we conclude that (again, assuming (I)),condition (II) is equivalent to the inequality (cid:96) ` ´ (cid:96) ` (cid:96) ` ´ (cid:96) ´ r i ě (cid:96) ` ´ p r ´ (cid:96) q , which is the same as (4.3). This proves the claim, and we conclude that (I) and(II) together imply that p L , p V , V qq is a linear linear series, and moreover, thatto see the converse, it is enough to prove that (4.3) implies condition (I).Thus, assume (4.3). Given (cid:96) P t , . . . , r u , let i be critical for D ‚ with deg D i “ a (cid:96) ,and choose (cid:96) , (cid:96) , (cid:96) , (cid:96) as above. Observe that r i ě t (cid:96) : a (cid:96) “ i u “ (cid:96) ` ´ (cid:96) sothat (I) implies that r ` ´ (cid:96) ´ (cid:96) ě (cid:96) ` ´ (cid:96) . It thus follows that r ě (cid:96) ` (cid:96) ě (cid:96) ` (cid:96) ,so r ´ (cid:96) ě (cid:96) . Thus, we find a r ´ (cid:96) ě a (cid:96) ě deg D b ´ i , IMIT LINEAR SERIES FOR CURVES NOT OF COMPACT TYPE 23 giving (I), and completing the proof of the lemma. (cid:3) The pseudocompact-type case
We conclude by generalizing the results of the previous section to arbitrary curvesof pseudocompact type, thereby providing a simultaneous generalization of the two-component case and the compact-type case. As before, we start with combinatorialpreliminaries.
Notation . If Γ is a graph, let ¯Γ be the graph obtained from Γ by collapsing allmultiple edges to single edges, while leaving the vertex set unchanged. We say Γ isa multitree if ¯Γ is a tree.Just as before we defined twists motivated by twisting at a component, in themultitree case we define twists motivated by twisting on one side or the other ofthe node(s) at which two components meet.
Definition 5.2.
If Γ is a multitree, and p e, v q a pair of an edge e and an adjacentvertex v of ¯Γ, given an admissible multidegree w , we define the twist of w at p e, v q to be obtained from w as follows: for each ˜ e of Γ over e , increase µ p ˜ e q by σ p ˜ e, v q .Now, decrease w Γ p v q by the number of ˜ e for which µ p ˜ e q had been equal to 0, andfor each ˜ e , if the new µ p ˜ e q is zero, increase w Γ p v q by 1, where v is the other vertexadjacent to v .Notice that if v is the other vertex adjacent to an edge e , then twisting at p e, v q is inverse to twisting at p e, v q . In addition, we observe that the twist of w at p e, v q may be obtained as a sequence of twists of w at vertices v , where v varies overthe set of vertices in the same connected component as v in ¯Γ (cid:114) t e u . Conversely,twisting of w at any v can also be obtained as a composition of twists at p e, v q ,where e varies over edges adjacent to v .Throughout this section, all twists will be with respect to pairs p e, v q , ratherthan vertices. Warning . Even though on a combinatorial level, twisting w by v can be obtainedby a sequence of twists at different p e, v q , the same does not hold on the level ofthe maps between the associated line bundles. Situation 5.4.
Suppose we are given a multitree Γ, and an admissible multidegree w , and let p w v q v P V p Γ q be a collection of elements of V p G p w qq such that:(I) each w v is concentrated at v ;(II) for each v, v P V p ¯Γ q connected by an edge e , the multidegree w v is obtainedfrom w v by twisting b v,v times at p e, v q , for some b v,v P Z ě . Definition 5.5.
In Situation 5.4, let V p ¯ G p w qq Ď V p G p w qq consist of admissiblemultidegrees w such that there exist v, v P V p ¯Γ q connected by some edge e , with w obtainable from w v by twisting b times at p e, v q , for some b with 0 ď b ď b v,v .There is an edge (cid:15) from from w to w in ¯ G p w q if there exist p e, v q in ¯Γ such that w is obtained from w by twisting at p e, v q .Thus, ¯ G p w q is a tree, obtained by subdividing every edge of ¯Γ into b v,v edges,and replacing each edge with a pair of directed edges in opposite directions. Notethat in general the edges of ¯ G p w q need not be edges of G p w q , but can be thoughtof as “compositions” of edges of G p w q . However, in the case that Γ has only two vertices, we have that G p w q and ¯ G p w q are both chains, with the only differencebeing that ¯ G p w q is bounded by w v and w v , while G p w q is unbounded. Thus,our notation is consistent with that of Notation 4.8.We now move on to the geometric definitions and statements. Definition 5.6.
