aa r X i v : . [ m a t h . AG ] N ov On linear series with negative Brill-Noether number
Nathan PfluegerJuly 22, 2018
Abstract
Brill-Noether theory studies the existence and deformations of curves in projective spaces; its basicobject of study is W rd,g , the moduli space of smooth genus g curves with a choice of degree d line bundlehaving at least ( r + 1) independent global sections. The Brill-Noether theorem asserts that the map W rd,g → M g is surjective with general fiber dimension given by the number ρ = g − ( r + 1)( g − d + r ),under the hypothesis that 0 ≤ ρ ≤ g . One may naturally conjecture that for ρ <
0, this map is genericallyfinite onto a subvariety of codimension − ρ in M g . This conjecture fails in general, but seemingly onlywhen − ρ is large compared to g . This paper proves that this conjecture does hold for at least oneirreducible component of W rd,g , under the hypothesis that 0 < − ρ ≤ rr +2 g − r + 3. We conjecture thatthis result should hold for all 0 < − ρ ≤ g + C for some constant C , and we give a purely combinatorialconjecture that would imply this stronger result. Throughout this paper, a curve will always mean a complete algebraic curve over C , with at worst nodes assingularities.Brill-Noether theory studies the ways curves can lie in projective spaces. One of the principle objects ofstudy is the moduli space W rd,g , which parameterizes curves of genus g together with a chosen line degree d line bundle with at least ( r + 1) independent global sections. The geometry of this space is well-understoodover general curves; in particular the Brill-Noether theorem [7] states that when r ≥ g − d + r ≥ W rd,g → M g is either empty or has dimension given by the Brill-Noether number ,traditionally denoted ρ and defined as follows. ρ ( g, d, r ) = g − ( r + 1)( g − d + r )Furthermore, the general fiber is empty if and only if ρ ≥
0. This paper considers the extension of theBrill-Noether theorem to the case ρ <
0, i.e. to non-general curves. The main result is the following.
Theorem 1.1.
Suppose that g, d, r are positive integers with g − d + r ≥ and > ρ ≥ − rr +2 g + 3 r − .Then W rd,g has a component of dimension dim M g + ρ , whose image in M g has codimension equal to − ρ ,and whose general member has rank exactly r . The assumptions r ≥ g − d + r ≥ g − d + r ≤ ρ >
0, while if g − d + r = 1or r = 0 (these situations are dual to each other) then W rd,g is empty if ρ < M g + ρ is a lower bound on the dimension of any component of W rd,g (we will see one proofin section 3, by combining formula 1 and lemma 3.5), so this theorem asserts that this bound is achieved fornot-too-negative values of ρ .The proof proceeds by induction on the genus. The statement of 1.1 is not suitable for induction; weinstead introduce the notion of twisted Weierstrass points , and prove a suitable generalization in this context.The method is based on the limit linear series techniques introduced by Eisenbud and Harris [4] to constructcertain Weierstrass points. As a second application of our techniques, we also prove that the naive dimension1stimate for the number of moduli of a Weiestrass point always fails when the Semigroup does not satisfy acombinatorial condition called primitivity (Theorem 7.2).The outline of this paper is as follows. Section 2 discusses background and previous results, and statessome conjectures. Section 3 introduced the notion of a twisted Weierstrass point corresponding to a partition P ; we define moduli spaces W g ( P ) of twisted Weierstrass points on genus g curves and show that studying g rd s on genus g curves is equivalent to studying twisted Weierstrass points on genus g curves correspondingto the “box-shaped” partition (( g − d + r ) r +1 ). Section 5 describes a construction, using the theory of limitlinear series, of twisted Weierstrass points in genus g + 1 from twisted Weierstrass points in genus g . Section6 defines a combinatorial invariant called the difficulty of a partition, and shows how bounding this invariantimplies the existence of dimensionally proper twisted Weierstrass points. Sections 7 and 8 demonstrate thistechnique by bounding the difficulty of two different sorts of partitions. Section 7 reproves a theorem ofEisenbud and Harris on dimensionally proper Weierstrass points , and then proves theorem 7.2, showingthat a primitivity hypothesis in that theorem cannot be removed. Finally, section 8 gives a bound on thedifficulty of box-shaped partitions sufficient to prove theorem 1.1. As the numbers g, d, r vary (constrained by g − d + r ≥ W rd,g exhibit two very different sortsof behavior. For 0 ≤ ρ ≤ g , the situation is well-understood: W rd,g is irreducible, maps subjectively to M g ,and has general fiber of dimension ρ . On the other hand, when − ρ ≫ g −
3) + ρ fails dramatically. Indeed, many natural families of curves (such as complete intersections, determinantalcurves, and curves on rational surfaces) have degree and genus such that ρ is extremely negative, and yetthese families have rather large dimension. This phenomenon, observed in numerous examples, has led tothe following folklore conjecture, sometimes called the rigid curves conjecture. Conjecture . For all r , there is a positive number C ( r ) such that whenever W rd,g is nonempty, all of itscomponents have dimension at least C ( r ) g .Observe that since dim M g is of course 3 g −
3, and a genus g curve has a a g -dimensional space of linebundles of degree d , the dimension of W rd,g is always less than 4 g . So another way to state this conjectureis the following: for there is a positive number C ( r ) such that whenever W rd,g is nonempty, C ( r ) < dim W rd,g g < W rd,g ).This conjecture predicts that there is a sort of “phase transition” as ρ moved from slightly negative valuesto very negative values, where the Brill-Noether dimension estimate begins to fail and the natural tendencyof embedded curves to vary in families of dimension linear in g (in addition to the dim P GL r +1 degrees offreedom from projective space itself) begins to dominate. The question this paper aims to address is: wheredoes this phase transition occur ?Anecdotal evidence suggests that the transition occurs at a constant multiple of g . For example, the sim-plest case of an embedded curve violating the Brill-Noether dimension estimate is the complete intersectionof a quadric and a quartic surfaces in P . In this case, ( g, d, r ) = (9 , ,
