aa r X i v : . [ m a t h . AG ] J a n GIT STABILITY OF WEIGHTED POINTED CURVES
DAVID SWINARSKI
In the late 1970s Mumford established Chow stability of smooth unpointed genus g curves em-bedded by complete linear systems of degree d ≥ g + 1, and at about the same time Giesekerestablished asymptotic Hilbert stability (that is, stability of m th Hilbert points for some large val-ues of m ) under the same hypotheses. Both of them then use an indirect argument to show thatnodal Deligne-Mumford stable curves are GIT stable. The case of marked points lay untoucheduntil 2006, when Elizabeth Baldwin proved that pointed Deligne-Mumford stable curves are asymp-totically Hilbert stable. (Actually, she proved this for stable maps, which includes stable curvesas a special case.) Her argument is a delicate induction on g and the number of marked points n ;elliptic tails are glued to the marked points one by one, ultimately relating stability of an n -pointedgenus g curve to Gieseker’s result for genus g + n unpointed curves.There are three ways one might wish to improve upon Baldwin’s results. First, one might wishto construct moduli spaces of weighted pointed curves or maps; it appears that Baldwin’s proof canaccommodate some, but not all, sets of weights. Second, one might wish to study Hilbert stabilityfor small values of m ; since Baldwin’s proof uses Gieseker’s proof as the base case, it is not easyto see how it could be modified to yield an approach for small m . Finally, the Minimal ModelProgram for moduli spaces of curves has generated interest in GIT for 2, 3, or 4-canonical linearsystems; due to its use of elliptic tails, Baldwin’s proof cannot be used to study these, as elliptictails are known to be GIT unstable in these cases.In this paper I give a direct proof that smooth curves with distinct weighted marked points areasymptotically Hilbert stable with respect to a wide range of parameter spaces and linearizations.Some of these yield the (coarse) moduli space of Deligne-Mumford stable pointed curves M g,n andHassett’s moduli spaces of weighted pointed curves M g, A , while other linearizations may give otherquotients which are birational to these and which may admit interpretations as moduli spaces. Thefull construction of the moduli spaces is not contained in this paper, only the proof that smoothcurves with distinct weighted marked points are stable, which is the key new result needed for theconstruction. For this I follow Gieseker’s approach to reduce to the GIT problem to a combinatorialproblem, though the solution is very different. Introduction
Let (
C, P , . . . , P n , A ) be a weighted pointed stable curve. That is, • C is a reduced connected projective algebraic curve with at worst nodes as singularities, • the points P i lie on C and are ordered (note we do not require that they be distinct, northat they be smooth points of C ), • A = ( a , . . . , a n ), where the a i are rational numbers between 0 and 1 inclusive, • a i = 0 if P i is a node, • a subset of the points is allowed to collide if the sum of their weights does not exceed 1,and • the Q -line bundle ω ( P a i P i ) is ample on C .Hassett introduced weighted pointed stable curves in [Hass]; the theory is extended to stable mapsby several people ([BM], [AG], [MM]).The goal of this paper is to describe linearizations for which the points of an appropriate spaceparametrizing embedded weighted pointed stable curves ( C ⊂ P N , P , . . . , P n , A ) are GIT stable.The main result of this paper, Theorem 7.1, does not say exactly this. Instead, for most of thispaper, we do the following: • We ignore the set of weights A and just study embedded pointed curves ( C ⊂ P N , P , . . . , P n ). • We assume that the curve C is smooth. • We assume that the points { P i } are distinct.Theorem 7.1 asserts that smooth pointed curves with distinct marked points are GIT stable withrespect to certain linearizations. Armed with this result, one may proceed to show that all weightedpointed stable curves are GIT stable for certain linearizations, justifying the title of this paper. Thisis not fully written out here, but it is discussed in Section 7.2.So, let x be a point parametrizing an embedded smooth pointed curve ( C ⊂ P N , P , . . . , P n ).Following Gieseker, the numerical criterion is reformulated in a way that permits a more combina-torial approach. A 1-PS λ of SL ( N +1) induces a weighted filtration of H ( C, O (1)) and a weightedfiltration of H ( C, O ( m )). The value of Mumford’s function µ L ( x, λ ) may be interpreted as the“minimum weight of a basis of H ( C, O ( m )) compatible with this filtration plus a contributionfrom the marked points.” (From now on, whenever we refer to a basis of H ( C, O ( m )), we alwaysimplicitly mean one that is compatible with the weighted filtration.) The numerical criterion saysthat if µ L ( x, λ ) is sufficiently small, then x is GIT stable with respect to λ . Any basis thereforegives an upper bound for µ L ( x, λ ), so the goal becomes: find a basis of sufficiently small weight.Our main tool for computing (a bound for) the weight of a basis is something I call a profile .This is a graph which may be associated to any filtration of a vector space such that the weightdecreases at each stage. Suppose ˜ F • is such a filtration of H ( C, O ( m )). (I use tildes for filtrations of H ( C, O ( m )); no tilde indicates a filtration of H ( C, O (1)).) Suppose the weight on the k th stage of˜ F • is ˜ r k . Then the profile associated to ˜ F • is just the decreasing step function in the first quadrant ofthe (codimension × weight)-plane whose value is ˜ r k over the interval [codim ˜ F k , codim ˜ F k +1 ) . Givenany profile, it is possible to choose a basis whose weight is less than the area under the profile.There is a notion of an absolute weight filtration on H ( C, O ( m )) (see Section 1.3); the areaunder its profile is the minimum weight of a basis. This is perhaps the most natural filtration toconsider, but it is too difficult to compute. So, like Gieseker, we study other filtrations.The action of a 1-PS λ induces a filtration V • of H ( C, O (1)). By considering specific spacesof degree m monomials in elements of V diagonalizing the λ -action, Gieseker produces a verystraightforward filtration ˜ V • of H ( C, O ( m )) as well as a second, slightly fancier filtration ˜ G • .Gieseker is able to show that the weight (or area) associated to ˜ G • is sufficiently small to establish λ -stability of smooth unpointed curves. Unfortunately, as we show with a concrete example, theanalogue of ˜ G • is not sufficient to establish λ -stability when there are marked points.One could try to improve ˜ G • , but it is too difficult (at least for me) to show that the sum ofits area and the marked points contribution is sufficiently small. Therefore I use ˜ V • as a starting IT STABILITY OF WEIGHTED POINTED CURVES 3 point to build a new filtration, ˜ X • , which is obtained by taking spans of carefully chosen spaces ofmonomials. The recipe is given in terms of the combinatorics of the base loci of the stages of thefiltration V • . Although ˜ X • is rather tedious to define, it has the virtue that we can bound the sumof its area and the marked points contribution sufficiently well to show that smooth curves withdistinct marked points are stable. The key new ingredients in my proof are the definition/choiceof ˜ X • ; an easy but important lemma (Lemma 3.2) which allows us to compute spans of spaces ofmonomials in the V j ’s using multiplicities of points in the base loci; and the combinatorial argument(see the proof of Lemma 6.1) which allows us to effectively bound the sum of the marked pointscontribution and the area of the profile associated to ˜ X • .Gieseker’s proof establishes stability for smooth unpointed curves embedded by complete linearsystems of degree d ≥ g + 1. (There are some misleadingly placed hypotheses in [Gies], but onecan check that everything works with the hypotheses just mentioned.) At the present time it isnecessary for me to make the hypotheses: • If n = 0, the parameter space satisfies N ≥ g − • If n ≥
1, then either the parameter space satisfies N ≥ g −
1, or else the linearizationsatisfies the following condition (the notation is explained in Section 1.1): γb > g − N .One might hope to do a little better (see Section 8.3), but at least this includes the important caseof bicanonically embedded pointed curves (i.e. pointed curves embedded by sections of( ω ( P + · · · P n )) ).Here is an outline of the paper: in Section 1 I describe the GIT problem carefully, specifyingthe parameter spaces and linearizations we will consider, and reformulate the numerical criterionin the form we shall use it. Profiles are also defined here. In Section 2 I review Gieseker’s proof,with a few enhancements, to fix notation; a reader familiar with Gieseker’s proof should be ableto read it very quickly. In Section 3 I give an example showing why his proof does not suffice formarked points, and a hint illustrating how we will go about fixing it.Throughout Sections 1–3 we steadily extract combinatorial data from the algebro-geometricaction of a 1-PS λ acting on the Hilbert point of a weighted pointed stable curve. The last resultof this type is Lemma 3.2, which allows us to compute codimensions of spans of monomial-typesublinear series of H ( C, O ( m )) using only the multiplicities of points in the base loci. After this,the problem becomes almost entirely combinatorial.In Section 4, I produce the filtration ˜ X • on H ( C, O ( m )) which is built using the filtration ˜ V • asscaffolding. The goal is now to show that the area under the profile for ˜ X • plus the contributionfrom the marked points is less than the bound specified by the numerical criterion.This is established in two steps: first, I describe a second, simpler graph called the virtual profile which is bounded above by the profile for ˜ X • . Basically it is the graph of the piecewise linearfunction connecting the left endpoints of the steps in the weight profile. (I’m oversimplifying thingsa little here—I’m glossing over some rounding errors.) The virtual profile is not really the profile ofany filtration, nor does it compute or bound the weight of a basis; the most rigorous interpretationI have for it is on the level of graphs. Again, while it is easy to compute the area of the profile (it’sa step function, after all!), when it is time to add the contribution from the marked points, it iseasier to do this with the virtual profile than with the profile. In Section 5 I bound the discrepancybetween the areas of the two graphs and show that this is relatively small when m is large. Thenin Section 6 I bound the sum of the area under the virtual profile for ˜ X • and the weight from the DAVID SWINARSKI marked points. Everything comes together in Section 7 to show that smooth pointed curves withdistinct marked points have GIT stable Hilbert points, and the application of this to constructionof moduli spaces is stated but not proven. Finally, this preprint concludes with a short section ofadditional remarks which are likely to be omitted from a published version.Here is a picture illustrating the profile and virtual profile associated to an example that isexplained in detail in Section 4.4. Note that I will always fill in the graphs of step functions toobtain staircase figures. CodimensionWeightIn summary: action of one 1-PS λ on a smooth pointed curve ⇓ a filtration V • of H ( C, O (1)) and a filtration ˜ V • of H ( C, O ( m )) ⇓ another filtration ˜ X • of H ( C, O ( m ))and two graphs associated to ˜ X • (a profile and a virtual profile ) ⇓ a basis of H ( C, O ( m )) of small weight ⇓ stability of the smooth pointed curve with respect to λ Two remarks on notation here may reduce anxiety for those skimming the proof:Note that from Section 4 onward it may appear at times as though we are using rational numbersas exponents of monomials. Although the resulting “virtual” spaces are usually nonsensical, in caseswhere they do make sense they are useful in motivating some definitions and calculations. However,such spaces are never used to produce basis elements in H ( C, O ( m )); to get basis elements, wealways round exponents appropriately.We will obtain two-dimensional arrays of integers c j,i . That is, j indexes the row, and i indexesthe column, opposite the usual alphabetic convention. There is nothing deep happening here; thereasons I made this choice are too silly to discuss further. Acknowledgements.
It is a great pleasure to thank my advisors, Ian Morrison and MichaelThaddeus, for their help with this work. I am also very grateful to Elizabeth Baldwin for sharingmuch of her early work with me, which got me interested in the problem and helped me getstarted. Finally, I would like to thank Johan de Jong and Brendan Hassett for their technical helpand encouragement. 1.
The GIT setup
The parameter spaces and linearizations we use.
In this chapter we investigate GITstability for the following general setup. Let P ( t ) := dt − g + 1 be a degree one polynomial. We form IT STABILITY OF WEIGHTED POINTED CURVES 5 the incidence locus I ⊂ Hilb ( P N , P ( t )) × Q n P N where the points in the projective space factorslie on the curve in P N parametrized by the point in the first factor. We study the GIT stabilityof points of I . Note two things: no sets of weights A appear in this paragraph; we will see inSection 7.2 that considering weighted marked points influences the choice of d , but otherwise playsno role in the stability proof. Also, we do not assume that C ⊂ P N is pluricanonically embedded,or even that the degree of C ⊂ P N matches the degree of the pluricanonical embedding— we caninvestigate GIT stability for more general setups than just those which have an obvious applicationto construction of moduli spaces of curves. All we need is that the embedding C ⊂ P N is by acomplete linear system, and some precise degree/dimension bounds in terms of the genus, whichwill be carefully stated at the end in Theorem 7.1. These will even allow some special embeddings.To do GIT, one must specify a linearization on the G -space (here, I ). Although not necessary,perhaps the easiest way to do this is to embed Hilb ( P N , P ( t )) × Q n P N in a high-dimensionalprojective space and use its O (1).Let C ⊂ P N be a subscheme with Hilbert polynomial P ( t ). For sufficiently large m, m ′ i , themaps ev mC : H ( P N , O ( m )) → H ( C, O C ( m ))ev m ′ i P i : H ( P N , O ( m ′ i )) → H ( P i , O P i ( m ′ i )) ∼ = C are surjective. The first map gives rise to an embedding of the Hilbert scheme in a Grassmannian,which in turn embeds in a projective space by the Pl¨ucker embedding. The maps in the second linecorrespond to m ′ i -uple embeddings of P N . Finally, a Segre embedding of all these projective spacesyields an embedding of Hilb ( P N , P ( t )) × Q n P N into a very large projective space, as desired.Now, to specify a linearization on I ⊂ Hilb ( P N , P ( t )) × Q n P N , it suffices to specify the ratiosbetween m and each m ′ i . I will do this as follows: let B = ( b , . . . , b n ) ∈ Q n ∩ [0 , n be a set ofweights, which I call the linearizing weights . Then set m ′ i = γb i m . (The coefficient γ will bespecified later, at least for the moduli spaces M g, A , where it is approximately 1/2; see Section 7.2.Factoring γ out of the ratios m ′ i /m like this now simplifies the statements of later results neededto construct the moduli spaces.) Finally, write b := P ni =1 b i .1.2. The numerical criterion for our setup.