Let X be a projective nodal curve, with dual graph Γ. X is of pseudocompact type if Γ is a multitree. Situation 5.7.
In Situation 5.4, suppose also that our Γ is obtained as the dualgraph of a given projective nodal curve X . Notation . In Situation 5.7, for each pair p e, v q of an edge and adjacent vertexof ¯Γ, let D p e,v q , . . . , D p e,v q b v,v ` be the sequence of effective divisors on Z v defined by D p e,v q “
0, and for i ě D p e,v q i ` ´ D p e,v q i “ ÿ ˜ e over e : σ p ˜ e, v q µ v p ˜ e q ” ´ i p mod n p ˜ e qq P ˜ e , where P ˜ e denotes the node of X corresponding to ˜ e , and µ v p‚q is obtained from w v .Our main result is the following. Theorem 5.9.
In the situation of Definition 2.21, suppose further that X is ofpseudocompact type, and we are in Situation 5.7. Then given a tuple p L , p V v q v P V p Γ q q ,for each pair p e, v q in ¯Γ , let a p e,v q , . . . , a p e,v q r be the multivanishing sequence of V v along D p e,v q‚ . Then the following are equivalent: (a) p L , p V v q v q is a limit linear series; (b) (2.1) has kernel of dimension at least r ` for every w P V p ¯ G p w qq ; (c) for any e P E p Γ q , with adjacent vertices v, v , we have: (I) for (cid:96) “ , . . . , r , if a p e,v q (cid:96) “ deg D p e,v q j with j critical for D p e,v q‚ , then (5.1) a p e,v q r ´ (cid:96) ě deg D p e,v q b v,v ´ j ;(II) there exist bases s p e,v q , . . . , s p e,v q r of V v and s p e,v q , . . . , s p e,v q r of V v such that ord D p e,v q‚ s p e,v q (cid:96) “ a p e,v q (cid:96) , for (cid:96) “ , . . . , r, and similarly for s p e,v q (cid:96) , and for all (cid:96) with (5.1) an equality, we have ϕ p e,v q j p s p e,v q (cid:96) q “ s p e,v q r ´ (cid:96) when we consider s p e,v q (cid:96) P V v p´ D p e,v q j q and s p e,v q r ´ (cid:96) P V v p´ D p e,v q b v,v ´ j q ,where j is as in (I), and ϕ j is as in Proposition 4.4. In (II) above, note that although Proposition 4.4 was only stated for two-component curves, since we are only interested in a given pair of adjacent verticesof Γ, the situation is no different in our present more general case.We first introduce some convenient notation. The following can be used to keeptrack of twisting at nodes:
IMIT LINEAR SERIES FOR CURVES NOT OF COMPACT TYPE 25
Notation . In Situation 5.4, given w P V p G p w qq , and p e, v q adjacent in ¯Γ, let t p e,v q p w q be the number of twists at p e, v q required to go from w v to w in a minimalnumber of twists.Note that t p e,v q p w q is well-defined, since the only way to cancel a twist at p e, v q is to twist at p e, v q , where v is the other vertex adjacent to v . In addition, we have t p e,v q p w q ` t p e,v q p w q “ b v,v .We can now define a notion of restriction of multidegrees to subcurves. Ofcourse, one can always restrict naively, but this turns out not to be well behavedwith respect to limit linear series, so instead we make the following definition. Definition 5.11.
In Situation 5.4, let X be a connected subcurve of X , withdual graph Γ . Then for any w P V p G p w qq , define the restriction of w to X as follows: starting from w , let w be the admissible multidegree obtained by, foreach pair p e, v q in ¯Γ where v P Γ but the other vertex v adjacent to e is not inΓ , twisting t p e,v q p w q times at p e, v q . Then, the restriction of w to X is the naiverestriction of w .The reason for this choice of restriction, rather than the more naive one, is thatif we naively restrict an arbitrary w , it will no longer be obtainable as a twist ofthe restrictions of the w v . With our choice of restriction, even though we modify w , we will be able to understand the kernel of (2.1) for a given w in terms of thekernels of the restrictions to subcurves covering X ; see the proof of Theorem 5.9below.Note that if w P V p ¯ G p w qq , say between w v and w v , and if X contains Z v and Z v , then in fact the restriction of Definition 5.11 is simply the same as naiverestriction. Proof of Theorem 5.9.