3) so ρ = − W , is 17, but an elementary calculation shows that in fact dim W r , = 18. So this counterexampleoccurs at ρ = − g + 2.Eisenbud and Harris [5] proved that when ρ = −
1, the space W rd,g is irreducible of the expected dimension,and that its image in M g is a divisor. Edidin [3] showed that in the case W rd,g has all components of theexpected dimension, mapping finitely to M g . Eisenbud and Harris claimed in their initial paper on limitlinear series [6] a result of the same form as our theorem 1.1 was forthcoming, but never published a proof. The proof is essentially the same as theirs, although we circumvent the analysis of limit canonical series. This conjecture is usually phrased in terms of components of the Hilbert scheme, but this form is essentially the same. ρ = − g . Itsmain defect is that it only asserts the existence of some component of W rd,g that behaves as expected. Thereason for this restriction is that our method of proof proceeds by smoothing certain reducible curves; thismethod cannot detect and components of W rd,g whose images in M g are compact.Our theorem 7.2 is the first step towards answering a different but very analogous question about Weier-strass points. Since non-primitive semigroups occur in every genus with weights as low as roughly g , thisshows that the analogous phase transition for Weierstrass points seems to occur when the expected codi-mension is roughly g . We elaborate considerably on the analogous questions for Weierstrass points in[13].We conclude this section with a general conjecture uniting questions about W rd,g with questions aboutWeierstrass points. See the following section for the definition of W g ( P ) and an explanation of how it isrelated to W rd,g . Conjecture . Let P be a partition and g a positive integer. Let X be any component of W g ( P ), regardedas a subvariety of P ic g × M g M g, . There exist two positive functions A ( r ) and B ( r ) of r with 0 < A ( r )
4, such that: • codim X ≤ min( B ( r ) g, | P | ). • If | P | ≤ A ( r ) g , then codim X = | P | . Question . In the range A ( r ) g ≤ | P | ≤ B ( r ) g , is there a purely combinatorial procedure to determine if W g ( P ) has any components of codimension | P | ? Theorem 1.1 can be deduced from a slightly stronger result about pointed curves. This section defines therelevant notion, that of twisted Weierstrass points , and discusses some basic aspects of their moduli and theconnection to theorem 1.1.Every point p on a smooth curve C determines a numerical semigroup called the Weierstrass semigroup of the point; it consists of those integers n such that C has a rational function of degree n whose only poleis at p . For all but finitely many points on a given curve C , this semigroup is { , g + 1 , g + 2 , · · · } ; the otherpoints are called Weierstrass points . See [2] for history and applications of Weierstrass points. Consider thefollowing generalization.Let C be a smooth curve, L a degree 0 line bundle on C , and p ∈ C a point. The twisted Weierstrasssequence of the triple ( C, L , p ) is the following set of nonnegative integers. S ( C, L , p ) = { n ∈ Z ≥ : h ( L ( np )) > h ( L (( n − p )) } In other words, the twisted Weierstrass sequence is the set of possible pole orders at p of rational sectionsof L that are regular away from p . In the special case L = O C , the twisted Weierstrass sequence is the classicalWeierstrass semigroup. By the Riemann-Roch formula, the complement of S has precisely g elements, where g is the genus of C . If twisted Weierstrass sequences are given the obvious partial ordering, then they areupper semi-continuous families; therefore the general twisted Weierstrass sequence is simply S = { g, g + 1 , g + 2 , · · · } . A triple ( C, L , p ) with a different sequence is called a twisted Weierstrass point. We can and will describe a twisted Weierstrass sequence using the (equivalent) data of a partition.Namely, the twisted Weierstrass partition P ( C, L , p ) is given by the multiset { ( n + g ) − s n } (restricted topositive entries), where the twisted Weierstrass sequence is s < s < s < · · · . Alternatively, one canidentify twisted Weierstrass sequences with Schubert cycles, which are identified with partitions in the usualway. This connection will be made more explicit after the definition below.3 g ( g − d + r )( r + 1) W g ( P ) ∼ = M g, W g ( P ) ∼ = { Weierstrass points } W g ( P ) ∼ = W rd,g × M g M g, Figure 1: Three examples of partitions and the geometric interpretation of W g ( P ). Definition 3.1.
Given a nonnegative integer g and a partition P , let ˜ W g ( P ) denote the moduli space oftriples ( C, L , p ), where C is a smooth curve, L is a line bundle of degree 0 and p ∈ C , such that P ( C, L , p ) = P .Let W g ( P ) denote the closure of ˜ W g ( P ) in P ic g × M g M g, .The space ˜ W g ( P ) can also be described in terms of Schubert cycles, as follows. Any family of curveswith marked point and line bundle corresponds to the following data: • A family of curves π : C → B , • A section s : B → C , and • A line bundle L on C .These data determine a filtration of vector bundles on B , given by E k = π ∗ ( L (( k − / L ( − Σ)), whereΣ is the divisor given by the image of s . By Grauert’s theorem ([9] corollary 12.9), each E k is a vector bundleof rank k on B . In addition to these, there is also a rank g vector bundle on B given by F = π ∗ ( L ((2 g − p )),with an obvious inclusion F ֒ → E g . This inclusion induces a section t : B → G to the Grassmannian bundle G of g -planes in E g . The filtration of E g given by E ⊂ E ⊂ · · · ⊂ E g defines, for each partition P , anopen Schubert cycle ˜Σ P ⊆ G , of codimension | P | (see [8] section 1 . B corresponding to points of ˜ W g ( P ) are precisely the inverse image t − (Σ P ).The description of ˜ W g ( P ) in terms of Schubert cycles gives the following bound on its local dimensionat any point. dim ( C, L ,p ) ˜ W ( P ) ≥ (4 g − − | P | (1) Definition 3.2.