By being a little more explicit, we obtain a usefulreformulation of the numerical criterion.In Gieseker’s paper and this paper we use Grothendieck’s convention that if V is a vector space,then P ( V ) is the collection of equivalence classes under scalar action of the nonzero elements ofthe dual space V ∨ . One consequence of this convention is that the numerical criterion takes theopposite sign from how it appears in [GIT].Let X be a projective algebraic scheme with the action of a group G linearized on a very ampleline bundle L . Let λ : G m → G be a 1-PS of G . Choose a basis { e , . . . , e N } of H ( X, L )diagonalizing the λ action and ordered so that the weights r ≤ · · · ≤ r N ∈ Z increase. The weightson the dual basis then have the opposite signs: − r , . . . , − r N .A point x ∈ X is represented by some non-zero ˆ x = P Ni =0 x i e ∨ i ∈ H ( X, L ) ∨ . Define µ L ( x, λ ) := min { r i | x i = 0 } . Then, with our sign conventions, we have the following characterization of semistability:
DAVID SWINARSKI
Theorem 1.1 (cf. [GIT] Theorem 2.1) . x ∈ X ss ( L ) ⇐⇒ µ L ( x, λ ) ≤ for all 1-PS λ = 0 x ∈ X s ( L ) ⇐⇒ µ L ( x, λ ) < for all 1-PS λ = 0 . In our situation X is the incidence scheme I , the point x ∈ X parametrizes an embedded pointedcurve ( C ⊂ P N , P , . . . , P n ), the scheme I is embedded in P ( V P ( m ) Sym m V ⊗ N n Sym m ′ i V ) where V = H ( P N , O (1)), and L is the O (1) on this very large projective space. Let λ be a 1-PS of SL ( V ).One particularly nice basis of V P ( m ) Sym m V ⊗ N n Sym m ′ i V is given by elements of the form(1) ( M ∧ · · · ∧ M P ( m ) ) ⊗ ( M ′ ) ⊗ · · · ⊗ ( M ′ n ) , where each M j is a monomial of degree m and each M ′ i is a monomial of degree m ′ i in the basiselements of V diagonalizing λ .The numerical criterion may be translated as follows: a point of I is stable with respect to λ ifand only if there is a basis element of the form (1) such that(1) the images of the M ℓ under the evalution map form a basis of H ( C, O C ( m )),(2) M ′ i does not vanish at P i ,(3) the SL ( N + 1) weights satisfy P ( m ) X ℓ =1 wt λ ( M ℓ ) + n X wt λ ( M ′ i ) < λ weights so that they decrease to 0 and sumto 1. If s N , . . . , s are the original weights, (so s N ≥ · · · ≥ s and P s j = 0), then the desiredtransformation is r j = ( s N − j − s ) / (( N + 1) | s | ). Also, we write A := P ( m ) X ℓ =1 wt λ ( M ℓ ) T := P ( m ) X ℓ =1 wt λ ( M ℓ ) + n X wt λ ( M ′ i )for parts of the left hand side of condition 3 . above. We may rewrite condition 3 . as follows. Lemma 1.2.
Condition . above with the unnormalized weights s j is equivalent to the followingcondition: . ′ With the normalized weights r j , the following inequality is satisfied: T := P ( m ) X ℓ =1 wt λ ( M ℓ ) + n X wt λ ( M ′ i ) < (cid:18) g − N + 1 (cid:19) m + 1 N + 1 n X m ′ i − g − N + 1 m = (cid:18) g − γbN + 1 (cid:19) m − g − N + 1 m. (2) Proof.
Suppose that we have the required collection of monomials satisfying P ( m ) X ℓ =1 wt λ ( M ℓ ) + n X wt λ ( M ′ i ) < s j . Let w , . . . , w N be a basis of H ( C, O (1)) diagonalizing the λ action. If M ℓ = w f ℓ, · · · w f ℓ,N N , then wt λ ( M ℓ ) = P Nj =0 f ℓ,j s j . IT STABILITY OF WEIGHTED POINTED CURVES 7
Let j ( i ) be the function whose value for each i = 1 , . . . , n is the largest index (hence giving thesmallest weight) such that the section w j ( i ) does not vanish at P i . Then wt λ ( M ′ i ) = m ′ i s j ( i ) . Thuscondition 3 . may be rewritten P ( m ) X ℓ =1 N X j =0 f ℓ,j s j + n X i =1 m ′ i s j ( i ) < ⇔ P ( m ) X ℓ =1 N X j =0 f ℓ,N − j (( N + 1) | s | r j + s ) + n X i =1 m ′ i (( N + 1) | s | r N − j ( i ) + s ) < . We proceed to divide by | s | . Note that our conventions imply that s < P ( m ) X ℓ =1 N X j =0 f ℓ,N − j (( N + 1) r j −
1) + n X i =1 m ′ i (( N + 1) r N − j ( i ) − < ⇔ ( N + 1) P ( m ) X ℓ =1 N X j =0 f ℓ,N − j r j − P ( m ) X ℓ =1 N X j =0 f ℓ,N − j + ( N + 1) n X i =1 m ′ i r N − j ( i ) − n X i =1 m ′ i < ⇔ P ( m ) X ℓ =1 N X j =0 f ℓ,N − j r j + ( N + 1) n X i =1 m ′ i r N − j ( i ) < N + 1 ( P ( m ) X ℓ =1 N X j =0 f ℓ,N − j + n X i =1 m ′ i )But we have P Nj =0 f ℓ,N − j = m since each M ℓ is a monomial of degree m . Hence we obtain P ( m ) X ℓ =1 N X j =0 f ℓ,N − j r j + ( N + 1) n X i =1 m ′ i r N − j ( i ) < dm − g + 1 + P ni =1 m ′ i N + 1Finally, we apply the relation m i = γb i m associated to the linearization and use b = P b i : P ( m ) X ℓ =1 N X j =0 f ℓ,N − j r j + ( N + 1) n X i =1 m ′ i r N − j ( i ) < dm − g + 1 + γbm N + 1(3)Now, if we let v j = w N − j , then the term P Nj =0 f ℓ,N − j r j is the weight of the monomial v f ℓ, · · · v f ℓ,N N .Also, v N − j ( i ) is the smallest weight section among the v j ’s which does not vanish at P i . Thus wemay interpret the left hand side of (3) as: the r -weight of a collection of monomials restricting tothe basis of H ( C, O ( m )) plus the r -weight of a collection of degree m ′ i monomials which do notvanish at P i .This argument can be run in reverse, so given a collection of monomials satisfying 3 . ′ we canproduce a collection of monomials satisfying 3. (cid:3) Note that property 1 . above requires a set of monomials in H ( P ( V ) , O ( m )) which map to abasis of H ( C, O ( m )) of small weight. We want to turn things around, and instead start on thecurve in H ( C, O ( m )) and work our way back to H ( P ( V ) , O ( m )). The action of a 1-PS λ of SL ( V ) on the Hilbert point of a curve induces a weights on elements of H ( C, O C ( m )) (cf. [HM] p.208). Briefly, take a basis of H ( P N , O (1)) diagonalizing the λ action. There is an obvious way todefine the weight of any degree m monomial, the weight of any degree m homogeneous polynomialis defined to be the maximum weight of its constituent monomials, and the weight of an elementof H ( C, O C ( m )) is the minimum of the weights of its preimages in H ( P N , O ( m )). DAVID SWINARSKI
The next proposition says that to establish GIT stability, it is enough to show that there exists any basis of H ( C, O ( m )) of small weight. Lemma 1.3.
If there exist a basis of H ( C, O ( m )) of λ -weight W , and monomials M ′ , . . . , M ′ n satisfying condition (2) above, and together these satisfy W + X wt λ M ′ i ≤ (cid:18) g − γbN + 1 (cid:19) m − g − N + 1 m, then there are monomials M , . . . , M P ( m ) which together with M ′ , . . . , M ′ n satisfy conditions 1, 2,and 3’ of the numerical criterion.Proof. Let q , . . . , q P ( m ) be a basis of H ( C, O ( m ))satisfying W + X wt λ M ′ i ≤ (cid:18) g − γbN + 1 (cid:19) m − g − N + 1 m. We may assume that the q ’s are in order of decreasing weight. Let p , . . . , p P ( m ) be a set ofpreimages of the q ’s of minimal weight (that is, wt p i = wt q i for each i ). Let { M i,j } be themonomials constituting p i , so that p i = P j i j =1 α i,j M i,j .Write the list of monomials { M i,j } in order of decreasing weight. If there are ties, choose anyorder on the tied entries. Write y = { M i,j } . Form the ( P ( m ) × y )-matrix whose entry in row i andthe column labelled by M i,j is the coefficient of M i,j in p i . Each row has a leading monomial (themonomial corresponding to the leftmost column with a nonzero entry in that row). Row reduce thismatrix to upper triangular form; this can only lower the leading weight in each row. Now choosethe leading monomials in each row. Either these map to a basis of H ( C, O ( m )) having weight lessthan or equal to the weight of the basis given by q , . . . , q P ( m ) , or else there is a relation betweenthese terms after restriction to the curve. If this happens, delete the column corresponding tothe leftmost monomial appearing in the relation, and begin again (row reduce to upper triangularform, check whether the leading terms in each row give a basis...). Eventually we must arrive ata set of monomials which give a basis for H ( C, O ( m )) (since { ρ ( p i ) } is a basis of H ( C, O ( m )))and the weight of this set of monomials is less than or equal to the weight of the basis given by q , . . . , q P ( m ) . (cid:3) Generalities on profiles.
As mentioned in the introduction, the main tool for computingthe weight of a basis is something I call a profile . (Gieseker uses profiles in his proof, but he doesn’tuse the word “profile.”) We define this abstractly now.Let V be a vector space such that every element of V has a weight associated to it. Let F • be adecreasing weighted filtration on W . That is, V = F ⊃ F ⊃ · · · ⊃ F N = 0, and there is a (finite)decreasing sequence of weights r > r > · · · > r N = 0 such that all the elements of F h have weightless than or equal to r h . Definition 1.4.
The profile of a decreasing weighted filtration F • as described above is the graph ofthe decreasing step function in the ( codimension × weight ) -plane whose value is r h over the interval [codim F h , codim F h +1 ) . This is like a distribution function bounding how many linearly independent elements have atmost a given weight. Indeed, given a profile, it is possible to choose a basis whose weight is nogreater than the area under the profile. We will sometimes speak of the “weight of a filtration” or
IT STABILITY OF WEIGHTED POINTED CURVES 9 “weight of a profile”; of course what we mean by this is the area underneath the profile, which isa bound for the weight of a basis adapted to this filtration.Now, there is a notion of an absolute weight filtration . It may be described as follows: For eachpossible weight r h , form Ω( r h ) := Span { v : v ∈ V, wt( v ) ≤ r h } . Then the profile associated to Ω • can be used to choose a basis of minimum weight, as it tellsexactly how many elements of high weight must be added to the basis before elements of lowerweight may be added.In this paper, we will encounter filtrations of H ( C, O (1)) and H ( C, O ( m )). To help keep trackof the ambient vector space of the filtration, we will use tildes for filtrations of H ( C, O ( m )). Thefiltration of greatest importance for us, ˜ X • (to be defined in Section 4), is of this type.2. A review of Gieseker’s proof
Let us quickly review Gieseker’s proof from [Gies], viewing it as the n = 0 case of the abovesetup. We have recast the numerical criterion to say: the m -th Hilbert point of a smooth curve isGIT stable if and only if there exists a basis of H ( C, O C ( m )) such that the sum of its weights isless than (1 + ǫ ) m .As discussed before Lemma 1.3, the action of a 1-PS λ of SL ( N + 1) on the Hilbert point of acurve induces a weights on elements of H ( C, O C ( m )) (cf. [HM] p. 208). Now, it is probably mostnatural to consider the absolute weight filtration on H ( C, O ( m )). If one could compute its profile,then one could compute Mumford’s function µ L ( x, λ ) on the nose. However, this is too difficult tocompute, so Gieseker considers another filtration instead.Here is a brief and slightly simplified description of the weighted filtration ˜ G • Gieseker uses andits profile. Given: a curve and a 1-PS λ . As before, renormalize the λ -weights so that they aredecreasing and sum to 1. Let { w i } be a basis of H ( C, O C (1)) ∼ = H ( P N , O (1)) diagonalizingthe λ action (and compatible with the order of the r i ). Let V i := span( { w j | j ≥ i } ) ⊆ V . Thenormalization ensures that all the points ( im, r i m ) lie in the first quadrant. Form the lower envelopeof these points, and let 0 = i , i , . . . , index the subsequence of points lying on the lower envelope.Then in H ( P ( V ) , O P ( V ) ( m )) ∼ = Sym m V we have the following filtration:(4) Sym m V = V mi V i ⊃ V m − i V i ⊃ · · · ⊃ V m − pi V pi ⊃ · · · ⊃ V i V mi V mi V i ⊃ V m − i V i ⊃ · · · ⊃ V m − pi V pi ⊃ · · · ⊃ V i V mi etc.The image of this filtration under restriction to the curve gives a filtration ˜ G • of H ( C, O C ( m )).We can compute the dimension of each stage of the filtration in H ( C, O C ( m )), and we know theweight of each stage, so this is the data of a profile. The profile is the graph of a step function; itsleft endpoints lie on the lower envelope of the set of points { ( im, r i m ) } . Here is a picture:CodimensionWeight (Looking ahead, the lower envelope here is the inspiration for what I will later call the virtualprofile .)Any basis adapted to this filtration will establish stability, as the area A under the profile is veryclose to the area under the lower envelope, and the area under the lower envelope is less than 1 m ,by a combinatorial lemma due to Morrison ([Morr], Section 4).2.1. The weighted filtration on H ( C, O (1)) . For speed, the previous subsection oversimplifiedsome details of Gieseker’s proof. We will now take the opportunity to begin building up thedefinitions and notation we need; I have grouped these in this section with his proof, because mostof the ideas here are extracted from his proof or follow easily from it.As we have observed already, the action of the 1-PS λ induces most fundamentally a weightedfiltration on H ( C, O (1)), but to establish stability we need to find a basis of H ( C, O ( m )) of smallweight. We will be going back and forth between these two vector spaces for the rest of the proof.We begin with H ( C, O (1)), and see what our knowledge of this filtration tells us about filtrationson H ( C, O ( m )). Once we find formulas for the area under the profile for a certain filtration on H ( C, O ( m )), we will ultimately bound the weight of the basis by relating quantities back to theircounterparts in H ( C, O (1)).Let V • be the weighted filtration on H ( C, O (1)) induced by the action of the 1-PS λ . That is,the stages of the filtration are distinguished by decreasing weight. Let z j be the size of the j th stage of the filtration, so z j = codim V j +1 − codim V j , and let r j be the weight. Assume that theweights r j have been normalized so that they are decreasing to zero and sum to 1 (that is, r N = 0and P z j r j = 1). Let D j be the base locus of the sublinear series V j , and let d j = deg D j . Let Q , ..., Q q be the points in Supp D N . (There will be a natural way to order them, but the order isimmaterial.) The marked points P i may or may not show up among the Q i ; set(5) B i = (cid:26) b k , Q i = P k for some k , Q i = P k for any k .(Note I am already assuming that the marked points are distinct, so Q i can only equal P k for atmost one k .) Let c j,i be the multiplicity of Q i in D j . (Note that the indices are not in alphabeticorder, opposite the usual convention. The reasons I have made this choice are too silly to discuss.)In general V j is contained in but not equal to H ( C, O (1)( − D j )). My experience with this problemleads me to conjecture that the maximum of Mumford’s µ L ( x, λ ) function occurs for 1-PS whereequality holds at every stage.2.2. Relating codegrees and codimensions in H ( C, O (1)) . We have one obvious bound onthe weights: P z j r j = 1. We will need to relate codegrees d j = P ni =1 c j,i and codimensions P j − τ =0 z τ .Near the top of the weighted filtrations, the base loci have low degree, so O (1)( − D j ) has highdegree, and the dimension/codimension of H ( C, O (1)( − D j )) may be computed using Riemann-Roch. More precisely: if deg D j > d − g + 1, then codim V j > N − g . So if codim V j ≤ N − g ,then deg D j ≤ d − g + 1, so deg O (1)( − D j ) > g −
2, so h ( O (1)( − D j )) = 0. Since V j ⊆ H ( C, O (1)( − D j )), we get a bound: the codegree of O (1)( − D j ) cannot exceed the codimension of V j . Recall from the definition of the z j ’s that codim V j = P j − τ =0 z τ . Writing D j = P qi =1 c j,i Q i , wehave: deg D j = P qi =1 c j,i . We thus obtain:(6) if P j − τ =0 z τ ≤ N − g , then P qi =1 c j,i ≤ P j − τ =0 z τ . IT STABILITY OF WEIGHTED POINTED CURVES 11
I call this the
Riemann-Roch region of the filtration. Write j RR for the largest index j which satisfies P j − τ =0 z τ ≤ N − g .On the other hand, if O (1) itself is special, or for stages of the filtration of high codimension (thatis, near the bottom), the line bundles O ( − D j ) have low degree, and we might have h ( O ( − D j )) = 0.Here we can use Clifford’s Theorem to get the following bound:(7) if P j − τ =0 z τ > N − g , then P qi =1 c j,i ≤ P j − τ =0 z τ + (cid:16)P j − τ =0 z τ − ( N − g ) (cid:17) − h ( C, O (1)).I call this the Clifford region of the filtration and write j Cliff for the smallest index j which satisfies P j − τ =0 z τ > N − g . (So of course j Cliff = j RR + 1.)Note that in the case of principal interest (when d = ν (2 g − a ) and ν is large, so that N isalso large), the Riemann-Roch region accounts for the lion’s share of the filtration.2.3. Passing to H ( C, O ( m )) . We want to use the base loci D j to control how multiples of the V j intersect, and this would work best if V j = H ( C, O (1)( − D j )). Gieseker observed that if we passfrom H ( C, O (1)) to H ( C, O ( m )) (which is where we ultimately need to produce a basis anyway),then we will be able to treat an arbitrary 1-PS λ as if it were of this form. Most of the proof ofLemma 2.1 below comes from pages 54–55 of [G2]. However, I want to add a few comments toGieseker’s proof, so I will run through the argument here.Let ( V u − ws V wt V ) v denote the subspace of H ( C, O (( u + 1) v )) generated by expressions of theform x · · · x v ( u − w ) y · · · y vw z · · · z v where the x ’s come from V s , the y ’s come from V t , and the z ’scome from V . Lemma 2.1.