First observe that because V p ¯ G p w qq Ď V p G p w qq , the im-plication (a) implies (b) is trivial. We will prove that (b) implies (c) and (c) implies(a), by induction on the number of components of X . The base case is that X has two components, which is precisely Lemma 4.6, together with Proposition 4.9.Now, for the induction step, the basic observation is that condition (c) is imposedon a pair of nodes at a time, so that (c) holds if and only if for each pair v , v ofadjacent vertices of Γ, the restriction p L w | Z v Y Z v , p V v , V v qq also satisfies (c) forthe curve Z v Y Z v , where w is any element of V p ¯ G p w qq lying between w v and w v . Note that deg L w | Z v Y Z v is not in general equal to d , but is independent ofthe choice of w .Thus, to see that (b) implies (c), we suppose that (2.1) has kernel of dimensionat least r ` w P V p ¯ G p w qq , and we will show that if v , v P V p Γ q areadjacent, then p L w | Z v Y Z v , p V v , V v qq satisfies (c). But suppose w P V p ¯ G p w qq lies between w v and w v . Then note that w agrees with both w v and w v awayfrom v and v and the edges between them, so arguing as in Proposition 3.3, thekernel of (2.1) for X injects into the kernel of (2.1) for Z v Y Z v under restrictionto Z v Y Z v . Thus, by Lemma 4.6 we conclude that p L w | Z v Y Z v , p V v , V v qq satisfies (c), as desired.On the other hand, if (c) is satisfied, we prove the desired statement by inductionon the number of components. Given w P V p G p w qq , there are two cases to consider.First, if for some p e, v q , we have t p e,v q p w q ă
0, let X be the subcurve of X corresponding to the connected component ¯Γ (cid:114) t e u containing v . Then if w is the multidegree obtained from w in Definition 5.11, because t p e,v q p w q ă
0, we have amap L w Ñ L w which is injective on r X ; let Y be the subcurve of r X on whichit is injective, and Z the subcurve on which it vanishes. Thus, r X Ď Y , and r X “ Y Y Z , and also Y and Z have no components in common. We thus have aninclusion L w | Y Ñ L w | Y whose image vanishes at Y X Z , and it follows that wecan extend by zero to get an inclusion H p Y, L w | Y q ã Ñ H p r X , L w q . On the other hand, by construction we observe that L w is trivial on componentson Y not contained in r X , so we have H p r X , L w | Ă X q “ H p Y, L w | Y q , inducing an inclusion H p r X , L w | Ă X q ã Ñ H p r X , L w q . Now, we have by hypothesis that (c) is satisfied on r X , so by the induction hypoth-esis, the kernel of (2.1) for r X has dimension at least r ` w , andusing the above inclusion, we get the same for r X in multidegree w , as desired.The second case is that t p e,v q p w q ě p e, v q , in which case we necessarilyhave 0 ď t p e,v q p w q ď b v,v . In this case, choose v P V p Γ q which is only adja-cent to one other v P V p Γ q (i.e., which is a leaf of ¯Γ). Let X be the closure ofthe complement of Z v in X ; then by hypothesis, (c) is satisfied for the restric-tions p L w | Z v Y Z v , p V v , V v qq and p L w | X , p V v q v ‰ v q , where w is any elementof V p ¯ G p w qq not lying between w v and w v . By the induction hypothesis, we con-clude that (2.1) has kernel of dimension at least r ` Z v Y Z v in multidegree w and for X in multidegree w , where w and w are the restrictions of w . Butbecause 0 ď t p e,v q p w q ď b v,v for all p e, v q , the kernel of (2.1) for X in multidegree w is simply the fibered product of the above two kernels over V v , and hence alsohas dimension at least r `
1, as desired. (cid:3) A smoothing theorem
In this section, we prove the following theorem, which says that – just as inthe Eisenbud-Harris case – when the space of limit linear series on a curve ofpseudocompact type has the expected dimension, then every limit linear series arisesas the limit of linear series on smooth curves. In fact, our theorem is stronger evenin the compact-type case, as it is not restricted to refined limit linear series. Ourproof is fundamentally different from that of Eisenbud and Harris, although it stillrelies in the end on obtaining a lower bound on the dimension of a relative modulispace. The key ingredient is the theory of linked determinantal loci, developed inAppendix A. We also use a portion of Theorem 5.9, in essence to reduce to thetwo-component case.