A point ( C, L , p ) ∈ ˜ W g ( P ) where equality holds in 1 is called a dimensionally proper point . Example . Let P = ( g ). Then ( C, L , p ) ∈ ˜ W g ( P ) if and only if h ( L ) = 1 and h ( L ( gp ) = 1). This is istrue if and only if L = O C and p is not a Weierstrass point. So ˜ W g ( P ) is isomorphic to the complement in M g, of the locus of Weierstrass points, and W g ( P ) ∼ = M g, . Therefore the local dimension at each point is(3 g −
2) = (4 g − − | P | , so every point is dimensionally proper. Example . Let P = ( g W g ( P ) consists of triples ( C, L , p ) such that h ( L ) = 1 , h (( g − p ) = 1,and h ( gp ) = 2. In other words, this is the locus in M g, of simple Weierstrass points. This is ´etale-locallyisomorphic to M g , so every point has local dimension (3 g −
3) = (4 g − −| P | , so all points are dimensionallyproper.We will now study twisted Weierstrass points with the particular type of partition that will be relevant totheorem 1.1. Let P = ( m n ) (i.e. the number m occurs n times). This partition corresponds to the followingtwisted Weierstrass sequence. S = { g − m, g − m + 1 , · · · , g − m + n − , g − m + n − , g + n, g + n + 1 , g + n + 2 , · · · } C, L , p ) ∈ ˜ W g ( P ) if and only if the following conditions hold. • h ( L (( g − m − p )) < h ( L ( g − m ) p ) = 1 • h ( L ( g − m + n − p ) = h ( L ( g + n − p ) = n These conditions are equivalent to saying that L ′ = L (( g + n − p ) is a line bundle of degree ( g − m + n − n −
1, such that p is not a ramification point for either the complete linear series |L ′ | or its dual | ω C ⊗ L ′∧ | . See [1] appendix C for a definition of ramification points, and a proof that there are finitelymany of them for a given linear series. Note that this is not true in positive characteristic.Since any linear series has a finite number of ramification points, this means that for any line bundle M on C of degree d = ( g − m + n −
1) and rank r = ( n − C, M ( − dp ) , p ) is a point of ˜ W g ( P ) forall but finitely many points p ∈ C . The upshot of this is the following. Lemma 3.5.
Let g, d, r be integers, and let ˜ W rd,g ⊂ P ic dg consist of those pairs ( C, L ) where L is a linebundle on C with degree d and h ( L ) = r + 1 . Let P be the partition (( g − d + r ) r +1 ) . Then there is a map f : ˜ W g ( P ) → ˜ W rd,g ( C, L , p ) ( C, L ( dp )) which is surjective, and whose fiber over any point ( C, L ) ∈ ˜ W rd,g is isomorphic to C with finitely manypunctures. Notice that, in the notion of the lemma, | P | = ( r + 1)( g − d + r ) = g − ρ ( g, d, r ). It follows from thisthat the map f in the lemma sends dimensionally proper points to dimensionally proper points. Thus tostudy dimensionally proper line bundles on curves is equivalent to studying dimensionally proper twistedWeierstrass points given by “box-shaped” partitions. This will be the object of the remainder of the paper. Remark . Notice that twisted Weierstrass points have a duality property: namely if P ∗ is the dual partitionof P (that is, P ∗ n = |{ m : P m > n | ), then ˜ W g ( P ) ∼ = ˜ W ( P ∗ ), via the map ( C, L , p ) ( C, ω C ( − (2 g − p ) ⊗L ∧ , p ). This generalizes the fact that ˜ W rd,g ∼ = ˜ W g − d + r − g − − d,g via L 7→ ω C ⊗L ∧ (via the correspondence discussedabove), since the dual partition of (( g − d + r ) r +1 ) is (( r + 1) g − d + r ). This duality is reflected, for example,in the two perspectives by which one typically studies classical Weierstrass points: in terms of pole order ofrational functions or in terms of ramification of the canonical series. Question . Let µ ( P, g ) be the maximum codimension of a component of W g ( P ) (or −∞ if there are none).When is µ ( P, g ) < | P | ? Is there a purely combinatorial description of which partitions P and integers g givestrict inequality?We will define in section 6 a function δ ( P ) of partitions such that µ ( P, g ) = | P | whenever g ≥ ( | P | + δ ( P )). Bounding this function will give theorem 1.1. First we describe the smoothing argument whichunderlies the definition of δ ( P ). To prove the existence of certain twisted Weierstrass points, it will be necessary to allow the curves todegenerate to singular curves, and to have a suitable notion of limits of the twisted Weierstrass points. Sucha notion is provided by limit linear series, as introduced by Eisenbud and Harris [6]. We begin by recallingthe relevant definitions; see [11] for an expository treatment.A linear series of degree d and rank r on a smooth curve C , also called a g rd , is a pair L = ( L , V ), where L is a degree d line bundle and V ⊆ H ( L ) is an ( r + 1)-dimensional vector space of sections. The moduli spaceof genus g curves with a chosen g rd is denoted G rd,g . Given a linear series L and a point p ∈ C , the vanishingsequence of L at p is the set of integers n such that V contains a section vanishing to order exactly n at p . The only difference from the definition of W rd,g is that here exact equality is required. r + 1) distinct integers; it is usually denoted a L ( p ) = ( a L ( p ) , a L ( p ) , · · · , a Lr ( p ))where a L ( p ) < a L ( p ) < · · · < a Lr ( p ). Equivalent to the vanishing sequence is the ramification sequence α L ( p ),given by α Li ( p ) = a Li ( p ) − i . Most authors work with the ramification sequence rather than the vanishingsequence; we will work almost entirely with the vanishing sequence since it is slightly more notationallyconvenient for our purposes.Let ˜ G rd,g ( a ) ⊆ G rd,g × M g M g, denote the space of triples ( C, L, p ) such that the vanishing sequence of L at p is precisely a . Let G rd,g ( a ) denote the space of such triples such that the vanishing sequence of L at p is atleast a . More generally, ˜ G rd,g ( a , a , · · · , a s ) ⊆ G rd,g × M g M g,s denotes the space of tuples ( C, L, p , · · · , p s )with vanishing sequence a i at p i .The theory of limit linear series works best for curves of compact type. A nodal curve X is called compacttype if its dual graph (that is, the graph whose vertices are the components of X and whose edges correspondto the nodes) has no cycles (equivalently, the Jacobian of X is compact). Recently, Amini and Baker gavea definition of limit linear series for arbitrary nodal curves, but there does not yet exist a moduli space forthese more general limit linear series. We will use the original definitions of Eisenbud and Harris. Definition 4.1.