Let u, v, w be nonnegative integers with ≤ w ≤ u and v ≥ . Suppose C is anarbitrary subscheme of P N with Hilbert polynomial dt − g + 1 and v ≥ d ( u + 1) − d ( u + 1)2 − g + 1 . Then ( V u − ws V wt V ) v = H ( C, O (( u + 1) v )( − ( u − w ) D s − wD t )) Remark.
Note that the bound on v depends on u and the Hilbert polynomial P ( z ) = dz − g + 1,but not on the curve C or the line bundle O C (1) embedding C into P N . Proof.
Let L s and L t be the line bundles generated by the sections in V s and V t . Here is the firstcomment to add to Gieseker’s proof: then L s = O C (1)( − D s ). We have( V u − ws V wt V ) v ⊂ H ( C, ( L u − ws L wt L ) v ) = H ( C, O (( u + 1) v )( − ( u − w ) D s − wD t )) . Now, since sections in V u − ws V wt generate L u − ws L wt , and V is very ample, we have that V u − ws V wt V is very ample, and hence determines an embedding C ֒ → P M . We have a short exact sequence0 → I ( v ) → O P M ( v ) → O C ( v ) → . (We now have two O C (1)’s in this proof, corresponding to the embeddings in P N and P M , but itis not difficult to tell them apart.) Write d s = deg D s , respectively for t ; then deg L s = d − d s anddeg L t = d − d t . Then the Hilbert polynomial for C ⊂ P M is P ( z ) = (( d − d s )( u − w ) + ( d − d t )( w ) + d ) z − g + 1 . The Gotzmann number for this Hilbert polynomial is m = (( d − d s )( u − w ) + ( d − d t )( w ) + d ) − (( d − d s )( u − w ) + ( d − d t )( w ) + d )2 − g + 1; recall that the Gotzmann number for a Hilbert polynomial has the property that it is the maximumregularity for any sheaf with that Hilbert polynomial ([Gotz] Lemma 2.9). Hence, H ( I ( v )) = 0since v is larger than the Gotzmann number. But then H ( P M , O ( v )) → H ( C, ( L u − ws L wt L ) v is surjective.Comparing this to the definition of ( V u − ws V wt V ) v , this says that( V u − ws V wt V ) v = H ( C, ( L u − ws L wt L ) v ) = H ( C, O (( u + 1) v )( − ( u − w ) D s − wD t ))as desired.Finally note that d − d s and d − d t are no larger than d ; hence taking v ≥ d ( u + 1) − d ( u + 1)2 − g + 1 . ensures that v is greater than or equal to the Gotzmann number for any V s and V t . (cid:3) Remark.
We will be applying this result when C is a smooth curve in P N ; for this application,the Gotzmann number is really much larger than we should need. I hope to improve this resultsignificantly, which should be helpful (if not necessary) when studying stability for small values of m .Let m = ( u + 1) v . Then there is a filtration ˜ V • of H ( C, O ( m )) by the subspaces ( V uj V ) v .Note however that if in the original filtration, there are two successive stages where the baselocus does not increase, now, after passing to H ( C, O ( m )), the second of these stages has risen upto replace the first of these two stages. Thus, in H ( C, O ( m )), we need only record the subsequenceof the j ’s where the degree of the base locus increases. I will index these by the letter k .The filtration ˜ V • may be further refined by using spaces of the form ( V u − wk V k +1 ) w V ) v . We willabuse notation and write ˜ V • for this refinement also. Thus, the index of the filtration ˜ V • may bethe single index k , or a pair ( k, w ).I will use tildes for quantities associated to ˜ V • . We have ˜ V k = H ( C, O ( m )( − ˜ D k )), where˜ D k = uvD j k . We write ˜ d k := uvd j k and ˜ c k,i := uvc j k ,i . Then˜ V k = H ( C, O ( m )( − ˜ c k, Q − · · · − ˜ c k,q Q q ))and elements of this space have weight ≤ ˜ r k := uvr j k + vr .Define ˜ N to be the smallest index giving the vr -weight space. We have:(8) Space Weight˜ V = H ( C, O ( m )) ˜ r ˜ V = H ( C, O ( m )( − ˜ c , Q − · · · − ˜ c ,q Q q )) ˜ r ˜ V = H ( C, O ( m )( − ˜ c , Q − · · · − ˜ c ,q Q q )) ˜ r ... ...˜ V ˜ N = H ( C, O ( m )( − ˜ c ˜ N , Q − · · · − ˜ c ˜ N,q Q q )) ˜ r ˜ N = vr We may extract the multiplicities of the points in the base loci in the weighted filtration ˜ V • andthe weights to obtain an ( ˜ N + 1) × ( q + 1) array: IT STABILITY OF WEIGHTED POINTED CURVES 13 (9) ˜ c , · · · ˜ c ,q ˜ r ˜ c , · · · ˜ c ,q ˜ r ... ... ... ...˜ c ˜ N, · · · ˜ c ˜ N,q ˜ r ˜ N = vr This array has the following properties: the ˜ c k,i ’s are all nonnegative integers; the ˜ r i ’s are rationalnumbers weakly decreasing to vr ; and in the first row the ˜ c ,i ’s are all zero. Furthermore we seethat the sum of the entries in row k is governed by either a Riemann-Roch bound (6) or a Cliffordbound (7). 3. Why Gieseker’s proof doesn’t cover marked points
To my knowledge, Elizabeth Baldwin first wrote down the straightforward generalization ofGieseker’s result to M g,n (unpublished), and it is not difficult to see that the analogue of Gieseker’sfiltration does not suffice to establish stability in cases where b i is more than a little larger than 0.Here is a counterexample:3.1. Example 1.
Purpose: to show that the profile associated to ˜ G • (which equals ˜ V • in thisexample) does not suffice to establish asymptotic Hilbert stability when there are marked points.Suppose n ≥
3. Consider the 1-PS λ which acts with linearly decreasing weights on the markedpoints. That is, λ induces the following weighted filtration:Space Weight V = H ( C, O (1)) V = H ( C, O (1)( − P )) V = H ( C, O (1)( − P − P )) V = H ( C, O (1)( − P − P − P )) 0The points ( im, r i m ) all lie on their lower envelope. Also, we have r + r + r = 1. Using γb i = 1 /
2, we have T ≈ m − m + γbm = 5 / m > (1 + ǫ ) m .So the straightforward adaptation of Gieseker’s proof is not enough to establish the stability ofsmooth pointed curves with respect to the linearizations we have specified.3.2. The key observation.
In fact it is not difficult to show that the 1-PS of Example 1 is notdestabilizing.We use the following easy linear algebra lemma:
Lemma 3.1.
Let V , . . . , V n be subspaces of a vector space V . Write V ij := V i ∩ V j , V ijk := V i ∩ V j ∩ V k , etc. Then codim Span { V , . . . , V n } = X codim V i − X i Soon we are going to put a lot of effort into minimizing multi-plicities. The following lemma shows that this makes easy work of computing spans of spaces ofthe form we have encountered. Lemma 3.2. Suppose we are given q subspaces E , . . . , E q of H ( C, O ( m )) of the form: E = H ( C, O ( m )( − d , Q − · · · − d ,q Q q ) E = H ( C, O ( m )( − d , Q − · · · − d ,q Q q )) ... E q = H ( C, O ( m )( − d q, Q − · · · − d q,q Q q )) The E i need not be distinct, and though the notation looks a little similar to that of filtrations above,we do not mean in any way to imply that the E i form a filtration—in the applications we have inmind, they do not.Suppose that E i minimizes the multiplicity of Q i —that is, the minimum in each column appearsalong the diagonal. Suppose also that q X i =1 max j d j,i < dm − g. Then Span( E , . . . , E q ) = H ( C, O ( m )( − q X i =1 d i,i Q i )) and codim Span( E , . . . , E q ) = d , + d , + · · · + d q,q . Proof. The condition q X i =1 max j d j,i Q i < dm − g. IT STABILITY OF WEIGHTED POINTED CURVES 15 ensures that the codimension of the intersection of any subset of these q spaces may be computedusing Riemann-Roch. Thus, for each subset I ⊆ { , . . . , q } , say I = { i , . . . , i k } we havecodim E i ··· i k = max( d i , , . . . , d i k , ) + max( d i , , . . . , d i k , ) + · · · + max( d i ,q , . . . , d i k ,q ) . Suppose j I . Then the term max( d i ,j , . . . , d i k ,j ) is cancelled by a term coming from I ∪ { j } .Being a subset of cardinality one greater, its codimension gets opposite sign from that of I . Andsince by hypothesis d j,j is the smallest term in column j , it drops out of max( d i ,j , . . . , d i k ,j , d j,j ),giving us exactly the cancellation we claimed. Given I , every j ∈ { , . . . , q } is either in I or notin I , so it is clear whether the term max( d i ,j , . . . , d i k ,j ) is cancelling or being cancelled. The onlyterms surviving are the d i,i since there are no double intersections of the form E ii in our setup tocancel them.Finally, the base locus of Span( E , . . . , E q ) must be P qi =1 d i,i Q i (since we can find sections thatvanish to each Q i to exactly order d i,i ). This gives(10) Span( E , . . . , E q ) ⊂ H ( C, O ( m )( − q X i =1 d i,i Q i )) . But the codimensions of the two spaces in line (10) are the same, so we must actually have equality:Span( E , . . . , E q ) = H ( C, O ( m )( − q X i =1 d i,i Q i )) . (cid:3) The filtration ˜ X • and its profile Subscript conventions. In the course of the proof we will need to keep track of a set ofsubsequences of a subsequence of a sequence. My first attempt, using several layers of subscripts,proved unsatisfactory; I know of no good convention for this kind of accounting, so I will use thefollowing notation and conventions.4.1.1. Tildes. Recall that k indexes a subset of the rows j of the original filtration V • . Quantitiesassociated to ˜ V • (like the multiplicities ˜ c and weights ˜ r are written with tildes and indexed by k ’s;quantities associated to V • (such as c and r ) have no tildes and are indexed by j ’s. When I wantto refer to a subsequence of c or r , rather than using nested subscripts and writing for instance r j k I will simply write r k ; this should cause no confusion, since the presence or absence of a tildeindicates whether a layer has been suppressed.4.1.2. Cases I-IV and the functions s ( k, i ) and t ( k, i ) . It is useful to define two functions s and t insome (but not all) situations. We will take the time now to define four cases, which will be referredto in this section and in Section 5.I. We have ˜ c k,i < ˜ c k +1 ,i < ˜ c k +2 ,i . That is, the multiplicity of the point Q i jumps at row k andagain at row k + 1. In this case we do not define s ( k, i ) and t ( k, i ) . II. We have ˜ c k,i = ˜ c k +1 ,i = ˜ c k +2 ,i . That is, the multiplicity of Q i does not jump at row k or atrow k + 1. Define s ( k, i ) to be the last row where this multiplicity jumped, and let t ( k, i ) bethe next row where it jumps, or else ˜ N if ˜ c k,i = ˜ c ˜ N,i . In symbols, s ( k, i ) is the largest indexstrictly (in Case II) less than k such that ˜ c s ( k,i ) ,i < ˜ c s ( k,i )+1 ,i , and t ( k, i ) is the smallestindex strictly (in Case II) greater than k such that ˜ c t ( k,i ) ,i < ˜ c t ( k,i )+1 ,i if this exists, or else ˜ N . Finally, the reader will see after reading Case III and Case IV that in Case II we have s ( k, i ) = s ( k + 1 , i ) and t ( k, i ) = t ( k + 1 , i ).III. We have ˜ c k,i = ˜ c k +1 ,i < ˜ c k +2 ,i . That is, the multiplicity of Q i does not jump at row k butjumps at row k + 1. Then as in Case II we define s ( k, i ) to be the last row where thismultiplicity jumped, and we define t ( k, i ) = k + 1.IV. We have ˜ c k,i < ˜ c k +1 ,i = ˜ c k +2 ,i . That is, the multiplicity of Q i jumps at row k but not atrow k + 1. We define s ( k, i ) = k , and as in Case II let t ( k, i ) be the next row where thismultiplicity jumps, or else ˜ N if ˜ c k,i = ˜ c ˜ N,i .Defining s and t differently in Cases II-IV as we have done permits us to treat these casessimultaneously in Section 5.2, which more than makes up for the extra work involved here. Thereare two reasons why Case I is treated separately from the other cases. First, there is an easy wayto deal with Case I that is not available in Cases II-IV. Second, if one tries to analyze Case I theway we analyze Cases II-IV, one obtains a coefficient which I can bound in Case II-IV which I havenot figured out how to bound in Case I. So, it is desirable to treat Case I separately.4.1.3. Eliminating redundancies. Rather than printing i redundantly in subscripts, whenever I canI will leave it off the second time. For example I will simply write ˜ c s ( k,i ) for ˜ c s ( k,i ) ,i .4.1.4. The functions j ( i, ℓ ) and k ( i, ℓ ) . We will also want to keep track of the subset of j ’s or k ’swhere the multiplicity of the point Q i in the base locus increases. I will do this as follows:Say the multiplicity of Q i jumps K i times between the top of the filtration and the bottom. Westart counting from zero, so these stages of the