Theorem 6.1.
Let π : X Ñ B be a smoothing family, with special fiber X acurve of pseudocompact type. Let n be a chain structure on X , and r π : r X Ñ r B anextension of π having fiber type p X , n q . Let p O v q v be the induced enriched structureon X .Given an admissible multidegree w on the resulting r X , and p w v q v as in Situ-ation 5.4, if the moduli space G r ¯ w p X , n , p O v q v q has dimension ρ at a given point, IMIT LINEAR SERIES FOR CURVES NOT OF COMPACT TYPE 27 then the corresponding limit linear series arises as the limit of linear series on thegeometric generic fiber of π .More precisely, if π : r X Ñ r B is any regular smoothing family of fiber type p X , n q , then the scheme r G r ¯ w p r X { r B, X , n , p O v q v q has universal relative dimensionat least ρ over B , and if the special fiber G r ¯ w p X , n , p O v q v q has dimension exactly ρ at a point, then r G r ¯ w p r X { r B, X , n , p O v q v q is universally open at that point. If also thespecial fiber is geometrically reduced at the given point, then r G r ¯ w p r X { r B, X , n , p O v q v q is flat at that point. In the above, we use the relative dimension terminology introduced in [Oss13].
Proof.
The idea is to give a slightly different construction of the relative limitlinear series moduli space r G r ¯ w p r X { r B, X , n , p O v q v q , taking ideas from the proofof Theorem 5.3 of [Oss06] and using the linked determinantal loci developed inAppendix A. We can work set-theoretically, since our goal is a dimension statement.As in our earlier construction, start with the scheme Pic w p r X q , which is smoothover B of relative dimension g , and let Ă M be the universal line bundle, with Ă M w the induced line bundle in multidegree w for each w P V p G p w qq . Next, choose asufficiently π -ample divisor D on r X ; using our sections of π , we may assume that D “ ř v P V p Γ q D v , where D v X X meets only Z v . Note that we do not need to twistup on the exceptional components, since they are rational and our multidegreesare always nonnegative on them. We then have for each w that p ˚ p Ă M w p D qq islocally free of rank d ` deg D ` ´ g , and commutes with base change. Let G bethe fibered product over Pic w p r X q of the schemes G p r ` , p ˚ p Ă M w v p D qqq , where v ranges over V p Γ q . This is thus smooth over B of relative dimension g ` | V p Γ q|p r ` qp d ` deg D ` ´ g ´ p r ` qq“ g ` | V p Γ q|p r ` qp d ` deg D ´ r ´ g q . For each v , let V v be (the pullback to G of) the universal subbundle of p ˚ p Ă M w v p D qq .Let G be the closed subset of G obtained by imposing that for each v , the composedmap V v Ñ p ˚ p Ă M w v p D qq Ñ p ˚ p Ă M w v p D q| D v q vanishes identically, and by intersecting, for each e P E p ¯Γ q having adjacent vertices v, v , with the linked determinantal locus associated to the chain p ˚ p Ă M w p D qq for w between w v and w v together with the subbundles V v and V v . Then our key claimis that G is equal to r G r ¯ w p r X { r B, X , n , p O v q v q . Given the claim, we are done: theformer conditions impose codimension at most p r ` qp ř v deg D v q “ p r ` q deg D ,and the latter impose, by Theorem A.3, codimension at most | E p ¯Γ q|p r ` qp d ` deg D ` ´ g ´ p r ` qq“ p| V p Γ q| ´ qp r ` qp d ` deg D ´ r ´ g q . Subtracting the above maximal codimensions from the relative dimension of G , weare left with g ` p r ` qp d ´ r ´ g q “ ρ , and according to Corollary 5.1 of [Oss13], wefind that r G r ¯ w p r X { r B, X , n , p O v q v q has universal relative dimension at least ρ over B , as desired. The assertions on universal openness and flatness in the case thatthe special fiber has dimension exactly ρ at a point then follow from Proposition3.7 of [Oss13]. We are thus reduced to proving the claim. On the level of points, we analyze firstthe generic fiber X η , and then the special fiber r X . Over the generic fiber, the mapsbetween the L w are all isomorphisms, so the linked determinantal conditions in thedefinition of G imply that the V v all map to one another under these isomorphisms,and the condition that each V v vanish on D v implies that they all vanish on allof D . Thus, for a fixed choice of v , we have that points of G on the genericfiber are all uniquely determined by a choice of V v contained in L w v , which isthe same as r G r ¯ w p r X { r B, X , n , p O v q v q . Next, on the special fiber, we are assertingthe following: given a line bundle L of multidegree w and a tuple p V v q v with V v Ď Γ p r X , L w v p D qq , if each V v vanishes on D v , and for each w P V p ¯ G p w qq between w v and w v , the map(6.1) Γ p r X , L w p D qq Ñ Γ p r X , L w v p D qq{ V v ‘ Γ p r X , L w v p D qq{ V v has kernel of dimension at least r `