Let X be a curve of compact type. A refined limit linear series L of degree d and rank r (or limit g rd ) on X consists of a g rd L C on each connected component C of X (called the C -aspect of L ), suchthat for each node p ∈ X joining components C and C , the following compatibility condition holds. a L C i ( p ) + a L C r − i ( p ) = d for i = 0 , , , · · · , r The vanishing sequence a L ( p ) of a limit series at a smooth point p is the vanishing sequence of the C -aspect of L , where p ∈ C .Eisenbud and Harris also define coarse limit series to be a collection of C -aspects such that the compat-ibility condition holds as an inequality. We will not need to consider coarse limits in this paper. Note thatOsserman [12] gave a different definition of limit linear series that is more suitable for the construction ofa global moduli scheme. His definition is equivalent to the Eisenbud-Harris definition in the special case ofrefined limit series. Eisenbud and Harris do not construct a global moduli space of limit g rd s over all of M g ,but instead construct a local moduli space. More precisely, they construct a moduli space of (refined) limitlinear series over a Kuranishi family of any curve of compact type. In either formalism, the existence of asuitable moduli space, plus a dimension bound on it coming from Schubert conditions, implies the following“regeneration theorem.” To state it first requires one more definition. Definition 4.2.
A marked curve (
X, p , · · · , p s ) with a linear series L of degree d and rank r is called dimensionally proper if the local dimension of G rd,g ( a L ( p ) , · · · , a L ( p s )) is exactlydim M g,s + ρ − s X i =1 r X j =0 ( a Lj ( p i ) − j ) . Theorem 4.3 (Corollary 3.7 of [6]) . Let L be a limit g rd on a curve X of compact type, and p , · · · , p s ∈ X are smooth points. Suppose that each component C of X is dimensionally proper with respect to all the pointsof C that are nodes in X and all the marked points p i that lie on C . Then there exists a smooth markedcurve ( X ′ , p ′ , · · · , p ′ s ) with a dimensionally proper g rd L ′ such that a L ′ ( p ′ j ) = a L ( p j ) for all j . This markedcurve and linear series lies in a one-parameter family whose limit is the marked curve ( X, p , · · · , p s ) withlimit linear series L . Definition 4.4.
A marked curve (
X, p , · · · , p s ) of compact type with a refined limit L satisfying thehypotheses of theorem 4.3 will also be called dimensionally proper .The following lemma reinterprets the data of a twisted Weierstrass point in a manner that makes thetheory of limit linear series applicable. Whenever we say that a sequence a is “at least” another sequence a ′ , we mean that a i ≥ a ′ i for each i . emma 4.5. Let r ≥ g − be an integer. Then ˜ W g ( P ) ∼ = ˜ G rr + g,g ( a ) , where a = ( a , a , · · · , a r ) is thesequence given by a i = i + P r − i , via the maps ( C, L , p ) ( C, |L (( r + g ) p ) | , p ) and ( C, ( L , H ( L )) , p ) ( C, L ( − ( r + g ) p ) , p ) .Proof. Since r + g ≥ g − |L (( r + g ) p ) | is indeed a g rr + g ; unraveling definitions shows that the vanishingsequence at p is a . So this is a well-defined map to ˜ G rr + g,g ( a ). In reverse, every g rr + g is necessarily complete,hence of the form |L| for some L ; then ( C, L ( − ( r + g ) p ) , p ) indeed lies in ˜ W g ( P ) by the same calculation.Therefore, we have the following notion of a limit twisted Weierstrass point : a curve X of compact type,with marked smooth point p and refined limit g rr + g L (where r ≥ g −
1) and the vanishing sequence a asdescribed above. Constructing such object, and proving that they are dimensionally proper (in the sense ofdefinition 4.4) will suffice to construct dimensionally proper twisted Weierstrass points (on smooth curves). The object of this section is to demonstrate how dimensionally proper twisted Weierstrass points on genus g curves give rise to dimensionally proper twisted Weierstrass points on curves of genus g + 1, with slightlymodified partitions. The construction proceeds by adjoining an elliptic curve to the genus g curve, andsmoothing the resulting nodal curve. The basic technical tool is the regeneration theorem for limit linearseries, as introduced by Eisenbud and Harris [6] (see [11] for a readable expository account and [12] for amore recent perspective that is more applicable in characteristic p ).The following lemma is a slight restatement of proposition 5.2 from [4]. It is the basic tool in our inductiveconstructions. Lemma 5.1.
Fix integers r, d and two sequences b = ( b , b , · · · , b r ) and c = ( c , c , · · · , c r ) such that b i + c r − i = d − for each index i . Then for any genus curve E with distinct points p, q and degree d line bundle L , thereexists a unique linear series L = ( L , V ) on E such that for all i the following inequalities hold. a Li ( p ) ≥ b i a Li ( q ) ≥ c i Proof.