filtration are the 0 th jump up through the ( K i − th jump. As a convention, we append ¯ N (the index of the last row of the filtration V • ) or ˜ N (the indexof the last row of the filtration ˜ V • ) as the K thi element of this sequence. We write two increasingset functions j ( i, • ) : { , . . . , K i } → { , . . . , ¯ N } and k ( i, • ) : { , . . . , K i } → { , . . . , ˜ N } and use these to index the rows where the multiplicity of the point Q i in the base locus increases.That is, the function j ( i, • ) takes values in the j ’s, and similarly k ( i, • ) takes values in the k ’s.Here is an example to give a little practice with this notation: j ( i, 0) means the index j where themultiplicity of Q i jumps for the 0 th time. This is the lowest row of the filtration where Q i is not inthe base locus, so r j ( i, is the least weight of a section not vanishing at Q i .As before, when i appears more than once in a subscript, we will leave it off the second time.Thus c j ( i, ,i becomes c j ( i, and we have c j ( i, = 0 while c j ( i, = c j ( i, > A consequence of these conventions. As a consequence, note that previously when goingbetween the filtrations V • and ˜ V • we had ˜ c k,i = uvc j k ,i . But now with our new notation we canwrite ˜ c k ( i,ℓ ) = uvc j ( i,ℓ ) . In this sense the definitions of j ( i, ℓ ) and k ( i, ℓ ) have eliminated some ofthe need for nested subscripts.Finally we note that although the notations are similar in format, j and k are somewhat differentin character from s and t . Briefly, j and k are “lookup” functions, whereas s and t are “previous”and “next” functions. IT STABILITY OF WEIGHTED POINTED CURVES 17 The filtration ˜ X • and its profile. Here we describe the filtration ˜ X • of H ( C, O ( m )) andits weight profile. ˜ X • is obtained from the filtration ˜ V • by taking spans of the stages of ˜ V • withother cleverly chosen spaces.The filtration ˜ X • will have ˜ N × u + 1 stages.For each k = 0 , . . . , ˜ N − 1, and for each w = 0 , . . . , u − X k,w .Our starting point is the space ( V u − wk V wk +1 V ) v . Elements of this space have weight less than orequal to v ( u − w ) r k + vwr k +1 + vr .Our goal: for each i from 1 to q , find subspaces of H ( C, O ( m )) whose weight is less than orequal to v ( u − w ) r k + vwr k +1 + vr , for which the multiplicity of Q i is less than the multiplicityin the base locus of ( V u − wk V wk +1 V ) v . We do this as described in the following definition. Also, it isconvenient to define certain quantities ˜ x ( k, i, w ) at this time; their role will be explained soon. Definition 4.1 (The filtration ˜ X • and its profile) . First, ˜ X , = H ( C, O ( m )) . For the remaining triples ( k, w, i ) with ( k, w ) = (0 , , where k = 0 , . . . , ˜ N − , w = 0 , . . . , u − ,and i = 1 , . . . , q , the contribution to the profile is found as follows: • If the multiplicity of Q i is zero in row k + 1 (and hence zero in row k also), there is nocontribution to ˜ X k,w , and ˜ x ( k, i, w ) = 0 . • If the multiplicity of Q i is nonzero in row k + 1 and we are in Case I as defined in Section4.1.2, so the multiplicity of Q i jumps at row k and row k + 1 , then we add no new spacesto ˜ X k,w , and the space ( V u − wk V wk +1 V ) v into ˜ X k,w , and ˜ x ( k, i, w ) is the multiplicity of Q i in ( V u − wk V wk +1 V ) v ; • If the multiplicity of Q i is nonzero in row k + 1 and we are in Case II, III, or IV as definedin Section 4.1.2, so the multiplicity of Q i jumps at no more than one of the rows k and k + 1 , let s ( k, i ) and t ( k, i ) be as defined there. For each w we find the smallest integer W = W ( u, v ; k, w, i ) such that ( V u − Ws ( k,i ) V Wt ( k,i ) V ) v has weight less than v ( u − w ) r k + vwr k +1 + vr .Then ( V u − Ws ( k,i ) V Wt ( k,i ) V ) v is added to ˜ X k,w , and ˜ x ( k, i, w ) is the multiplicity of Q i in the baselocus of ( V u − Ws ( k,i ) V Wt ( k,i ) V ) v .Then ˜ X k,w = Span { ( V u − wk V wk +1 V ) v , spaces of type ( V u − Ws ( k,i ) V Wt ( k,i ) V ) v if there are any } , and let ˜ x ( k, w ) be the codimension of ˜ X k,w .Note ˜ X k,w is the span of between and q + 1 distinct spaces; there may be fewer than q + 1 distinct spaces in the span, as there may be points Q i , which make no contribution, and/or repeatsmay occur among the spaces of the form ( V u − Ws ( k,i ) V Wt ( k,i ) V ) v .Finally, for the last stage of the filtration, define ˜ X ˜ N := ˜ V ˜ N .Thus, the profile associated to ˜ X • is the graph of decreasing step function whose value over theintervals [˜ x ( k, w ) , ˜ x ( k, w + 1)) is v ( u − w ) r k + vwr k +1 + vr , and whose value over the interval [codim ˜ X ˜ N , dim H ( C, O ( m ))] is vr . Note that the spaces used to construct each ˜ X k,w satisfy the degree hypothesis of Lemma 3.2:every space going into the span is either of the form ( V u − wk V wk +1 V ) v or ( V u − W ( k,w,i ) s ( k,i ) V W ( k,w,i ) t ( k,i ) V ) v .But the base locus of any space of this form is bounded by the base locus of ( V u ¯ N V ) v , which is uvc ¯ N, + · · · + uvc ¯ N,q . That is, max j { d j,i } ≤ uvc ¯ N,i , so we have q X i =1 max j { d j,i } ≤ q X i =1 uvc ¯ N,i ≤ uvd < uvd + ud − g = dm − g. However, it is not always true that ( V u − wk V wk +1 V ) v or ( V u − Ws ( k,i ) V Wt ( k,i ) V ) v always minimizes themultiplicity of Q i among these q spaces. (It is possible to find the minimum, but we will not dothis now. See Section 8.3 for a little more discussion.) Therefore, we cannot apply Lemma 3.2to conclude that ˜ x ( k, w ) = P qi =1 ˜ x ( k, w, i ). However, we may use Lemma 3.2 to conclude that˜ x ( k, w ) ≤ P qi =1 ˜ x ( k, w, i ), since the minimum multiplicity for the point Q i must be smaller than˜ x ( k, w, i ). Of course, this is not enough to bound ˜ x ( k, w + 1) − ˜ x ( k, w ). But since the ˜ r k ’s aredecreasing, the weight A of this profile will only decrease if some ˜ x ( k, w ) < P qi =1 ˜ x ( k, w, i ). Socomputing using equality at every stage gives the following upper bound for A :(11) A ≤ ˜ N − X k =0 u − X w =0 ( v ( u − w ) r k + vwr k +1 + vr )(˜ x ( k, w + 1) − ˜ x ( k, w )) + (dim ˜ X ˜ N ) vr . We have ˜ X ˜ N = H ( C, O ( m )( − uvD ¯ N )), and so we may compute dim ˜ X ˜ N = dm − uvd ¯ N − g + 1 =( d − d ¯ N ) uv + dv − g + 1. Substituting this into (11), we obtain(12) A ≤ ˜ N − X k =0 u − X w =0 ( v ( u − w ) r k + vwr k +1 + vr )(˜ x ( k, w + 1) − ˜ x ( k, w )) + (( d − d ¯ N ) uv + dv − g + 1) vr . Rather than trying to bound the right hand side of (12), we will follow a different approach. Wewill define a “virtual” profile whose graph has area A vir nearly the same as the area of the graph A of the actual profile, but which is computationally a little easier to work with. Let ∆ = A − A vir be the discrepancy. Also, for each i between 1 and q , recall that r j ( i, is the r j such that c j,i = 0and c j +1 ,i > 0. Then(13) T ≤ A vir + ∆ + n X i =1 γB i r j ( i, ( u + 1) v . We use the rest of this section to define the virtual profile. In the next section we bound ∆, andin Section 6 we bound A vir + P ni =1 γB i r j ( i, ( u + 1) v . Putting this all together with (13), we willget a bound for T .4.3. The virtual profile. The virtual profile simplifies the graph of the profile in three ways: • In the profile, we form a span of q spaces for all k and for all w , so the step function isdefined over ˜ N × u + 1 intervals; in the virtual profile, we only partition the domain (thecodimension axis) into ˜ N + 1 intervals. • In the profile, we round so that W = W ( u, v ; k, w, i ) is always an integer, so exponents,multiplicities, and codimensions are integers; in the virtual profile, their counterparts arerational numbers. • In particular the quantity ˜ f ( k ) (defined below) is the virtual counterpart to ˜ x ( k, x ( k, , uvr k + vr ) and (˜ x ( k +1 , , uvr k +1 + vr )are connected by a staircase; but in the virtual profile, we connect the two points ( ˜ f ( k ) , ˜ r k )and ( ˜ f ( k + 1) , ˜ r k +1 ) by straight line segments. IT STABILITY OF WEIGHTED POINTED CURVES 19 We will call the figure so obtained the virtual profile and use A vir , the area under the virtual profile,to approximate A . Definition 4.2 (The virtual profile) . For each k = 0 , . . . , ˜ N − , we define ˜ f ( k ) as follows. Webegin by defining ˜ f i ( k ) for each i . Fix i . Graph the set of points { (˜ r k ( i,ℓ ) , ˜ c k ( i,ℓ ) ) : ℓ = 0 , . . . , K i } andconnect these by straight line segments. Then ˜ f i ( k ) is the piecewise linear function whose value at k is the second coordinate of the point on this graph lying over ˜ r k .The picture described above translates into the following rules. We refer to Cases I-IV as definedin 4.1.2: If ˜ c k +1 ,i = 0 , then ˜ f i ( k ) = 0 . I. In Case I, we have ˜ c k +1 ,i = 0 and the multiplicity ˜ c k,i of Q i jumps at row k (that is, ˜ c k,i < ˜ c k +1 ,i ). Then ˜ f i ( k ) = ˜ c k,i . II,III,IV. Otherwise, let s ( k, i ) and t ( k, i ) be as defined in Section 4.1.2. Then ˜ f i ( k ) = (cid:18) ˜ r k − ˜ r t ( k,i ) ˜ r s ( k,i ) − ˜ r t ( k,i ) ˜ c s ( k,i ) + (1 − ˜ r k − ˜ r t ( k,i ) ˜ r s ( k,i ) − ˜ r t ( k,i ) )˜ c t ( k,i ) (cid:19) . Note that in Case IV the formula above just gives ˜ f i ( k ) = ˜ c k,i , since s ( k, i ) = k in Case IV.Finally, ˜ f ( k ) := q X i =1 ˜ f i ( k ) . The virtual profile is the graph of the piecewise linear function connecting the points { ( ˜ f ( k ) , ˜ r k ) } . Note the switch in the order of the coordinates that takes place: ˜ f i ( k ) is defined by a graph inthe (weight × multiplicity of Q i )-plane, whereas the virtual profile is graphed along with the profilein the (codimension × weight)-plane.The quantity ˜ f ( k ) is an approximate upper bound for the codimension of the ˜ r k -weight space in H ( C, O C ( m )). We have: A vir = ˜ N − X k =0 12 ( ˜ f ( k + 1) − ˜ f ( k ))(˜ r k +1 + ˜ r k ) + (dim ˜ V ˜ N ) vr = ˜ N − X k =0 12 ( ˜ f ( k + 1) − ˜ f ( k ))(˜ r k +1 + ˜ r k ) + ( d − d ¯ N ) uv + dv − g + 1 . (14)Also, for each i between 1 and q , recall that r j ( i, is the r j such that c j,i = 0 and c j +1 ,i > 0. Let T vir = A vir + ( u + 1) v γ P qi =1 B i r j ( i, denote the approximation to T obtained by approximating A by A vir . We have the following upper bound for T vir :(15) T vir ≤ ˜ N − X k =0 12 ( ˜ f ( k +1) − ˜ f ( k ))(˜ r k +1 + ˜ r k )+(( d − d ¯ N ) uv + dv − g +1) vr +( u +1) v γ n X B i r j ( i, . Before we proceed, I will illustrate the ideas described above by applying them to Example 1.4.4. Illustration: the profile and virtual profile for ˜ X • in Example 1. Recall that Example1 concerns the 1-PS with q = 3 which induces the following weight filtration: Space Weight V = H ( C, O (1)) V = H ( C, O (1)( − P )) V = H ( C, O (1)( − P − P )) V = H ( C, O (1)( − P − P − P )) 0After passing to H ( C, O ( m )) we obtain:Space Weight ˜ r ˜ V = H ( C, O ( m )) uv + v ˜ V = H ( C, O ( m )( − uvP )) uv + v ˜ V = H ( C, O ( m )( − uvP − uvP )) uv + v ˜ V = H ( C, O ( m )( − uvP − uvP − uvP )) v The virtual profile for Example 1. Let us compute the virtual profile first, as this requiresfewer calculations than computing ˜ X • and the profile. We can compute the virtual profile for anarbitrary u, v :For k = 0 there is nothing to compute.For k = 1, the multiplicity of P does not jump from row 1 to row 2. We are in Case II.Looking at where the multiplicity P jumps, we have s (1 , 1) = 0 and t (1 , 1) = 3, and we find that˜ f (1) = uv . The multiplicity of P jumps between row 1 and row 2; we are in Case IV, andwe have ˜ f (1) = ˜ c , = 0. Finally, since the multiplicity of P is zero in both row 1 and row 2,˜ f (1) = 0. Then ˜ f (1) = uv . Also, ˜ r = uv + v .For k = 2, the multiplicity of P does not jump from row 2 to row 3. We are in Case II, s (2 , 1) = 0and t (2 , 1) = 3, and ˜ f (2) = uv . The multiplicity of P does not jump between row 2 and row 3;we are in Case II, and s (2 , 2) = 1 and t (2 , 2) = 3, giving ˜ f (2) = uv . Finally, the multiplicity of P jumps at row 2; we are in Case IV, so ˜ f (2) = ˜ c , = 0. Then ˜ f (2) = uv . Also, ˜ r = uv + v .Finally, for k = ˜ N = 3 there is also nothing to compute.The area of the region under the graph connecting the points (0 uv, uv + v ), ( uv, uv + v ),( uv, uv + v ) and (3 uv, v ) is u v + uv . To this we add the weight of the vr region, whichis (dim ˜ V ˜ N ) vr = (( d − uv + dv − g + 1)( v ). We have: A vir = 12 u v + 12 duv + 12 dv − 12 ( g − v. Using γB i = , the contribution from the marked points is ( u v + 2 uv + v ). We have: T vir = 1 u v + ( 12 d + 1) uv + ( 12 d + 1) v − 12 ( g − v. Interpreting the vertices of the virtual profile. If we suppose that the integer uv is divisibleby 6, we can give a little more meaning to the calculations above.For k = 1 we can begin with the space ˜ V , which gives us the point (1 uv, uv + v ). To thiswe add the space V uv V uv V v to minimize the multiplicity of P . Similarly we add V uv V v tominimize the multiplicity of P . The multiplicity of P is zero in all the spaces of this weight. IT STABILITY OF WEIGHTED POINTED CURVES 21 The codimension of V uv V v is uv , and the codimension of V uv V uv V v is also uv . However, usingLemma 3.2, the codimension of their span is uv . In other words, the point (1 uv, uv + v ) in theprofile of ˜ V • slides left to ( uv, uv + v ) in the virtual profile for ˜ X • .A similar analysis for k = 2 yields the list of spaces V uv V uv V v , V uv V uv V v , and V uv V v minimizing the multiplicities of P , P , and P respectively. The codimension of their span is uv ,so the point (2 uv, uv + v ) in the profile of ˜ V • slides left to ( uv, uv + v ) in the virtual profilefor ˜ X • .It seems that for any fixed 1-PS λ we could choose uv sufficiently divisible to clear any de-nominators which may arise. However, we cannot do this across all 1-PS, so we will consider thisinterpretation of the vertices of the virtual profile as motivational, not part of the rigorous proof.Also, even when we have such divisibility, so that the virtual profile’s vertices have this interpre-tation, I see no rigorous way to interpret the straight line segments connecting the vertices. So, itseems best to regard the virtual profile merely as a graph and not an algebro-geometric object ofany kind.4.4.3. The filtration ˜ X • and its profile for Example 1. Now we compute the filtration ˜ X • and itsprofile. For this, we ought to specify u, v first. We choose u = 3 and v = 5. Of course, this valueof v is really too small to use with Lemma 2.1, but let us ignore this in the interest of presentinga reasonably sized example. Also, in this example, we will always have ˜ x ( k, w ) = P qi =1 ˜ x ( k, w, i ).(How do I know this? See Section 8.3 for a hint.)The filtration ˜ X • has ten stages. The first and the last are easy to compute—we have ˜ X , = H ( C, O ( m )) and ˜ X = ( V V ) . Let’s compute one of the middle stages, ˜ X , , as an example:The multiplicity of P does not increase from row 1 to row 2 to row 3, so we are in Case II, and s (1 , 1) = 0 and t (1 , 1) = 3. We find W = 2. (Here W may be computed from its defining properties,or by skipping ahead and using Formula (20) derived in Section 5.) Thus the contribution to ˜ X , from P is ( V V V ) , and ˜ x (1 , , 1) = 10. The multiplicity of P increases from row 1 to row2, but not from row 2 to row 3, so we are in Case IV, and s (1 , 2) = 1 and t (1 , 2) = 3. Here W = 1, and the contribution from P to ˜ X , is ( V V V ) , and ˜ x (1 , , 2) = 5. The multiplicityof P is zero in both row 1 and row 2, so P does not contribute to ˜ X , . We have: ˜ X , =Span { ( V V V ) , ( V V V ) , ( V V V ) } , and ˜ x (1 , 1) = 15 . Here is the filtration ˜ X • . I have left the spans unsimplified.Stage Space Codim Wt˜ X , = H ( C, O ( m )) 0 10˜ X , = Span { ( V V V ) , ( V V V ) } / X , = Span { ( V V V ) , ( V V V ) } / X , = Span { ( V V V ) , ( V V V ) , ( V V V ) } / X , = Span { ( V V V ) , ( V V V ) , ( V V V ) } 15 40 / X , = Span { ( V V V ) , ( V V V ) , ( V V V ) } 15 35 / X , = Span { ( V V V ) , ( V V V ) , ( V V V ) , ( V V V ) } 20 5˜ X , = Span { ( V V V ) , ( V V V ) , ( V V V ) , ( V V V ) } 30 25 / X , = Span { ( V V V ) , ( V V V ) , ( V V V ) , ( V V V ) } 40 20 / X = ( V V ) 45 15 / Notice that ˜ X , = ˜ X , = ˜ X , , and ˜ X , = ˜ X , . Nothing in our definitions prevents this, and itdoes not harm us either—all it means is that when we compute the area under the profile betweenthese stages of the filtration, we will obtain a complicated expression for zero.Here are the profile and virtual profile for ˜ X • in Example 1 with u = 3, v = 5. Tick marks onthe horizontal axis show units of 5; tick marks on the vertical axis show units of 2.5.CodimensionWeightIn this picture the area under the profile looks significantly larger than the area under the virtualprofile, but for larger values of u these areas become relatively closer. This is made rigorous in thenext section, but as an example, here are the profile and virtual profile for ˜ X • in Example 1 with u = 20, v = 5. Tick marks on the horizontal axis show units of 10; tick marks on the vertical axisshow units of 5. CodimensionWeight Progress report. We have at last defined all the key ingredients mentioned in the introduction:one filtration V • of H ( C, O (1)),two filtrations ˜ V • and ˜ X • of H ( C, O ( m )),and two graphs associated to ˜ X • In Sections 5, 6, and 7 it remains to study these filtrations and graphs more closely and show thatthey have the properties claimed.5. The discrepancy between the profile and virtual profile This section is devoted to showing that the areas of the profile and virtual profile are very closewhen m is large. That is, we bound the discrepancy ∆ := A − A vir . The strategy and methods ofthis section are extremely straightforward.We will bound ∆ by computing bounds for several terms which contribute to it. Roughlyspeaking, we will compute the discrepancy ∆ k,i for each k and i , but it takes a little care to sayexactly what we mean by this, as the regions of the graph may be offset a little bit. For instance,in the picture corresponding to Example 1 with u = 3, v = 5, we would partition the virtual profile IT STABILITY OF WEIGHTED POINTED CURVES 23 at codimension 17.5 (a breakpoint of the piecewise linear function) but the corresponding partitionfor the profile occurs at codimension 20.For the virtual profile this is straightforward. The area under the graph of the virtual profilemay be divided in an obvious way into ˜ N trapezoids and one final rectangle. Let us focus on thearea A vir k of the k th trapezoid: A vir k = 12 ( ˜ f ( k + 1) − ˜ f ( k ))(˜ r k +1 + ˜ r k )= 12 ( q X i =1 ˜ f i ( k + 1) − q X i =1 ˜ f i ( k ))(˜ r k +1 + ˜ r k )= q X i =1 12 ( ˜ f i ( k + 1) − ˜ f i ( k ))(˜ r k +1 + ˜ r k ) . I will write A vir k,i for the i th summand: A vir k,i = 12 ( ˜ f i ( k + 1) − ˜ f i ( k ))(˜ r k +1 + ˜ r k ) . We compute A vir k,i now.5.1. Computing A vir k,i . A vir k,i is the area of the trapezoid whose vertices are ( ˜ f i ( k ) , f i ( k + 1) , f i ( k + 1) , ˜ r k +1 ), and ( ˜ f i ( k ) , ˜ r k ). To compute ˜ f i ( k + 1) − ˜ f i ( k ), recall the definition of ˜ f i ( k ) givenin Definition 4.2. We use the four cases defined in Section 4.1.2.I. The multiplicity ˜ c • ,i jumps at row k and again at row k + 1. Then the spaces contributing tothe profile are V uvk V v at the k th vertex and V uvk +1 V v at the ( k + 1) th vertex, and in between,spaces of the form V ( u − w ) vk V wvk +1 V v are used. Thus in the virtual profile we are calculatingas if spaces of the form V αuvk V (1 − α ) uvk +1 V v were being used between these two vertices with α ranging from 0 to 1.II. The multiplicity ˜ c • ,i does not jump at row k or at row k + 1. Recall that we have( s ( k, i ) , t ( k, i )) = ( s ( k + 1 , i ) , t ( k + 1 , i )). In the profile, spaces of the form V ( u − W ) vs ( k,i ) V W vt ( k,i ) V v are being used between these two vertices. In the virtual profile, we are calculating as ifspaces of the form V αuvs ( k,i ) V (1 − α ) vt ( k,i ) V v were being used between these two vertices (thoughhere the range of α is a subinterval strictly in the interior of [0 , c • ,i does not jump at row k but jumps at row k + 1. Recall that t ( k, i ) = k + 1. Once again, in the profile, spaces of the form V ( u − W ) vs ( k,i ) V W vt ( k,i ) V v are being used inthis region. For this reason Case III is very similar to Case II. In the virtual profile, we arecalculating as if spaces of the form V αuvs ( k,i ) V (1 − α ) vt ( k,i ) V v were being used in this region, with α beginning at a value strictly smaller than 1 and decreasing to 0.IV. The multiplicity ˜ c • ,i jumps at row k but not at row k + 1. By the definition of s we have s ( k, i ) = k , and in the profile spaces of the form V ( u − W ( k,w,i )) vs ( k,i ) V W ( k,w,i ) vt ( k,i ) V v are being used inthis region. In the virtual profile, we are calculating as if spaces of the form V αuvs ( k,i ) V (1 − α ) vt ( k,i ) V v were being used in this region, with α starting at 1 and ending at a value strictly greaterthan 0. Computing A vir k,i , Case I. By Definition 4.2 we have ˜ f i ( k + 1) = ˜ c k +1 ,i and ˜ f i ( k ) = ˜ c k,i . Thus A vir k,i = 12 (˜ r k +1 + ˜ r k )( ˜ f i ( k + 1) − ˜ f i ( k ))= 12 ( uvr k +1 + vr + uvr k + vr )( uvc k +1 ,i − uvc k,i )= u v ( 12 ( r k +1 + r k )( c k +1 ,i − c k,i + uv ( r ( c k +1 ,i − c k,i )) . (16) Cases II, III, and IV. In Case II we have˜ f i ( k + 1) = ˜ r k +1 − ˜ r t ( k +1 ,i ) ˜ r s ( k +1 ,i ) − ˜ r t ( k +1 ,i ) ˜ c s ( k +1 ,i ) + (1 − ˜ r k +1 − ˜ r t ( k +1 ,i ) ˜ r s ( k +1 ,i ) − ˜ r t ( k +1 ,i ) )˜ c t ( k +1 ,i ) and ˜ f i ( k ) = ˜ r k − ˜ r t ( k,i ) ˜ r s ( k,i ) − ˜ r t ( k,i ) ˜ c s ( k,i ) + (1 − ˜ r k − ˜ r t ( k,i ) ˜ r s ( k,i ) − ˜ r t ( k,i ) )˜ c t ( k,i ) , and ( s ( k, i ) , t ( k, i )) = ( s ( k + 1 , i ) , t ( k + 1 , i )). Thus A vir k,i = 12 (˜ r k +1 + ˜ r k )( ˜ f i ( k + 1) − ˜ f i ( k ))= 12 (˜ r k +1 + ˜ r k )( ˜ r k − ˜ r k +1 ˜ r s ( k,i ) − ˜ r t ( k,i ) (˜ c t ( k,i ) − ˜ c s ( k,i ) ))= 12 ( uvr k +1 + uvr k + 2 vr )( uv r k − r k +1 r s ( k,i ) − r t ( k,i ) ( c t ( k,i ) − c s ( k,i ) ))= u v (cid:18) 12 ( r k +1 + r k )( c t ( k,i ) − c s ( k,i ) ) r k − r k +1 r s ( k,i ) − r t ( k,i ) (cid:19) + uv (cid:18) r ( c t ( k,i ) − c s ( k,i ) ) r k − r k +1 r s ( k,i ) − r t ( k,i ) (cid:19) (17)By a similar calculation, and using some of the information presented in paragraphs III and IVabove, we derive the same formula in Case III and Case IV.5.2. Computing bounds for A k,i . We have defined A vir k,i but have not yet defined a correspondingquantity A k,i . We do this now. Let A k,i denote the following sum:(18) A k,i := u − X w =0 (( u − w ) r k + wr k +1 + r )(˜ x ( k, w + 1 , i ) − ˜ x ( k, w, i )) . In pictures, P qi =1 A k,i is the area under the profile between ˜ x ( k, 0) and ˜ x ( k + 1 , A k,i . We split into Cases I-IV as in Section 5.1. Case I. Again, using Definition 4.1 we have ˜ x ( k, w + 1 , i ) = v ( u − ( w + 1)) c k,i + v ( w + 1) c k +1 ,i and ˜ x ( k, w, i ) = v ( u − w ) c k,i + vwc k +1 ,i , so ˜ x ( k, w + 1 , i ) − ˜ x ( k, w, i ) = c k +1 ,i − c k,i . We have: A k,i = u − X w =0 (( u − w ) r k + wr k +1 + r )(˜ x ( k, w + 1 , i ) − ˜ x ( k, w, i ))= u − X w =0 (( u − w ) r k + wr k +1 + r )( c k +1 ,i − c k,i )= u v ( 12 ( r k +1 + r k )( c k +1 ,i − c k,i )) + uv (( r + 12 ( r k +1 + r k ))( c k +1 ,i − c k,i )) . (19) IT STABILITY OF WEIGHTED POINTED CURVES 25 Cases II, III, and IV. The calculation is long; fortunately, we can treat Cases II, III, and IV. Also, from here to the end of Section 5.2, we will suppress the subscripts k,i as much as possible,as they do not change. We will reintroduce them at the end of this subsection in line (32).Recall that in Definition 4.1, for each w , we defined W = W ( k, w, i ) to be the smallest integersuch that the space ( V u − Ws V Wt V ) v has weight less than or equal to v ( u − w ) r k + vwr k +1 + vr .We use this property to get an expression for W in Case II or Case III: v ( u − W ) r s + vW r t + vr ≤ v ( u − w ) r k + vwr k +1 + vr ⇔ W ≥ u ( r s − r k ) + w ( r k − r k +1 ) r s − r t ⇒ W ( w ) = W ( k, w, i ) = (cid:24) u ( r s − r k ) + w ( r k − r k +1 ) r s − r t (cid:25) (20)It is useful to write ζ = ζ k,i := r k − r k +1 r s − r t (21) ξ = ξ k,i := r s − r k r s − r t . (22)Then(23) W = ⌈ uξ + wζ ⌉ . Also, since s < k < t , we have 0 ≤ ζ < ≤ ξ < x ( k, w, i ) = v ( u − W ( w )) c s + vW ( w ) c t ˜ x ( k, w + 1 , i ) = v ( u − W ( w + 1)) c s + vW ( w + 1) c t ⇒ ˜ x ( k, w + 1 , i ) − ˜ x ( k, w, i ) = v ( c t − c s )( W ( w + 1) − W ( w )) . Putting this into (18) we have: A k,i = u − X w =0 v (( u − w ) r k + wr k +1 + r ) v (( c t − c s )( W ( w + 1) − W ( w )))= v ( c t − c s ) u − X w =0 ( ur k + r − w ( r k − r k +1 ))( W ( w + 1) − W ( w )) ! = v ( c t − c s ) ( ur k + r ) u − X w =0 ( W ( w + 1) − W ( w )) − ( r k − r k +1 ) u − X w =0 w ( W ( w + 1) − W ( w )) ! . (24)5.2.1. Calculating pieces of (24). Before we continue computing A k,i it is helpful to work out thesums appearing in (24). We begin with the first sum, P u − w =0 ( W ( w + 1) − W ( w )). Let(25) h y i := y − ⌊ y ⌋ , so that h y i denotes the fractional part of y . Then: u − X w =0 ( W ( w + 1) − W ( w ))= u − X w =0 ( ⌈ uξ + wζ + ζ ⌉ − ⌈ uξ + wζ ⌉ )= u − X w =0 ( ⌈h uξ i + wζ + ζ ⌉ − ⌈h uξ i + wζ ⌉ ) . (26)Now, imagining the summation as a dynamic process, the sum in line (26) increases by one everytime the first summand passes an integer and the second summand hasn’t caught up yet. Thishappens ⌊ uζ + h uξ i⌋ times, so we have(27) u − X w =0 ( W ( w + 1) − W ( w )) = ⌊ uζ + h uξ i⌋ . It is helpful to have a nicer expression for ⌊ uζ + h uξ i⌋ . We write ⌊ uζ + h uξ i⌋ = uζ + h uξ i − h uζ + h uξ ii and define(28) η := h uξ i − h uζ + h uξ ii so that(29) ⌊ uζ + h uξ i⌋ = uζ + η. Note that − < η < P u − w =0 w ( W ( w + 1) − W ( w )). Simplifying as above, we have: u − X w =0 w ( W ( w + 1) − W ( w ))= u − X w =0 w ( ⌈h uξ i + wζ + ζ ⌉ − ⌈h uξ i + wζ ⌉ ) . I claim(30) u − X w =0 w ( ⌈h uξ i + wζ + ζ ⌉ − ⌈h uξ i + wζ ⌉ ) = uζ + η X ℓ =1 (cid:24) ℓ − h uξ i ζ (cid:25) , ζ = 00 , ζ = 0As before, the factor ( ⌈h uξ i + wζ + ζ ⌉ − ⌈h uξ i + wζ ⌉ ) is 0 except when the first summand hasjust passed an integer and the second summand has not caught up, and then this factor is 1. Wecan describe the values of w which are multiplied by nonzero coefficient: for each integer ℓ in theappropriate range, we have w = l ℓ −h uξ i ζ m .Note ζ appears in the denominator, and ζ can take the value 0. It could be forgetten all tooeasily that these two things do not happen at the same time, causing concern that this summand(or later quantities) is undefined, so I will write an indicator function ζ =0 = ζ k,i =0 to remind usthat when ζ = 0, we add 0. IT STABILITY OF WEIGHTED POINTED CURVES 27 The main calculation resumed. We now resume the main calculation by reprinting line (24),and then substituting in (27), (29), and (30): A k,i = v ( c t − c s ) ( ur k + r ) u − X w =0 ( W ( w + 1) − W ( w )) − ( r k − r k +1 ) u − X w =0 w ( W ( w + 1) − W ( w )) ! = v ( c t − c s ) (( ur k + r )( uζ + η ) − ( r k − r k +1 ) ζ =0 uζ + η X ℓ =1 (cid:24) ℓ − h uξ i ζ (cid:25)! = v ( c t − c s ) (( ur k + r )( uζ + η ) − ( r k − r k +1 ) ζ =0 uζ + η X ℓ =1 (cid:18) ℓζ − h uξ i ζ − (cid:28) ℓ − h uξ i ζ (cid:29) + 1 (cid:19)! = v ( c t − c s ) (cid:0) ( u r k ζ + ( ηr k + ζr ) u + r η − ζ =0 ( r k − r k +1 ) ζ ( 12 ( uζ + η )( uζ + η + 1))+ ζ =0 ( r k − r k +1 ) ζ ( h uξ i )( uζ + η ) − ζ =0 uζ + η X ℓ =1 (cid:18) − (cid:28) ℓ − h uξ i ζ (cid:29)(cid:19)! (31)In the last line, we have 0 ≤ − h ℓ −h uξ i ζ i . Since this quantity is subtracted, we obtain an upperbound for A k,i by replacing this by zero. We also begin grouping terms by their u -degree: A k,i ≤ v ( c t − c s ) (cid:18) ( r k ζ − ζ =0 ( r k − r k +1 ) ζ ζ ) u +( ηr k + ζr + ζ =0 ( r k − r k +1 ) ζ ( h uξ i − 12 (2 η + 1)) ζ ) u +( r η + ζ =0 ( r k − r k +1 ) ζ ( η h uξ i − η − η ))1 (cid:19) = v ( c t − c s ) (cid:18) ( 12 ( r k + r k +1 ) ζ ) u +( ηr k + ζr + ζ =0 ( r k − r k +1 )( h uξ i − η + 12 )) u +( ηr + ζ =0 ( r s − r t )( η h uξ i − η − η ))1 (cid:19) . Finally, we restore the k, i symbols which have been suppressed throughout this subsection,yielding: A k,i ≤ v ( c t ( k,i ) − c s ( k,i ) ) (cid:18) ( 12 ( r k + r k +1 ) ζ k,i ) u +( η k,i r k + ζ k,i r + ζ k,i =0 ( r k − r k +1 )( h uξ k,i i − η k,i + 12 )) u +( η k,i r + ζ k,i =0 ( r s ( k,i ) − r t ( k,i ) )( η k,i h uξ k,i i − η k,i − η k,i ))1 (cid:19) . (32)This completes our calculation of A k,i in Case II, III, or IV.5.3. Bounding the discrepancy. We now have all the ingredients we need to bound ∆.∆ := A − A vir ≤ ˜ N − X k =0 q X i =1 ( A k,i − A vir k,i ) . In Case I, by comparing (16) and (19) we see that∆ k,i := A k,i − A vir k,i = uv ( c k +1 ,i − c k )( 12 ( r k − r k +1 ) ≤ uv ( c k +1 ,i − c k )( 72 ) + v ( c k +1 ,i − c k )(3) . (33)Of course this last estimate is far from sharp, but it is useful to estimate this way to match whatappears in Cases II-IV.In Cases II-IV, by comparing (17) and (32) (and using the definition of ζ k,i at (21)) we see that∆ k,i := A k,i − A vir k,i ≤ uv (( c t ( k,i ) − c s ( k,i ) )( η k,i r k + ζ k,i =0 ( r k − r k +1 )( h uξ k,i i − η k,i + 12 )))+ v (( c t ( k,i ) − c s ( k,i ) )( η k,i r + ζ k,i =0 ( r s − r t )( η k,i h uξ k,i i − η k,i − η k,i )(34)Recall that the weights r j and the fractional parts of any quantity must be between 0 and 1, and − < η k,i < 1. Therefore we may make various coarse estimates: η k,i r k < h uξ k,i i − η k,i + 12 < ⇒ ζ k,i =0 ( r k − r k +1 )( h uξ k,i i − η k,i + 12 ) < 52 ; η k,i r < η k,i h uξ k,i i − η k,i − η k,i < − ⇒ ζ k,i =0 ( r s − r t )( η k,i h uξ k,i i − η k,i − η k,i ) < . (35)Combining these inequalities with (34) we obtain:(36) ∆ k,i ≤ uv (( c t ( k,i ) − c s ( k,i ) )( 72 ) + v (( c t ( k,i ) − c s ( k,i ) )(3) . Next, I claim that the estimates (33) and (36) yield(37) ˜ N − X k =0 ∆ k,i ≤ uv ( 72 c ¯ N,i ) + v (3 c ¯ N,i ) . IT STABILITY OF WEIGHTED POINTED CURVES 29 Refer back to the definition of s and t in Section 4.1. Equation (37) follows because the pairs k, k + 1 from Case I and the pairs ( s, t ) from Case II, III, and IV fit together in such a way thatwhen the estimates (33) and (36) are summed over k , the sum telescopes.Finally, using the estimates obtained in (37), we obtain(38) ∆ ≤ q X i =1 ˜ N − X k =0 ∆ k,i ≤ uv ( 72 d ) + v (3 d ) . Observe that ∆ is of order uv and not of order u v .6. Bounding T vir The reader is strongly encouraged to review the subscript notations introduced in Section 4.1,especially the definitions of j ( i, ℓ ) and k ( i, ℓ ), before proceeding.6.1. Setting up a comparison. Recall that in line (15) we obtained the following bound on T vir : T vir ≤ ˜ N − X k =0 12 ( ˜ f ( k + 1) − ˜ f ( k ))(˜ r k +1 + ˜ r k ) + (( d − d ¯ N ) uv + dv − g + 1) vr +( u + 1) v γ q X B i r j ( i, = ˜ N − X k =0 12 ( ˜ f ( k + 1) − ˜ f ( k ))(˜ r k +1 + ˜ r k ) + (( d − d ¯ N ) uv + dv − g + 1) vr (39) + u v γ q X B i r j ( i, + q X γB i r j ( i, (2 uv + v )Everything in this sum is in terms of k (it is, after all, the weight of a basis of H ( C, O ( m ))).Almost the only bound available is that the weights sum to 1: P ¯ Nj =0 z j r j = 1. Our goal in thissubsection is to rewrite (39) in a form that makes it easy to compare to P z j r j .We focus on the first term of (39): ˜ N − X k =0 12 ( ˜ f ( k + 1) − ˜ f ( k ))(˜ r k +1 + ˜ r k ) = ˜ N − X k =0 12 ( q X i =1 ˜ f i ( k + 1) − q X i =1 ˜ f i ( k ))(˜ r k +1 + ˜ r k )= ˜ N − X k =0 q X i =1 12 ( ˜ f i ( k + 1) − ˜ f i ( k ))(˜ r k +1 + ˜ r k )(40)Let A vir k,i denote the area of the region described in Definition 4.2. Then we have: ˜ N − X k =0 q X i =1 12 ( ˜ f i ( k + 1) − ˜ f i ( k ))(˜ r k +1 + ˜ r k ) = ˜ N − X k =0 q X i =1 A vir k,i = q X i =1 ˜ N − X k =0 A vir k,i (41)where in the last line we have changed the order of summation. Let A vir i = P ˜ N − k =0 A vir k,i . Observethat, for a fixed i , it may not be necessary to partition this region into ˜ N vertical trapezoids to compute the area A vir i ; a partition corresponding to the domains of definition of the piecewise linearfunction f i , which may be coarser than that given by the full set of k ’s, will do.Recall that k ( i, • ) indexes the rows k where the multiplicity ˜ c • ,i jumps. Then we may compute: A vir i = ˜ N − X k =0 12 ( ˜ f i ( k + 1) − ˜ f i ( k ))(˜ r k +1 + ˜ r k )= K i − X ℓ =0 12 (˜ c k ( i,ℓ +1) − ˜ c k ( i,ℓ ) )(˜ r k ( i,ℓ +1) + ˜ r k ( i,ℓ ) )= u v K i − X ℓ =0 12 ( c j ( i,ℓ +1) − c j ( i,ℓ ) )( r j ( i,ℓ +1) + r j ( i,ℓ ) ) ! + uv ( c ¯ N,i r )(42)We develop the coefficient of the u v term of (42): K i − X ℓ =0 12 ( c j ( i,ℓ +1) − c j ( i,ℓ ) )( r j ( i,ℓ +1) + r j ( i,ℓ ) ) ! = K i X ℓ =1 12 ( c j ( i,ℓ ) − c j ( i,ℓ − ) r j ( i,ℓ ) + K i − X ℓ =0 12 ( c j ( i,ℓ +1) − c j ( i,ℓ ) ) r j ( i,ℓ ) ! = K i − X ℓ =1 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) r j ( i,ℓ ) + 12 c j ( i, r j ( i, ! . (43)Once again, c j ( i, is the first nonzero multiplicity of Q i in a base locus in V • , and r j ( i, is the leastweight of a section not vanishing at Q i . Putting (43), (42), and (41) into (40), we have: T vir ≤ u v q X i =1 K i − X ℓ =1 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) r j ( i,ℓ ) + ( 12 c j ( i, + γB i ) r j ( i, ! (44) + q X i =1 γB i r j ( i, ! (2 uv + v ) + (( d − d ¯ N ) uv + dv − g + 1) vr + uv ( q X i =1 c ¯ N,i r )It is convenient to define I j to be the set of i ’s where the multiplicity jumps at row j , and notfor the first or last time:(45) I j := { i | ∃ ℓ = 0 , K i s . t . j = j ( i, ℓ ) } . We switch the order of summations in (44) to obtain: T vir = u v N X j =0 X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) + X i : j = j ( i, ( 12 c j ( i, + γB i ) r j (46) + q X i =1 γB i r j ( i, ! (2 uv + v ) + ( duv + dv − g + 1) vr which is of the form we desired. IT STABILITY OF WEIGHTED POINTED CURVES 31 Comparing. The next lemma gives a bound for the coefficient of u v in (46). Lemma 6.1. ¯ N X j =0 X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) + X i : j = j ( i, ( 12 c j ( i, + γB i ) r j ≤ ¯ N X j =0 Z j r j , where Z j := z j , j < j RR z j + ( j X τ =0 z τ − ( N − g )) , j = j RR z j , j ≥ j Cliff Idea of proof (Wall Street version). Think of j as being time in days, the Z j ’s as daily income,and the coefficient of r j on the left hand side as daily losses. We will show that every time youhave a losing day, you have enough in the bank to see you through. Idea of proof (algebraic geometry version). The Z j ’s defined above bound the change in degreeof the base loci from V j to V j +1 . The only way there can be a jump larger than this is if d j lagsbehind the maximum allowable degree for this codimension. In this case, we are using more smallweights and fewer large weights than we conceivably could, so the weight of the resulting basis willnot be maximal. Proof. We may rewrite the desired inequality as ¯ N X j =0 Z j − X i : j = j ( i, ( 12 c j ( i, + γB i ) − X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) r j ≥ . We work successively on each index j where Z j − X i : j = j ( i, ( 12 c j ( i, + γB i ) − X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) < . If there are no such j , we are done. So suppose there is at least one such index, and let the setof these be indexed j e beginning with e = 1. By the definition of j we have Z j − X i : j = j ( i, ( 12 c j ( i, + γB i ) − X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) > j < j , so j − X j =0 Z j − X i : j = j ( i, ( 12 c j ( i, + γB i ) − X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) r j ≥ j − X j =0 Z j − X i : j = j ( i, ( 12 c j ( i, + γB i ) − X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) r j and j − X j =0 Z j − X i : j = j ( i, ( 12 c j ( i, + γB i ) − X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) ≥ . We wish to establish that j X j =0 Z j − X i : j = j ( i, ( 12 c j ( i, + γB i ) − X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) r j ≥ j X j =0 Z j − X i : j = j ( i, ( 12 c j ( i, + γB i ) − X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) r j (which is easy) and that j X j =0 Z j − X i : j = j ( i, ( 12 c j ( i, + γB i ) − X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) ≥ . We rewrite this last inequality as(47) j X j =0 Z j − j X j =0 X i : j = j ( i, ( 12 c j ( i, + γB i ) + X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) ≥ . We study the second sum in (47) above. Each i falls into exactly one of the following cases: Case 0. If c • ,i does not jump before or at j —that is, j ( i, > j —then this i does not contribute. Case 1. If c • ,i jumps exactly once before or at j —that is, j ( i, ≤ j < j ( i, i contributes 12 c j ( i, + γB i ≤ c j +1 ,i + 12 ≤ c j +1 ,i , since c j ( i, = c j +1 ,i and γB i ≤ and c j +1 ,i ≥ Case 2. If c • ,i jumps exactly twice before or at j —that is, j ( i, ≤ j < j ( i, c j ( i, + γB i + 12 c j ( i, ≤ c j +1 ,i . This follows because c j ( i, = c j +1 ,i and c j ( i, ≥ c j ( i, + 1. Case 3. If c • ,i jumps three or more times before or at j , then some telescoping occurs, and thecontribution is 12 c j ( i, + γB i + 12 c t ( j ,i ) + 12 c s ( j ,i ) − c j ( i, ≤ c j +1 ,i . Here I am abusing notation a little (according to Section 4.1 the first argument of s ( • , i ) or t ( • , i ) issupposed to be a k , not a j ). Here s ( j , i ) denotes the largest index less than or equal to j where c • ,i jumps, and t ( j , i ) denotes the smallest index strictly greater than j index where c • ,i jumps.Thus, c t ( j ,i ) = c j +1 ,i and c s ( j ,i ) ≤ c j ,i .To summarize, in each case, we see that the contribution is no more than c j +1 ,i .If j < j RR , so that j + 1 is in the Riemann-Roch region, then by (6) we have q X i =1 c j +1 ,i ≤ j X j =0 z j , IT STABILITY OF WEIGHTED POINTED CURVES 33 so the left hand side of (47) is indeed nonnegative: j X j =0 Z j − j X j =0 X i : j = j ( i, ( 12 c j ( i, + γB i ) + X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) ≥ j X j =0 z j − j X j =0 z j = 0 . We have thus dealt with the first index, if it falls inside the Riemann-Roch region. We mayrepeat the argument at each j e in the Riemann-Roch successively, stopping when either the j e ’sare exhausted or we reach the Clifford region. At each step we need to show two things in order toproceed to the next step: first, j e X j =0 Z j − X i : j = j ( i, ( 12 c j ( i, + γB i ) − X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) r j ≥ j e X j =0 Z j − X i : j = j ( i, ( 12 c j ( i, + γB i ) − X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) r j (which is always easy to check), and second, j e X j =0 Z j − X i : j = j ( i, ( 12 c j ( i, + γB i ) − X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) ≥ . Next suppose that j e = j RR , so j e + 1 = j Cliff . Then by (7) we have q X i =1 c j e +1 ,i ≤ j e X j =0 z j + j e X j =0 z j − ( N − g ) j e X j =0 Z j − j e X j =0 X i : j = j ( i, ( 12 c j ( i, + γB i ) + X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) ≥ j e X j =0 z j + j e X j =0 z j − ( N − g ) − j e X j =0 z j + j e X j =0 z j − ( N − g ) = 0 . Finally suppose that some j e + 1 falls within the Clifford region. Then by (7) we have q X i =1 c j e +1 ,i ≤ j e X j =0 z j + j e X j =0 z j − ( N − g ) . Using the definitions given in the statement of the lemma, we compute j e X j =0 Z j = j RR − X j =0 Z j + Z j RR + j e X j = j Cliff Z j = j RR − X j =0 z j + z j RR + j RR X j =0 z j + ( N − g ) + 2 z j Cliff + · · · + 2 z j e = 2 j e X j =0 z j − ( N − g )and once again the left hand side of (47) is nonnegative: j e X j =0 Z j − j e X j =0 X i : j = j ( i, ( 12 c j ( i, + γB i ) + X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) ≥ j e X j =0 z j − ( N − g ) − j e X j =0 z j + j e X j =0 z j − ( N − g ) = 0 . Again, proceed to the next j e until the set of these has been exhausted. (cid:3) Ideally, we would now show that the bound obtained in Lemma 6.1 is smaller than what isrequired in the numerical criterion. Unfortunately, this is not always true. Lemma 6.1 is sufficientfor most, but not all, sets of linearizing weights B . Below I have listed five cases which exhaust allpossibilities. This partitioning may look strange, but it is in order of difficulty of proof. In CasesA-C, I can prove asymptotic stability of smooth curves. In Cases D and E, I cannot prove stability,so I will ultimately impose hypotheses to ensure that these cannot occur.Choose any sufficiently small value ǫ > 0. (The size of ǫ allowed will become clear in Cases Band C below, and the role of ǫ will become clear in the proof of Theorem 7.1.) Then we considerthe following five cases:(48) Case A. n ≥ γb ≥ g − N + ǫ ( N + 1) . Case B. n ≥ γb < g − N + ǫ ( N + 1) < Case C. n = 0 and N ≥ g − Case D. n = 0 and N < g − Case E. n ≥ γb < g − N + ǫ ( N + 1) ≥ Let us proceed first with Case A: To apply Lemma 6.1 to our problem, we need to bound P Z j r j .Let r N − g +1 , . . . , r N − , r N = 0 be the last g weights (that is, ignore the index j and list the smallestweights as many times as indicated by their multiplicities). Then we have X Z j r j ≤ X z j r j + r N − g +1 + · · · + r N ≤ r N − g +1 + · · · + r N Now we bound r N − g +1 + · · · + r N : Lemma 6.2. r N − g +1 + · · · + r N ≤ g − N .Proof. Recall that r N = 0, so we may omit it from all the following sums. We argue similarlyto [Morr] Theorem 4.1. We wish to maximize r N − g +1 + · · · + r N − , which is linear in the r ’s, IT STABILITY OF WEIGHTED POINTED CURVES 35 subject to the constraints P ¯ N − j =0 z j r j = 1 and that the r ’s are decreasing. In the affine hyperplanein ( N − r -space determined by the equation P ¯ N − j =0 z j r j = 1, the condition that the r ’s are decreasing defines an ( N − r = · · · = r h > r h +1 = · · · = r N − = 0 . The function must take its maximum at (at least) one of these vertices, and it is easy to check thatthe maximum occurs when r = · · · = r N − > , or r j = N for all j , yielding a maximum value of g − N . (cid:3) Also, the defining hypothesis of Case A at line (48) may be written as follows. γb ≥ g − N + ǫ ( N + 1) ⇔ g − N ≤ g − γbN + 1 − ǫ Therefore, as a trivial extension of Lemma 6.2, we have:(49) r N − g +1 + · · · + r N ≤ g − γbN + 1 − ǫ We combine (49) with the bound found in (46) to obtain:(50) T vir ≤ (cid:18) g − γbN + 1 − ǫ (cid:19) u v + q X i =1 γB i r j ( i, ! (2 uv + v ) + ( duv + dv − g + 1) vr . Note that the leading coefficient 1 + g − γbN +1 − ǫ is less than the leading coefficient 1 + g − γbN +1 of thenumerical criterion (2) by ǫ . This completes our discussion of Case A.Next we turn to Cases B and C, defined in line (48). In these cases, the bound given in Lemma6.2 is too large to use with the numerical criterion. Fortunately, if we examine the proof of Lemma6.1 closely, we can improve the bound there a little bit. Lemma 6.3. (1) Suppose a sufficiently small ǫ > has been chosen and n ≥ and γb < g − N + ǫ ( N + 1) < , so that we are in Case B. Then ¯ N X j =0 X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) + X i : j = j ( i, ( 12 c j ( i, + γB i ) r j ≤ ¯ N X j =0 Z j r j − (cid:18) − γb (cid:19) r N − , where the Z j are as in Lemma 6.1, and r N − = (cid:26) , z ¯ N > r ¯ N − , z ¯ N = 1 . (2) Suppose n = 0 . Then ¯ N X j =0 X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) + X i : j = j ( i, ( 12 c j ( i, + γB i ) r j ≤ ¯ N X j =0 Z j r j − r N − , where the Z j are as in Lemma 6.1, and r N − = (cid:26) , z ¯ N > r ¯ N − , z ¯ N = 1 . Proof. Note this is a trivial extension of Lemma 6.1 if z ¯ N > 1, as then r N − = 0. So suppose z ¯ N = 1; then P ¯ N − j =0 z j = N − 1. By the proof of Lemma 6.1 we know that ¯ N − X j =0 Z j − X i : j = j ( i, ( 12 c j ( i, + γB i ) − X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) r j ≥ ¯ N − X j =0 Z j − X i : j = j ( i, ( 12 c j ( i, + γB i ) − X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) r ¯ N − and ¯ N − X j =0 Z j − X i : j = j ( i, ( 12 c j ( i, + γB i ) − X I j 12 ( c j ( i,ℓ +1) − c j ( i,ℓ − ) ≥ . So if(51) ¯ N − X j =0 Z j − q X i =1 c ¯ N,i ≥ − γb, then we are done. Note the left hand side of (51) is a nonnegative integer. So suppose the left handside of (51) is zero; we will explain how to improve the estimates used in the proof of Lemma 6.1by at least − γb .First, if n = 0, there are no marked points, and B i = 0 for all i . Since we estimated γB i ≤ ,we have the improvement we need.So suppose n ≥ 1. If there is at least one point Q i appearing in a base locus in V • which is notone of the marked points P i , then similarly since B i = 0 and we always estimated γB i ≤ , wehave the improvement we need. So we may suppose that every Q i is a P j (hence q < n ).If there are no points Q i —that is, the base locus of V ¯ N is empty—then the weight vr space hascodimension 0 in H ( C, O ( m )), and we can easily show T vir is smaller than what is required by thenumerical criterion.So suppose there is at least one point Q in the base locus of V ¯ N . But now, on the one hand wehave by hypothesis that γB i ≤ γb < g − N +1 + ǫ ( N + 1) ≤ ; but in the proof of Lemma 6.1 we onlyestimated γB i ≤ ; so we see that we may improve our estimate by at least the desired amount. (cid:3) We proceed with Case B. We may argue just as we did in Lemma 6.2 to get(52) r N − g +1 + · · · + r N − ( 12 − γb ) r N − ≤ (cid:18) g − − ( 12 − γb ) (cid:19) N Combining (52) with (46), we obtain:(53) T vir ≤ g − + γbN ! u v + q X i =1 γB i r j ( i, ! (2 uv + v ) + ( duv + dv − g + 1) vr . IT STABILITY OF WEIGHTED POINTED CURVES 37 We desire that the leading coefficient should be smaller than what is required by the numericalcriterion by ǫ . That is, we want: g − + γbN ≤ g − γbN + 1 − ǫ ⇔ ǫ ≤ N ( N + 1) ( N − g + 3 − γb ) . (54)The right hand side of (54) is positive because the hypotheses of Case B imply that N ≥ g − γb < . Thus, when ǫ is sufficiently small (depending on N , ν , and B ) then (54)is satisfied.Next we consider Case C. Lemma 6.3.2 covers this situation, and we may argue just as we didin Lemma 6.2 to get(55) r N − g +1 + · · · + r N − r N − ≤ (cid:18) g − (cid:19) N Then, we want to arrange that g − N ≤ g − N + 1 − ǫ ⇔ ǫ ≤ N ( N + 1) ( N − g + 3) . (56)Since N ≥ g − ǫ sufficiently small.This completes our discussion of Cases B and C.Unfortunately, in Cases D and E, I know of no way to improve the bound of Lemma 6.1 in orderto get the leading coefficient of T vir small enough to use with the numerical criterion in this case!Therefore, at present I am forced to make the following hypotheses to ensure that Cases D and Edo not occur:(1) If n = 0, then N ≥ g − n ≥ g ≥ γb ≥ g − N + ǫ ( N + 1) or else γb < g − N + ǫ ( N + 1) < .Note that for n ≥ g = 0 or g = 1 and b > 0, we always have γb ≥ g − N + ǫ ( N + 1), so thishypothesis does not impose any restriction on d or N in these cases; we only need the linear systemembedding the curve to be complete.7. GIT stability of smooth pointed curves The stability theorem. We are ready to prove the main result: Theorem 7.1. Let γ = ν/ (2 ν − . Choose any ǫ > which is sufficiently small depending on d , g , and n . If n = 0 assume N ≥ g − . If n ≥ and g ≥ then suppose γb ≥ g − N + ǫ ( N + 1) orelse γb < g − N + ǫ ( N + 1) < . Consider a point in the incidence locus I parametrizing a smoothpointed curve ( C, { P i } ) embedded in P N by any (i.e. not necessarily pluricanonical) complete linearsystem of degree d . Assume also that the points P i are distinct.If n ≥ , suppose each b i ∈ B satisfies γb i < (this may not be covered by the previous as-sumptions). Let m = ( u + 1) v . Then for certain large values of m , the point of I parametriz-ing ( C, { P i } , C ⊂ P N ) is GIT stable for the SL ( N + 1) -action with the linearization specified by m ′ i = γb i m for each i . More precisely, there exist: (1) a positive integer u depending on d , g , n , and B , but not on the curve C , the points P i , orthe embedding C ⊂ P N (2) a function v ( u ) whose domain is all integers greater than u , and which depends on u , d , g , B and ǫ but not on the curve C , the points P i , or the embedding C ⊂ P N such that for any