1, then in fact each V v is contained in Γ p r X , L w v q ,and the map(6.2) Γ p r X , L w q Ñ à v Γ p r X , L w v q{ V v has kernel of dimension at least r ` w P V p G p w qq . Our first observationis that for all v, v , we must have V v mapping into V v Ď Γ p r X , L w v p D qq under thenatural twisting maps. Because the maps L w v Ñ L w v always factor as a sequenceof such maps between adjacent vertices, it is enough to prove this when v, v areadjacent. In this case, we consider (6.1) in the case w “ w v , noting that the kernelis necessarily contained in V v . Then our hypothesis implies that the kernel is all of V v , and hence that V v maps into V v , as desired. Our next observation is that for w P V p ¯ G p w qq , under our hypotheses we have that the kernel of (6.1) is identifiedwith the kernel of(6.3) Γ p r X , L w p D qq Ñ à v Γ p r X , L w v p D qq{ V v . Indeed, this follows from the first observation, together with the fact that if w liesbetween w v and w v , then for any v the map L w Ñ L w v always factors througheither L w v or L w v .It then follows that the kernel of (6.3) vanishes on D for each w , since for each v , the map L w Ñ L w v is injective on Z v , so if V v vanishes on D v the kernel of(6.3) vanishes on D v as well. Since the D v are disjoint, we conclude that the kernelvanishes on D . Considering the case w “ w v , we conclude in particular that each V v vanishes on D , as desired. It follows that the kernel of (6.3) is identified withthe kernel of (6.2), so we have proved the desired statement for w P V p ¯ G p w qq .Moreover, if we set V v to be the image of V v in Γ p Z v , L v q , we see that the kernelof (6.2) is identified with the kernel of (2.1), so the equivalence of (a) and (b) inTheorem 5.9 then yields the desired statement for all w P V p G p w qq . (cid:3) Remark . Note that despite the pseudocompact type hypothesis, our proof ofthe smoothing theorem was built around our general definition of limit linear seriesrather than the equivalent definition of §
5. In fact, we expect that a similar proofshould be possible in full generality, with the main difficulty being the need fora much more general theory of linked Grassmannians. In our proof, due to thespecial form of curves of pseudocompact type, we were able to inductively reduceto what was, in essense, the “two-component” version of the linked Grassmannian,
IMIT LINEAR SERIES FOR CURVES NOT OF COMPACT TYPE 29 but in general no such reduction is possible. There is some evidence, in the form ofexamples and of parallel results for local models of certain Shimura varieties (see,for instance, Goertz [Goe01]), that such a general theory of linked Grassmanniansshould exist, but we expect that it will be substantially more difficult than thespecial case we have used here.We conclude with a scheme structure comparison result involving the construc-tion carried out in the proof of Theorem 6.1. This relates our construction tothe related definitions for the higher-rank case given in § Notation . Now suppose that we are in the situation of Theorem 6.1, or of The-orem 5.9, in which case we take B “ r B to be a point. Let r G r ¯ w p r X { r B, X , n , p O v q v q be the closed subscheme of the space r P rw ‚ p r X { r B, X , n , p O v q v q defined by the inter-section of the p r ` q st vanishing loci of the maps (3.4), as w varies over V p ¯ G p w qq .Thus, a priori we have that r G r ¯ w p r X { r B, X , n , p O v q v q is a closed subscheme of r G r ¯ w p r X { r B, X , n , p O v q v q , and Theorem 5.9 tells us that they are supported on thesame subset. Proposition 6.4.