For all pairs of indices ( i, j ) with i + j < d , define the following vector space of sections of L . W i,j = im (cid:0) H ( L ( − ip − jq )) ֒ → H ( L ) (cid:1) That is, W i,j consists of those sections vanishing to order at least i at p and at least order j at q . W i,j has dimension d − i − j , by Riemann-Roch.Separate the indices { , , , · · · , r } into the longest intervals intervals I k = { u k , u k +1 , · · · , v k } , such that b u k , b u k +1 , · · · , b v k are consecutive integers. Let m be the number of these intervals, so that { , , · · · , r } is adisjoint union of I , I , · · · , I m . Then u = 0 , v m = r , and v k + 1 = u k +1 . Define, for each k ∈ { , , · · · , m } , V k := W b uk ,c r − vk . Observe that the dimension of V k is d − b u k − b r − v k = d − b u k − ( d − b v k = 1+ b v k − b u k =1 + v k − u k = | I k | .Let V be the sum of all the spaces V k . We claim that V satisfies the conditions of the lemma, andthat it is the unique such vector space of sections. First, we verify that V satisfies the conditions of thelemma. By the Riemann-Roch formula, each vector space V k has the following orders of vanishing at p : { b u k , b u k +1 , · · · , b v k − , b ′ v k } , where b ′ v k = (cid:26) b v k + 1 if L ∼ = O E (( b v k + 1) p + ( c r − v k ) q ) v k otherwise.7n all cases, b ′ v k < b u k +1 , so the sections of any two different spaces V k have disjoint sets of orders ofvanishing at p . It follows that the orders of vanishing at p of sections in V is the disjoint union m [ k =1 { b u k , b u k +1 , · · · , b v k − , b ′ v k } . In particular, the dimension of V is P mk =1 | I k | = r + 1, and its vanishing sequence at p is at least b u , b u +1 , · · · , b v , b u , · · · , b v , · · · , b v m , which is identical to b , b , · · · , b r . Symmetric reasoning shows thatthe orders of vanishing of V at q are at least c , c , · · · , c r . So V satisfies the conditions of the lemma.Now suppose that V ′ satisfies the conditions of the lemma. Then the sections of V ′ vanishing to order atleast a u k have codimension at most u k , and those vanishing to order at least b r − v k at q have codimension atmost r − v k , hence V ′ ∩ V k has codimension at most r +( u k − v k ) and thus dimension at least 1+ v k − u k = | I k | .Therefore this intersection must be all of V k . Thus V ′ ⊇ V , and dim V ′ = dim V , so in fact V ′ = V . So V is the unique such vector space of sections.In fact, examining the end of the proof of lemma 5.1, we have actually proved the following. Lemma 5.2.
Let L = ( L , V ) be a linear series as described in lemma 5.1. Then the actual orders ofvanishing of L are as follows: a Li ( p ) = (cid:26) b i + 1 if ( b i + 1) ∈ Λ and b i +1 > b i + 1 b i otherwise a Li ( q ) = (cid:26) c i + 1 if ( c i + 1) ∈ ( d − Λ) and c i +1 > c i + 1 c i otherwisewhere Λ is the arithmetic progression { n : L ∼ = O E ( np + ( d − n ) q ) } . (cid:3) For notational convenience, we make the following definition. In the following definition and the remainderof this paper, an arithmetic progression will be a proper subset Λ of the integers such that the set of differencesof elements of Λ is closed under addition. In particular, Λ may be empty or have only a single element, butit may not be all of Z . Definition 5.3.
Let a = ( a , a , a , · · · , a r ) be a strictly increasing sequence of integers, and let Λ be anarithmetic progression (as defined above). Define the upward displacement a +Λ and downward displacement a − Λ of a with respect to Λ as follows.( a +Λ ) i = ( a i + 1 if a i + 1 ∈ Λ and a i +1 > a i + 1 a i otherwise( a − Λ ) i = ( a i − a i ∈ Λ and a i − < a i − a i otherwiseIn these expressions i is an index in { , , · · · , r } and for notational convenience a − = −∞ and a r +1 = ∞ (when these appear on the right side). This definition is interpreted visually, using partition notation, infigure 2.Informally, the upward displacement “attracts” the sequence upward to the progression Λ, while thedownward displacement “repels” the sequence downward away from Λ. Another interpretation is that dis-placement forgets, for each pair { λ − , λ } (where λ ∈ Λ) which of these two numbers is in the sequence,remembering only how many (0, 1, or 2) are present.The following lemma reformulates the previous two lemmas in the language of limit linear series.8 emma 5.4.
Let C be a smooth curve, p ∈ C a point, E a genus curve, and p , q two distinct points on E . Let X be the the nodal curve obtained by attaching C and E at p and p . Let L C = ( L C , V C ) be a g rd on C , and L E a degree ( d + 1) line bundle on E . Then there exists a unique limit g rd +1 L on X with thefollowing properties.1. The C -aspect of L is L C + p (that is, L C with a base point added at p ).2. The E -aspect of L has line bundle L E .3. For all i ∈ { , , · · · , r } , a Li ( q ) ≥ a L C i ( p i ) .Let Λ = (cid:8) n : L E ∼ = O E ( nq + ( d + 1 − n ) p ) (cid:9) ; then the vanishing sequence of L is precisely a L ( q ) = (cid:0) a L C ( p ) (cid:1) +Λ ,and L is a refined limit linear series if and only if ( a L C ( p )) − Λ = a L C ( p ) .Proof. For i ∈ { , , · · · , r } , let b i = d − a L C r − i ( p ) and let c i = a L C i ( p ). Then of course b i + c r − i = ( d + 1) − i , so a suitable E -aspect for L exists and is unique by lemma 5.1. Lemma 5.2 shows that vanishingsequence of this E -aspect (and therefore of L ) is c +Λ as claimed. The vanishing sequence of the E -aspect at p is b +( d +1 − Λ) , and L is refined if and only if this is equal to b . But observe that since b i = d − c r − i , this isequivalent to c − Λ = c , as claimed.By allowing the curve X and limit linear series L to vary, this construction on two-component curvesgives the following result on dimensionally proper linear series with specified ramification. Proposition 5.5.