integers u ≥ u and v ≥ v ( u ) , the point of I parametrizing ( C, { P i } , C ⊂ P N ) isGIT stable for the SL ( N + 1) -action with the linearization specified by m ′ i = γb i m for each i .Proof. By (50) and (38) we have T = T vir + ∆ ≤ (cid:18) g − γbN + 1 − ǫ (cid:19) u v + q X i =1 γb i r j ( i, ! (2 uv + v )+( duv + dv − g + 1) vr + 72 duv + 3 dv = (cid:18) g − γbN + 1 − ǫ (cid:19) u v + n X i =1 γb i r j ( i, + 72 d ! uv + n X i =1 γb i r j ( i, + 3 d ! v ≤ (cid:18) g − γbN + 1 − ǫ (cid:19) u v + (cid:18) γb + 72 d (cid:19) uv + (2 γb + 3 d ) v ≤ (cid:18) g − γbN + 1 − ǫ (cid:19) u v + (cid:18) n + 72 d (cid:19) uv + ( n + 3 d ) v (57)Note that this bound depends on d , g , and n . Therefore, in the important special case when d = ν (2 g − a ), it also depends on ν and a . But we emphasize that in every case, this bounddoes not depend on the particular curve C , the points P i , the embedding C ⊂ P N , or the 1-PS λ .Recall the bound required in the numerical criterion:(58) (cid:18) g − γbN + 1 (cid:19) m − g − N + 1 m = (cid:18) g − γbN + 1 (cid:19) ( u v + 2 uv + v ) − g − N + 1 ( uv + v ) . We want to show that (57) is less than (58), or equivalently that0 ≤ (cid:18)(cid:18) g − γbN + 1 − ( g − γbN + 1 − ǫ ) (cid:19) u + (cid:18) g − γbN + 1 − γb − d (cid:19) u + (cid:18) g − γbN + 1 − γb − d (cid:19)(cid:19) v − (cid:18) g − N + 1 ( u + 1) (cid:19) v. (59)But the coefficient of u in the coefficient of v is ǫ > 0. So for all sufficiently large u , thepolynomial ǫu + (cid:18) g − γbN + 1 − γb − d (cid:19) u + (cid:18) g − γbN + 1 − γb − d (cid:19) is positive; but then for all sufficiently large v , the polynomial (cid:18) ǫu + (cid:18) g − γbN + 1 − γb − d (cid:19) u + (cid:18) g − γbN + 1 − γb − d (cid:19)(cid:19) v − (cid:18) g − N + 1 ( u + 1) (cid:19) v is positive, too. Once again, we emphasize that the size of u required depends on d , g , B , and ǫ but not on the particular curve C , the points P i , the embedding C ⊂ P N , or the 1-PS λ . Similarly IT STABILITY OF WEIGHTED POINTED CURVES 39 the size of v required depends on d , g , B , ǫ and u but not on the particular curve C , the points P i ,the embedding C ⊂ P N , or the 1-PS λ . (cid:3) Remark. Theorem 7.1 as stated does not establish stability for all large values of m , only forsome large values of m . Similarly, Gieseker’s stability proof ([Gies], Theorem 1.0.0) only establishesstability for some, not all, large values of m . In both cases it seems possible that one may be ableto use variation of GIT arguments to conclude stability for all sufficiently large values of m , but Ihave not checked this.7.2. Application to the construction of moduli spaces. My motivation for studying thisproblem was to give GIT constructions of moduli spaces of weighted pointed stable curves. Wedescribe the parameter spaces and linearizations for this application now.Let ( C, P , . . . , P n , A ) be a weighted pointed stable curve with n marked points. Write a := P a i ,and assume that 2 g − a > 0. Then for ν sufficiently large, ( ω C ( P a i P i )) ⊗ ν =: O C (1) is a veryample line bundle. Write V ν, A = H ( C, ( ω C ( X a i P i )) ⊗ ν ) = H ( C, O C (1)) d = deg O C (1) = ν (2 g − a ) N + 1 = dim V ν, A = ν (2 g − a ) − g + 1 P ( t ) = h ( C, O C ( t )) = dt − g + 1 . Then ( C, P , . . . , P n , A ) is represented by a point (in fact, many) inside the incidence locus I ⊂ Hilb ( P ( V ν, A ) , P ( t )) × Q n P ( V ν, A ) where the points in the second factor land on the curve in the firstfactor. In fact, ( C, P , . . . , P n , A ) lies in a locally closed subscheme of I corresponding to weightedpointed curves embedded by ( ω C ( P a i P i )) ⊗ ν .It is very important to note that d , N , and P ( t ) all depend on g , n , A and ν . So, even if g and n are held constant, if A or ν varies, one is moving between loci in different Hilbert schemes—that is,one is using different parameter spaces—and this is not variation of GIT in the sense of Thaddeusand Dolgachev and Hu. On the other hand, if g , ν , and A are held constant and only B varies, thisis VGIT in the sense of Thaddeus and Dolgachev and Hu.I claim the following theorem, although the proof is not completely written down yet: Theorem 7.2. Suppose g , n , d , ν , A , and B fit the setup of this paper and satisfy the hypothesesof Theorem 7.1. Let γ = ν/ (2 ν − . Suppose ν ≥ and d = ν (2 g − a ) , and let J be the locusin I where O (1) ∼ = ( ω ( P a i P i )) ν . Then: • If A = B and b i ≤ , then J//SL ( N + 1) ∼ = M g, A . • In particular, if A = B and + ǫ < b i < γ for each i = 1 , . . . , n , then J//SL ( N +1) ∼ = M g,n . How much of Theorem 7.2 has been checked? I believe all that is needed is extremely minorchanges to the Potential Stability Theorem of [BS]. It should still say that nothing “bad” can beGIT stable; the argument is very long, so I have not checked all of it, but it is also extremelyrobust, and I am very confident that it will work. One can easily write down the “Basic Inequality”when there are weighted marked points. I have done this, and checked that the condition on pointscolliding agrees exactly with the definition of M g, A , and that the argument that J ss is closedinside I ss still goes through. It then follows that all weighted pointed stable curves are GIT stable,justifying the title of this paper and completing the proof of Theorem 7.2. If g , ν and A are held fixed and the set of linearizing weights B is allowed to vary sufficiently farfrom A , the quotient may undergo a flip. Identifying these quotients is a project I am currentlyworking on. 8. Additional remarks (Director’s cut) In the course of my research I have learned a little bit more about this problem than just whatappears in this paper. In particular, I relate my proof to Gieseker’s in the unpointed case, andthis leads to a conjecture about the worst 1-PS. Next, I mention two suggestions for improving themain result, one that I expect would not work, and one that probably would.8.1. Comparison to Gieseker and Morrison’s results, and the worst 1-PS. We may inter-pret Gieseker’s proof ([Gies], Theorem 1.0.0) as the n = 0, q = 1 case of Theorem 7.1. This easilyleads to a coarse upper bound for T . The bound so obtained is not quite as good as the boundgiven in [Morr], Section 4 and used in Gieseker’s proof. However, after running the proof here, onecan perform their analysis on top of that, and the resulting bounds for the leading coefficient wouldthen agree.Kempf and Rousseau showed that when x is GIT-unstable, there is a “worst 1-PS” destabilizing x . This suggests the following strategy for proving stability: suppose for purposes of contradictionthat x is unstable, then find the worst 1-PS, then show that it is actually not destabilizing. Morrisonand I have never gotten this strategy to work in our situation (we can’t find the worst 1-PS, forthe same reason that we can’t compute the absolute weight filtration discussed in Section 1.3).However, we can describe the 1-PS for which it is most difficult to prove stability using ourmethods: it is the 1-PS for which there is only one point Q = P i in the base locus of V ¯ N , where b i is the largest value in B , every stage of the filtration is a complete sublinear series of H ( C, O (1)),and the weights are linearly decreasing (hence, uniquely determined by the conditions that theydecrease to zero and sum to 1).Of course, just because it is hard for us to show that this 1-PS is stable does not mean it isactually the worst 1-PS, but it certainly is a candidate. I believe it would be an interesting toshow either that this is the worst 1-PS, or exhibit another 1-PS which is worse. In the meantime,I mention this 1-PS for its value as a heuristic test for GIT stability for parameter spaces andlinearizations where this is currently unknown, and for testing putative stability proofs.8.2. Can we improve these results if we use a more complicated filtration than ˜ V • asscaffolding? Q: We only take the span of “three-layer” spaces V αuvs V (1 − α ) uvt V v . Could we getany further improvement by defining a filtration using spaces of the form V αuvs V βuvt V (1 − α − β ) uvw V v ? A: There may be room for improvement of our results, but when m is large, adding more layerswill not buy you anything. We never really asked what is the best way to produce a basis. Wealways began with a space of the form V αuvk V (1 − α ) uvV v k +1 having weight αr k uv + (1 − α ) r k +1 uv + vr and asked the question: for what choice of β j for j = 0 to N willcodim span( V αuvk V (1 − α ) uvk +1 , V β uv V β uv · · · V β N uvN V v ) be minimized?There are constraints. First, P Nj =0 β k = 1. Also, the weight of the second space in the spanshould be less than or equal to that of the first, so β r + · · · + β N r N ≤ αr k + (1 − α ) r k +1 . IT STABILITY OF WEIGHTED POINTED CURVES 41 These conditions give a polytope in β -space. Minimizing the multiplicity of each P i meansminimizing the linear function f ( β , . . . , β N ) = c ,i β + c ,i β + · · · + c N, β N over this polytope. The minimum must occur on the boundary, specifically at one (or more) of thevertices of the polytope, and these are precisely the “three-layer” spaces.The argument just given should be approximately true when m is very large and divisible (sothat all the exponents are integers), but it could break down badly for small m . So, for small m stability, we might want to consider filtrations which are much more complicated than those usedin this paper.8.3. Lower convex envelopes might give better bounds for T . Recall from Section 4.2 thatin the definition of ˜ X • , we do not minimize the multiplicity of each Q i . In fact, it is not hard tofind the minima; instead of using the functions s ( k, i ) and t ( k, i ), defined as “ ‘previous’ and ‘next’among values where the multiplicity of Q i jumps,” we should instead use σ ( k, i ) and τ ( k, i ), definedas “ ‘previous’ and ‘next’ among values where the multiplicity of Q i jumps which lie on the lowerenvelope of these.” That is, there are q lower envelopes to keep track of.It is possible that if one defines a filtration ˜ Y • using lower envelopes like this, one might be able toprove stability under a weaker hypotheses than those used in this paper. In particular I believe thatthis might yield a proof of asymptotic stability of canonically embedded smooth nonhyperellipticcurves. The obstacle is the proof of the analogue of Lemma 6.1. I can’t figure out how to get thisto work if you use lower envelopes instead of just the next value; instead of relating everything to c j +1 one would need to work with much later c ’s, and I don’t see how to do this. References [AG] Alexeev, V. and G. M. Guy. “Moduli of weighted stable maps and their gravitational descendants.”math.AG/0607683.[BM] Bayer, A. and Y. Manin. “Stability conditions, wall-crossing and weighted Gromov-Witten invariants.”math.AG/0607580.[BS] Baldwin, E. and D. Swinarski. “A geometric invariant theory construction of moduli spaces of stablemaps.” arXiv:0706.1381.[Gies] Gieseker, D. Lectures on Moduli of Curves. Tata Institute Lecture Notes, Springer, 1982.[G2] Gieseker, D. “Geometric invariant theory and applications to moduli problems.” 45–73, LNM ,Springer, 1983.[Gotz] Gotzmann, G. “Eine Bedingung f¨ur die Flachheit unda das hilbertpolynom eines graduierten Ringes.” Math. Z. (1978), 61–70.[HM] Harris, J. and I. Morrison. Moduli of Curves. Graduate Texts in Mathematics , Springer, 1998.[Hass] Hassett, B. “Moduli spaces of weighted pointed stable curves.” Adv. Math. no. 2 (2003), 316–352.[Morr] Morrison, I. “Projective Stability of Ruled Surfaces.” Inv. Math. (1980), 269–304.[MM] Mustat¸˘a, A. and A. Mustat¸˘a. “Intermediate moduli spaces of stable maps.” Invent. Math. no. 1(2007), 47–90.[GIT] Mumford, D., Fogarty, J. and F.C. Kirwan. Geometric Invariant Theory. Third Edition. Springer, 1994.[Mum] Mumford, D. “Stability of Projective Varieties.” Enseignement Math. (2) (1977), no. 1-2, 39–110. Department of Mathematics, Columbia University, New York NY 10027, USA E-mail address ::