The moduli scheme r G r ¯ w p r X { r B, X , n , p O v q v q is proper over Pic w p r X { r B q ,and in the case that π is a smoothing family, its generic fiber is naturally identifiedwith G rd p X η q .Moreover, the set-theoretic construction of r G r ¯ w p r X { r B, X , n , p O v q v q described inthe proof of Theorem 6.1 yields a scheme structure agreeing with r G r ¯ w p r X { r B, X , n , p O v q v q .Proof. The proof of the first part is the same as for Proposition 3.14. For the secondpart, we need to show that the set-theoretic analysis in the proof of Theorem 6.1works on the level of T -valued points if we consider only w P V p ¯ G p w qq . Thus,suppose we are given a T -valued tuple p L , p V v q v q , where each V v is a subbundle of p ˚ L w v p D q , and for any w P V p ¯ G p w qq between w v and w v , the map(6.4) p ˚ L w p D q Ñ pp p ˚ L w v p D qq{ V v q ‘ pp p ˚ L w v p D qq{ V v q has p r ` q st vanishing locus equal to T , and also that the composed maps V v Ñ p ˚ L w v p D q Ñ p ˚ p L w v p D q| D v are zero for each v . We want to show that in fact all the V v vanish on all of D , andfor all w P V p ¯ G p w qq , the p r ` q st vanishing locus of (3.5) is all of T .First, given v, v adjacent, setting w “ w v in (6.4), we see by Proposition B.3.4and Lemma B.2.3 (iv) of [Oss14c] that the kernel must be equal to V v , and thusthat V v maps into V v . Traversing ¯Γ in this way we conclude that each V v mapsinto each V v for any v ‰ v . We then observe that for any w , and any v , the map p ˚ p L w p D q| D v q Ñ p ˚ p L w v p D q| D v q is an isomorphism, so since V v maps into V v and V v vanishes on D v , we conclude that V v likewise vanishes on D v . Since D “ ř v D v , we find that each V v vanishes on all of D , and may be considered asa subbundle of p ˚ L w v . Similarly, we see that the kernel of (6.4) is (universally)identified with the kernel of(6.5) p ˚ L w Ñ pp p ˚ L w v q{ V v q ‘ pp p ˚ L w v q{ V v q , so by Proposition B.3.2 of [Oss14c] we have that the p r ` q st vanishing loci of thetwo maps agree. But then, again using that each V v maps into each other V v , andthe map from L w to L w v factors through L w v or L w v if w lies between w v and w v , we see that the kernel of (6.5) is also universally identified with the kernel of(3.5), giving the desired statement.Note that neither the construction from Theorem 6.1 nor our analysis of itsscheme structure depended on B being positive-dimensional, and in particular wealso conclude the desired statement in the case that B is a point. (cid:3) Appendix A. Linked determinantal loci
In this appendix, we develop a theory of “linked determinantal loci,” which arein essence a determinantal locus analogue of the linked Grassmannian developed inAppendix A of [Oss06]. A preliminary definition is the following:
Definition A.1.
Let S be a scheme, and d, n be positive integers. Suppose that E , . . . , E n are vector bundles of rank d on S and we have morphisms f i : E i Ñ E i ` , f i : E i ` Ñ E i for each i “ , . . . , n ´
1. Given s P Γ p S, O S q , we say that E ‚ “ p E i , f i , f i q i is an s -linked chain if the following conditions are satisfied:(I) For each i “ , . . . , n , f i ˝ f i “ s ¨ id , and f i ˝ f i “ s ¨ id . (II) On the fibers of the E i at any point with s “
0, we have that for each i “ , . . . , n ´ f i “ im f i , and ker f i “ im f i . (III) On the fibers of the E i at any point with s “
0, we have that for each i “ , . . . , n ´ f i X ker f i ` “ p q , and im f i ` X ker f i “ p q . This is precisely the condition required for the ambient chain of vector bundlesin the definition of a linked Grassmannian in [Oss06], although the terminology wasintroduced later, in [OT14]. We then define:
Definition A.2.
Let E ‚ be an s -linked chain on a scheme S . Given r ą
0, suppose F , F n are rank- r subbundles of E and E n respectively. Then the associated linked determinantal locus is the closed subscheme of S on which the morphisms(A.1) E i Ñ p E { F q ‘ p E n { F n q have rank less than or equal to d ´ r for all i “ , . . . , n .In Definition A.2, the necessary morphisms E i Ñ E j are obtained simply bycomposing the f i or f i , as appropriate.Thus, a linked determinantal locus is by definition an intersection of n determinalloci in S , for morphisms from vector bundles of rank d to vector bundles of rank2 d ´ r . The standard codimension bound for determinantal loci then implies that(each irreducible component of) a linked determinantal locus has codimension atmost n p d ´ p d ´ r qqp d ´ r ´ p d ´ r qq “ nr p d ´ r q . However, the structure imposedby our hypotheses implies that in fact, the codimension is far smaller. Our maintheorem is the following. IMIT LINEAR SERIES FOR CURVES NOT OF COMPACT TYPE 31
Theorem A.3.
