Kernel Bundles, Syzygies Of Points, and The Effective Cone of C_{g-2}
aa r X i v : . [ m a t h . AG ] M a y KERNEL BUNDLES, SYZYGIES OF POINTS, AND THE EFFECTIVECONE OF C g − YUSUF MUSTOPA
Abstract.
We obtain a complete description of the effective cone of C g − when C is ageneral curve of genus g ≥ , as well as a new bound in the case where C is a smooth planequintic. In addition, we obtain a new virtual bound for the effective cone of C g − m whichis a genuine bound when m = 2 , and we also characterize certain natural divisors on C g − as subordinate loci associated to adjunctions of kernel bundles. Dedicated To The Memory Of Sandra Samelson
Introduction
Let C be a smooth projective curve of genus g. The d − th symmetric power C d of C is a smooth projective variety of dimension d which is a fine moduli space parametrizingeffective divisors of degree d on C ; consequently, the “degree- d aspect” of maps from C toother varieties is encoded in the geometry of C d . A rich source of subvarieties of C d is provided by the so-called subordinate loci associatedto linear series on C. Given a linear series (
L, V ) of degree n ≥ d and dimension r ≤ d on C, the subordinate locus Γ d ( L, V ) (we omit V when the linear series is complete) is an r − dimensional determinantal subvariety of C d supported on the set { D ∈ C d : V ∩ H ( C, L ( − D )) = { }} These loci have been used to study the structure of algebraic cycles on the Jacobian of C (e.g. [Iz1], [Iz2] [Herb], [vdGK]) and—more importantly for us—have also been used tostudy cones of divisor classes on C d (e.g. [Kou],[Pac], [Deb], [Chan]). Our first result, whichgeneralizes (iii) of Theorem F in [Mus], continues directly in the latter vein. heorem I . Let C be a general curve of genus g ≥ . Then the cycle D supported on [ Γ g − ( L, V ) (where the union is taken over all linear series ( L, V ) of dimension 1 and degree g − ) isan effective divisor which spans a boundary ray of the effective cone of C g − . It is not immediately clear that D is of codimension 1 in C g − . We will address this issuelater in this introduction; for now, we point out that the linear series (
L, V ) of dimension1 and degree g − C of genus g ≥ g − − dimensional variety (this follows from Theorem 2.1) and that the loci Γ g − ( L, V ) arecurves which cover D as ( L, V ) varies.Let us also mention an important property of D which will be used later: whenever D ∈ C g − − D , we have that dim | D | = 0 and the residual series | K C ( − D ) | is a basepoint-free pencil (compare Corollary V).The notion of subordinate locus introduced above may be generalized in a straightforwardmanner to define subordinate loci associated to coherent systems on C, that is, pairs ( F, V )which consist of a vector bundle F on C and a subspace V ⊆ H ( C, F ) . One of the maingoals of this paper is to demonstrate how these higher-rank subordinate loci occur in thestudy of the effective cone of C d . Our results on these loci center around the well-known kernel bundle M L := φ ∗ L Ω P N (1) , where L is a globally generated line bundle on C and φ L : C → P N is the associatedmorphism. Theorem II . Let C be a general curve of genus g ≥ . Then for all p ∈ C we have theequality of cycles (0.1) Γ g − ( K C ⊗ M K C ( − p ) ) = D + X p where X p is the reduced divisor supported on { D ′ + p : D ′ ∈ C g − } . ubordinate loci of the form Γ d ( K C ⊗ M L ) admit a natural interpretation in terms ofmultiplication maps. Pulling back the Euler sequence on P N via φ L yields the exact sequence(0.2) 0 → M L → H ( C, L ) ⊗ O C → L → D ∈ C d . Twisting by K C ( − D ) and taking cohomology yields the exactsequence −−−−−→ H ( C, K C ⊗ M L ( − D )) −−−−−→ H ( C, L ) ⊗ H ( C, K C ( − D )) µ L,KC ( − D ) −−−−−−−−−→ H ( C, K C ⊗ L ( − D )) where µ L,K C ( − D ) is the multiplication map. It then follows that Γ d ( K C ⊗ M L ) parametrizesall D ∈ C d for which µ L,K C ( − D ) fails to be injective.Any serious consideration of multiplication maps is closely related to Koszul cohomology,and the reader familiar with the latter has undoubtedly noticed its relevance to the presentwork. While we make almost no explicit use of it here, it is