Suppose that a = ( a , a , · · · , a r ) is a strictly increasing sequence of nonnegative integers,and Λ is an arithmetic progression (as defined above) such that a − Λ = a and a +Λ differs from a in at most twoplaces. If ˜ G rd,g ( a ) has a dimensionally proper point, belonging to a component mapping to M g, with generalfiber dimension d , then ˜ G rd +1 ,g +1 ( a +Λ ) has a dimensionally proper point, belonging to a connected componentmapping to M g, with general fiber dimension at most d + 1 (if a = a +Λ ) or at most d (if a = a +Λ ).Proof. Assume without loss of generality that Λ is as small as possible. This means that if a +Λ differs from a in two places, then Λ is the progression generated by those two values ( a +Λ ) i that are greater than a i ; if a +Λ differs from a in one place, then Λ is a single element; and if a +Λ = a , then Λ is empty.Let ( C, L C , p ) be a dimensionally proper point of ˜ G rd,g ( a ). Let ( E, L , p , q ) be a twice-pointed ellipticcurve, chosen in the following way. • If Λ is infinite, say { n : n ≡ m mod d } for some m and d , then let L = O E ( mq + ( d + 1 − m ) p ) andselect p , q so that ( p − q ) is a d -torsion point on Pic ( E ). • If Λ has a single element m , then let L = O E ( mq + ( d + 1 − m ) p ) and choose p , q so that ( p − q ) isnot torsion. • If Λ is empty, then choose L distinct from all line bundles O E ( mq + ( d + 1 − m ) p ) and choose p , q arbitrarily.Let X be the nodal curve described in lemma 5.4, L the limit g rd +1 on X described in that lemma, and L E its E -aspect. Since a − Λ = a , this series is refined. We shall show that ( X, L, q ) is dimensionally proper in the sensedescribed in section limits. By assumption, the C -aspect L C + p , with the marked point p , is dimensionallyproper. So it suffices to prove that ( E, L E , p , q ) is dimensionally proper. Let δ be the number of places where a +Λ differs from a . An elementary calculation shows that this is equivalent to showing that the local dimensionof G rd +1 , (cid:16) a L E ( p ) , a L E ( q ) (cid:17) at ( E, L E , p , q ) is 3 − δ . Now, the map f : G rd +1 , (cid:16) a L E ( p ) , a L E ( q ) (cid:17) → Pic d +11 , (that is, to the moduli space of twice-marked genus 1 smooth curves with a chosen degree ( d + 1) line bundle)is set-theoretically injective by lemma 5.1. By lemma 5.2, the image of f consists of all ( E ′ , L ′ , p ′ , q ′ ) suchthat the arithmetic progression Λ ′ = { n : L ′ ∼ = O E ′ ( nq ′ + ( d + 1 − n ) p ′ } contains Λ. By a little casework,the dimension of the image is 3 − δ . It follows that ( E, L E , p , q ) is dimensionally proper, and therefore so9s ( X, L, q ). By theorem 4.3, ˜ G rd +1 ,g +1 ( a +Λ ) has a dimensionally proper point. The bound on the dimensionof fibers over M g, follows by considering the semicontinuity of fiber dimension for the map from the spaceof limit linear series (on a 1-parameter family degenerating to ( X, p )) over ¯ M g, . As before, we will use the following convention: an arithmetic progression will mean a proper subset Λ ⊂ Z such that Λ − Λ is closed under addition. In particular, Λ may be empty or have a single element, butit cannot be all of Z . Also, we adopt the following notational conventions: the partition elements are P ≥ P ≥ · · · ≥ P n , and P k is defined to be 0 for k > n and ∞ for k < Definition 6.1.
Let P be a partition and Λ an arithmetic progression. Then define the upward displacement P +Λ and downward displacement P − Λ of P with respect to Λ as follows.( P +Λ ) i = (cid:26) P i + 1 if ( P i − i ) ∈ Λ and P i − > P i P i otherwise( P − Λ ) i = (cid:26) P i − P i − i − ∈ Λ and P i +1 < P i P i otherwiseThis definition is much easier to understand visually; it is illustrated in figure 2. Here the partition P is represented by its Young diagram, and the arithmetic progression Λ is represented by an evenly spacedsequence of diagonal lines. Then the two displacements are obtained by finding all places where the line ofΛ meet the corners of P , and either “turning the corners out” (in the case of P +Λ or “turning the corners in”in the case of P − Λ ).Observe that if P ′ is any other partition such that P − Λ ≤ P ′ ≤ P +Λ , then the upward and downwarddisplacements of P ′ are the same as those of P (with respect to Λ). So displacement can be regarded as asort of projection to the nearest partition that is stable with respect to the given arithmetic progression.Call two partitions P , P linked if there is an arithmetic progression Λ (proper but possibly empty orsingleton) such that P is the upward displacement of P and P is the downward displacement of P . Notethat this implies that P is its own downward displacement and P is its own upward displacement. Saythat P and P are k - linked if they are linked and | P | − | P | = k .It is easy to verify that if P , P are any two partitions with P ≤ P , then P can be connected to P bya sequence of 1-linked partitions. Indeed, the arithmetic progressions can be taken to be singletons.As we saw in the previous section, we are particularly interested in 2-linked partitions. More specifically,we are interested in partitions that can be joined by a path of 1-linked and 2-linked pairs, using as few1-linked pairs as possible. Therefore make the following definition. Definition 6.2.
Call a sequence of partitions of increasing sum valid if any two adjacent partitions in thesequence are 1-linked or 2-linked. Define the difficulty δ ( P ) of a partition P to be the fewest number of1-linked adjacent pairs in a valid sequence from the empty partition to P .With this definition, we can now state the following lemma, which relates difficulties of partitions todimensionally proper twisted Weierstrass points. Lemma 6.3.