Each irreducible component of a linked determinantal locus hascodimension at most r p d ´ r q in S .Remark A.4 . Notice that set-theoretically, the linked determinantal locus is the setof points of S at which the kernel of (A.1) has dimension at least r , or equivalently,the set of points such that the fiber of E i contains at least an r -dimensional spacewhich maps into F inside E and into F n inside E n . In particular, the case i “ F mapping into F n ,and the i “ n case implies that F n must map into F .Now, in order to see that Theorem A.3 is plausible, consider points of S overwhich s is nonzero. On this locus, all the maps are isomorphisms, and our hypothe-ses imply that F maps into F n if and only if F n maps into F , and that moreoverthe linked determinantal locus consists precisely of the points on which F mapsinto F n . Hence, on this locus it is clear that the codimension is at most r p d ´ r q ,and we see that the interesting part of the theorem is the locus on which s van-ishes, or, crucially for our application to smoothing theorems, the global situationin which s vanishes at some points but not others.The strategy of our proof parallels the proof of the corresponding statementfor determinantal varieties: we first consider the universal case and conclude thedesired statement by realizing the linked determinantal locus as the image of alinked Grassmannian, and then conclude the statement of the theorem by pullingback from the universal case.We next recall the definition of the linked Grassmannian. Definition A.5.
Let S be a scheme, E ‚ an s -linked chain on S , and r ą
0. Thenthe linked Grassmannian LG p r, E ‚ q is the closed subscheme of G p r, E q ˆ S ¨ ¨ ¨ ˆ S G p r, E n q consisting of tuples p F , . . . , F n q such that for i “ , . . . , n ´ f i p F i q Ď F i ` and f i p F i ` q Ď F i .The relationship between linked Grassmannians and linked determinantal loci isdescribed by the following proposition. Proposition A.6.
Let S be any scheme, and ¯ E ‚ an s -linked chain on S . Let S “ G p r, ¯ E q ˆ S G p r, ¯ E n q , and let E ‚ be the pullback of ¯ E ‚ to S , with F Ď E and F n Ď E n the pullbacks of the universal bundles on G p r, ¯ E q and G p r, ¯ E n q respectively.Then the linked determinantal locus associated to E ‚ and F , F n is precisely theimage of the linked Grassmannian LG p r, ¯ E ‚ q under the projection morphism G p r, E q ˆ S ¨ ¨ ¨ ˆ S G p r, E n q Ñ G p r, E q ˆ S G p r, E n q . Proof.
It is clear from the definitions that the image of LG p r, ¯ E ‚ q is contained in thelinked determinantal locus, so we need only prove the converse. Since the statementis set-theoretic, we may work on the level of k -valued points with k a field, and wesee that what we want to prove is the following: given d -dimensional k -vector spaces E , . . . , E n , maps f i and f i making an s -linked chain on Spec k , and r -dimensionalsubspaces F Ď E and F n Ď E n such that the kernel of (A.1) has dimension atleast r for i “ , . . . , n , then there exist choices of r -dimensional subspaces F i Ď E i for i “ , . . . , n ´ f i and f i .Now, let K i Ď E i be the kernel of (A.1) for i “ , . . . , n ´
1. Then by hypothesis,dim K i ě r for all i , and it is also clear that f i p K i q Ď K i ` and f i p K i ` q Ď K i for all i . We claim that as long as dim K i ą r for some i , we can replace some K i by aproper subspace while preserving the above conditions; iterating this process yieldsthe desired statement. Now, let i be minimal such that dim K i ą r ; we claim thatthe span of the images of K i ´ and K i ` in K i must be strictly smaller than K i .Indeed, by condition (III) of s -linkage, the image of K i ` in K i also injects into K i ´ , but maps into the kernel of f i ´ . Because dim K i ´ “ r , we conclude thatthe span of the images of K i ´ and K i ` in K i must have dimension at most r ,so we can replace K i by any r -dimensional subspace containing this span; this willpreserve the linkage condition, and thus proves the claim. (cid:3) We next need to set up the relevant universal spaces. We have the following:
Proposition A.7.
Given d ą , let ¯ U d be the scheme of pairs of d ˆ d matrixes A and B over Z r t s such AB “ BA “ tI d . Let U d be the open subscheme of ¯ U d on which rk A ` rk B ě d. Then U d is smooth over Spec Z r t s of relative dimension d .Proof. We first observe that the fibers are smooth of dimension d : over pointswith t ‰
0, this is clear, as U d is simply isomorphic to GL d ; on the other hand,where t “ U d is reduced of dimension d , and if wefix the ranks of A and B (necessarily adding to d ), we obtain an open subset of U d which is an orbit of the action of GL d ˆ GL d , and must therefore be smooth.Thus, it is enough to show that U d is flat over Spec Z r t s . For this, we appeal toLemma 4.3 of [HO08], which asserts that it is enough to check that for any basechange of U d to Spec R with R a discrete valuation ring, no component of the basechange is supported in the special fiber. This then amounts to the assertion that ifwe are given a discrete valuation ring R , and an element x of R , that the scheme ofpairs of d ˆ d matrices A, B over R with AB “ BA “ xI d and with rk A ` rk B ě d does not have components supported over the closed point of R . But if we are givensuch A, B over the residue field k of R , with rk A “ d and rk B “ d , there aretwo cases to consider: if x is a unit, then A and B are invertible, so we may chooseany lift of A to R , and set B “ xA ´ . On the other hand, if x maps to 0 in k ,then up to change of basis on both sides, we may assume A is diagonal with thefirst d diagonal entries equal to 1, and the remaining entries 0, and B is diagonalwith the first d ´ d “ d entries equal to 0, and the remaining entries equal to 1.We may then lift to R simply by replacing the diagonal 0s with x . This shows thatevery point in the closed fiber is in fact contained in a section, yielding the desiredstatement. (cid:3) Finally, we recall the relevant theorem on linked Grassmannians from [Oss06].
Theorem A.8.
Suppose that S is integral and Cohen-Macaulay, and E ‚ is an s -linked chain on S . Then every component of LG p r, E ‚ q has codimension p n ´ q r p d ´ r q inside G p r, E q ˆ S ¨ ¨ ¨ ˆ S G p r, E n q , and if s is nonzero, then LG p r, E ‚ q isirreducible. We are now ready to prove our main theorem.
IMIT LINEAR SERIES FOR CURVES NOT OF COMPACT TYPE 33
Proof of Theorem A.3.
Let T be the product of n ´ U d over Spec Z r t s ,and let S univ0 be the open subscheme of T on which ker A i ` X im A i “ p q andker B i X im B i ` “ p q for i “ , . . . , n ´
2. Then we have an s -linked chain E univ ‚ on S univ0 (with s “ t ) by taking n copies of the trivial bundle, and using the A i and B i to define our maps. Let S univ be obtained from S univ0 as in PropositionA.6. We claim that it is enough to prove the theorem for the corresponding linkeddeterminantal locus on S univ . Indeed, given any S and E ‚ , the theorem is local on S ,so we may assume that the E i are trivialized, and our s -linked chain and subbundles F and F n then induce a morphism to S univ under which they are obtained as thepullbacks of E univ ‚ and the universal subbundles. Moreover, under this morphismwe have that the linked determinantal locus on S is the preimage of the linkeddeterminantal locus on S univ . Now, by Proposition A.7 we have that S univ0 andhence S univ is smooth over Spec Z r t s , and hence regular, and it then follows byTheorem 7.1 of [Hoc75] that if every component of the linked determinantal locusin S univ has codimension at most r p d ´ r q , then the same is true in S .But according to Proposition A.6, the linked determinantal locus in S univ isthe image of the linked Grassmannian LG p r, E univ ‚ q over S univ0 . By Theorem A.8,we know that LG p r, E univ ‚ q is irreducible of codimension p n ´ q r p d ´ r q , and it isclear that it maps generically finitely onto its image in S univ , since for t ‰ F uniquely determines all the other subbundles. We thus conclude byProposition 5.6.5 of [GD65] that the image – that is, the linked determinantal locus– has codimension r p d ´ r q , as desired. (cid:3) References [AB] Omid Amini and Matthew Baker,
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