Let P be any partition and Λ an arithmetic progression (proper and possibly empty or sin-gleton). If | P +Λ | − | P − Λ | ≤ and ˜ W g ( P − Λ ) has a dimensionally proper point lying in a fiber over M g, ofdimension d , then ˜ W g ( P +Λ ) has a dimensionally proper point lying in a fiber of dimension at most ( d + 1) ,and at most d if P +Λ = P − Λ .Proof. Without loss of generality, let P = P − Λ . Let ( C, L , p ) ∈ ˜ W g ( P ) be a dimensionally proper point.By lemma 4.5, this can also be regarded as a dimensionally proper point of ˜ G rr + g,g ( a ), where r = g , for a i = i + P r − i . Let Λ ′ = Λ + ( r + 1). Then it follows that, again by lemma 4.5, ˜ W g ( P +Λ ) ∼ = ˜ G rr + g,g ( a +Λ ′ ). Since10 = (8 , , , , P − Λ = (8 , , , P +Λ = (9 , , , , { } .11 differs from a +Λ in at most 2 places, proposition 5.5 implies that ˜ W g ( P +Λ ) has a dimensionally proper point,lying in a fiber over M g, of dimension at most d + 1 (at most d if P +Λ = P − Λ ). Corollary 6.4.
Let P be any partition. Then for all g ≥ ( | P | + δ ( P )) , ˜ W g ( P ) has a dimensionally properpoint, lying in a fiber over M g, of dimension at most max(0 , g − | P | ) . To prove theorem 1.1, we are interesting in bounding the difficulty of “box-shaped” partitions, i.e. parti-tions of the form ( a b ). The table below shows some experimental data about the difficulties of these partitionsfor various values of a and b . 2 3 4 5 6 7 8 9 10 11 122 2 4 4 6 6 6 6 8 8 10 103 4 5 6 7 6 7 8 7 6 7 64 4 6 4 6 6 8 4 6 6 6 45 6 7 6 7 6 5 6 5 4 5 66 6 6 6 6 6 4 4 4 4 4 47 6 7 8 5 4 7 4 5 6 5 68 6 8 4 6 4 4 4 4 4 6 49 8 7 6 5 4 5 4 5 4 5 410 8 6 6 4 4 6 4 4 4 4 611 10 7 6 5 4 5 6 5 4 7 4On the basis of these experimental data, we make the following conjecture. Conjecture . There is a constant C such that for all positive integers a, b ≥ δ (( a b )) ≤ C .The assumption a, b ≥ a = 2 then the corresponding twisted Weierstrass pointsdetect g d s, whose moduli are well-understood. Remark . It is apparent from the definition that δ ( P ) = δ ( P ∗ ) where P ∗ is the conjugate partition. Thisis not surprising, in light of the duality ˜ W g ( P ) ∼ = ˜ W g ( P ∗ ). Remark . Corollary 6.4 is equivalent, in the special case of box-shaped partitions, to saying that if P = (( r + 1) g − d + r ), and ρ ≥ − g + δ ( P ), then ˜ W rd,g has a dimensionally proper point. So if conjecture 6.5is true, it would should that that the “phase transition” from dimensionally proper to improper occurs veryclose to ρ = − g , as we expect. We suspect that conjecture 6.5 is tractable, but do not yet have a proof. Insection 8, we prove a somewhat weaker bound, linear in a and b , that is sufficient to give theorem 1.1. As an example and first application of the techniques described above, we will prove one of the main resultsof [4] on the existence of dimensionally proper Weierstrass points.To state the result requires a bit of terminology. A subset S ⊆ Z ≥ that contains 0 and is closed underaddition is called a numerical semigroup . The size of the complement is called the genus . The sum of theelements of the complement, minus (cid:0) g +12 (cid:1) , is called the weight . A semigroup is called primitive if twice thesmallest positive element is greater than all the gaps; this is equivalent to saying that for all sets S ′ ≥ S whose complement is size g , S ′ is also a semigroup. Let C S ⊆ M g, be the locus of Weierstrass points withsemigroup S ; a point of C S is dimensionally proper if the local codimension of C S in M g, is equal to theweight of S . Theorem 7.1 (Eisenbud and Harris) . Let S be a primitive numerical semigroup of weight at most g − .Then C S has dimensionally proper points . In fact the primitivity assumption is not an artifact of the proof, but is a crucial assumption. Our methodalso gives an easy proof of the following. These theorem was improved by Komeda [10], who replaced g − g −
1; see remark 7.4.
Theorem 7.2. If S is a non-primitive semigroup, then the moduli space C S of pointed curves with Weierstrasssemigroup S has no dimensionally proper points. Although this fact is not explicitly proved in [4], there are other ways to establish it; we give anotherargument in [13], based on the notion of the effective weight of a semigroup.First we re-express theorem 7.1 using the notation of this paper. Notice that C S ∼ = ˜ W g ( P ), where P isthe partition given by P n = ( g + n ) − s n (where 0 = s < s < s < · · · are the elements of S ). That S is aprimitive semigroup is equivalent to saying that 2( g + 1 − P ) ≥ ( g + P ∗ ) (where P ∗ is the dual partition),which is equivalent, using the fact that P = g , to P − P ∗ ≥ P −
2. Thus theorem 7.1 follows from thefollowing, by lemma 6.4.
Lemma 7.3.
Let P be any partition such that P − P ∗ ≥ P − and | P | ≤ P − . Then δ ( P ) = 2 P − | P | .Proof. First, notice that any valid sequence of partitions ending in P must have at least P steps, since P can increase by at most 1 at each step. This means that ( | P | + δ ( P )) ≥ P , i.e. δ ( P ) ≥ P − | P | . So itsuffices to show the opposite inequality.The opposite inequality follows by induction on P . As the base case, consider the case P = 0. Then | P | = P = δ ( P ), so the result follows. So assume that P >
0. Let k ≥ P k = P . Then let Λ be the arithmetic progression generated by P − P k − k −
1. The correspondingdiagonal lines meet the Young diagram of P at only two corners, both outward, at the ends of rows 0 and k (see figure 3). Thus P +Λ = P and P − Λ differs in exactly two places from P : P and P k are both decreaseed by1. Now, it is immediate that | P − Λ | ≤ P − Λ ) −
2. It remains to show that ( P − Λ ) − ( P − Λ ) ∗ ≥ P − Λ ) −
2. Since P decreased by 1 under the displacement, the only way that this inequality could fail is if P ∗ is unchanged, P is unchanged, and the inequality was sharp before, i.e. P − P ∗ = 2 P −
2. This would mean that P = P and the Young diagram meets the third diagonal in figure 3; see figure 4. But in this case, we would have | P | ≥ P + 2 P + ( P ∗ −
3) = 2 P −
1, which contradicts the assumption that | P | ≤ P −
2. Hence P − Λ satisfies the hypotheses of the lemma. Also, it is clear that 2 P − | P | is unchanged and δ ( P − Λ ) ≥ δ ( P ), sothe desired inequality follows by induction. completing the induction. Remark . Notice that the proof above very nearly shows the existence of all dimensionally proper Weier-strass points of weight less than g (rather than g − P = P , P = 1, and P ∗ = P − P + 2 (i.e. the area enclosed by the dashed line in figure 4 is empty). A different displacementworks in this case, namely by turning in the first and last outward corner, unless P = 0. So the onlypartitions that cannot be treated this way are P = ((2 m − m m ). Komeda [10] proved, by a different13igure 4: The only situation where primitivity may fail after displacement, in the proof of theorem 7.1method, that dimensionally proper Weierstrass points corresponding to these partitions exist. So by addingthese partitions as an additional base case, Komeda extended theorem 7.1 to all primitive semigroups ofweight less than g .Using the same technique of displacement along elliptic curves, we can also prove the non-existence ofdimensionally proper Weierstrass points. This result is substantially generalized, by a different method,in [13]. We include this proof because it demonstrated the capability of the technique of displacement todisprove the existence of dimensionally proper points as well. Proof of theorem 7.2.
Suppose for the sake of contradiction that S = { , s , s , · · · } is a non-primitive semi-group such that C S has a dimensionally proper point. Let P be the corresponding partition, so that P = g and ˜ W g ( P ) has a dimensionally proper point. Define P k to be the partition given by P k = P + k and P ki = P i otherwise. By displacing repeatedly along singleton arithmetic progressions, it follows that ˜ W g + k ( P k ) hasa dimensionally proper point (for each k ). This corresponds to a dimensionally proper Weierstrass point in C S k , where S k = { , s + k, s + k, · · · } . Since S is not primitive, there exists a positive integer f > s suchthat f S . Let k = f − s . Then s + k ∈ S k , but 2( s + k ) = f + k S k . This is a contradiction; so C S cannot have any dimensionally proper points. In order to prove the existence of a reasonably large class of dimensionally proper linear series, it suffices,by lemma 6.4 to bound the displacement difficulty of box-shaped partitions. We shall prove the followingbound, which is likely to be very far from optimal, but is strong enough to give theorem 1.1.
Lemma 8.1.
Let P be the partition ( a b ) , i.e. the partition of the number ab into b equal parts, where a, b ≥ . Then δ ( P ) ≤ a + 3 b − . The proof appears at the end of this section. This lemma, together with lemma 6.4 is sufficient tocomplete the proof of the main theorem.
Proof of theorem 1.1.
Suppose that g, d, r are integers such that r ≥ g − d + r ≥
2, and 0 > ρ ≥− rr +2 g + 3 r −
3. Let a = g − d + r , b = r + 1, and P = ( a b ). Note that a, b ≥
2. By lemma 3.5, ˜ W rd,g hasa dimensionally proper point if and only if ˜ W g ( P ) has a dimensionally proper point. By lemmas 6.4 and8.1, it suffices to show the g ≥ ( ab + a + 3 b − a, b ≥ r = 1 iswell-understood (by the duality mentioned in remark 3.6, the roles of a and b can be interchanged, so eithercan be taken to be r + 1). 14 k, P k,i P k, a Figure 5: The intermediate partitions P k,i used in the proof of lemma 8.1, together with the progressionsΛ k,i . The partition is it’s own upward displacement for all values of i except possibly one (shown in themiddle).Now, ρ ≥ − rr +2 g + 3 r − g − ab ≥ b − b +1 g + 3 b −
6, i.e. bb +1 g ≥ ab + 3 b −
6. This is equivalentto 2 g ≥ ( a + 3)( b + 1) − b +1) b = ab + a + 3 b − − b . Since b ≥
3, this implies that 2 g ≥ ab + a + 3 b − Proof of lemma 8.1.
The proof will be by explicit construction of a sequence of partitions. First considerthe case where a is even .Define the following intermediate partitions: P k,i = ( a k ( i + a ) i ) (see figure 5), for k ≥ i ∈{ , , · · · , a } .Let Λ k,i denote the arithmetic progression generated by the two diagonals shows in figure 5. That is,Λ k,i = { n : n ≡ i − k − a + 1) } . Observe that if 1 ≤ i ≤ a , then Λ k,i does not meet the otheroutward-facing corner of the Young diagram, so it follows that( P k,i ) − Λ k,i = P k,i − when i > . Now consider the upward displacement. The only inward-turned corner that Λ k,i can meet is the one atthe end of the first row of the Young diagram; this corresponds to the value P − a . From this we canconclude that ( P k,i ) +Λ k,i = P k,i unless k > a ≡ i − k − a + 1) . For a fixed positive value of k , there is at most one value i ∈ { , , · · · , a } such that congruence aboveholds. Therefore the sequence of partitions P k, < P k, < · · · < P k, a is nearly a valid sequence of partitions; at most one adjacent pair is invalid. By inserting an intermediatepartition at that place (if necessary), we obtain a valid sequence of partitions with at most two stepsincreasing the sum by only 1. Therefore δ ( P k, a ) ≤ δ ( P k, ). For k = 0, the original sequence is valid, so δ ( P , a ) ≤ δ ( P , ).Since P k, a = P k +1 , , it follows from this analysis that δ ( P b − , ) ≤ b −
2) + δ ( P , ) . Now, P b − , ≤ ( a b ) with | ( a b ) | − | P b − , | = a and | P , | = a . From this it follows (by a sequence ofdisplacements along singleton progressions) that δ (( a b )) ≤ a + 2 b − a is even.15ow, if a is odd, then δ ((( a − b )) ≤ a + 2 b −
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