almost certainly the naturalframework for further progress along some of the lines indicated by our results on higher-ranksubordinate loci. We refer to [Gr] for a foundational treatment of Koszul cohomology andto [ApFa] for a nice survey of some recent developments.Let us now return to the issues surrounding Theorem I. We begin by introducing the basicdivisor classes on C d . For each p ∈ C the image X p of the embedding i p : C d − ֒ → C d , D ′ D ′ + p is an ample divisor on C d whose numerical class (which is independent of p ) will be denotedby x. The pullback of a theta-divisor on Pic d ( C ) via the natural map a d : C d → Pic d ( C ) , D
7→ O C ( D )is a nef divisor on C d whose numerical class will be denoted by θ. (When 2 ≤ d ≤ g, we alsohave that θ is big.) The classes x and θ are linearly independent in the real N´eron-Severispace N R ( C d ) , and when C is general N R ( C d ) is generated by x and θ. he following result plays an important role in the proofs of Theorems I and II for reasonsthat will soon be evident. It also gives a new bound for the effective cone of C g − . Proposition III . Let C be a general curve of genus g ≥ , and let m be a positive integerwhich is at most g − . Then there is a virtual divisor D m on C g − m supported on the set [ Γ g − m ( L , V ) (where the union is taken over all linear series ( L , V ) of degree g − m and dimension 1) withclass proportional to θ − (1 + mg − m ) x, and it is an effective divisor when m = 1 or m = 2 . We now proceed to list some consequences in the case m = 1 . Theorem 3 of [Kou] saysthat for all C and for all d ≥ , the class∆ = 2( − θ + ( g + d − x )of the diagonal locus on C d (which parametrizes effective divisors of degree d having multi-plicity) spans a boundary ray of the effective cone of C d . Combining this result with TheoremI and Proposition III yields
Corollary IV . If C is a general curve of genus g ≥ then the effective cone of C g − isspanned by the classes ∆ and θ − (1 + g − ) x. Recall that for nonnegative integers s and e there exists a fine moduli variety G se ( C )parametrizing linear series of degree e and dimension s on C. In [Mus] we made heavy useof the fact that for a general curve C of genus g ≥ ≤ d ≤ g − , the rational map τ : C d G g − d − g − − d ( C ) taking a general D ∈ C d to the residual linear series | K C ( − D ) | is anisomorphism of smooth varieties in codimension 1. In particular, τ preserves the effectivecone, so that Corollary IV implies orollary V . If C is a general curve of genus g ≥ , then the effective cone of G g ( C ) isspanned by class of the divisor D ′ supported on { ( L, V ) ∈ G g ( C ) : ( L, V ) has a base point } and the class of the divisor ∆ ′ supported on the closure of { ( L, V ) ∈ G g ( C ) : V = H ( L ) and | K C ⊗ L − | has multiplicity } Let us now consider the case m ≥ , where D m admits a nice interpretation in termsof secant planes. It can be shown that for a general curve C and general D ∈ C g − m , theresidual series | K C ( − D ) | is basepoint-free of dimension 2 m − φ K C ( − D ) is birational onto its image (we refer to the proof of Proposition III for details).The Riemann-Roch Theorem then implies that for m ≥ , the support of D m may becharacterized as the Zariski closure of the set { D ∈ C g − m : dim | K C ( − D ) | = 2 m − φ K C ( − D ) ( C ) admits an m − secant ( m − − plane } The structure of D m as a virtual divisor on C g − m comes from the fact that it is supportedon the projection of a locus in C g − m × C m which is shown to be of pure dimension g − m − , namely { ( D, E ) ∈ C g − m × C m : E fails to impose independent conditions on | K C ( − D ) |} Showing that D m is an honest divisor is therefore equivalent to showing that for general D ∈ C g − m , there are at most finitely many E ∈ C m which fail to impose independentconditions on | K C ( − D ) | . For recent progress on problems of this nature, see [Cot], [Fa], andthe references therein.We now turn from general curves to special curves. In the hyperelliptic case, the non-diagonal boundary ray of the effective cone of C g − (resp. C g − ) in the ( θ, x ) − plane isspanned by θ − x (resp. θ − x ); this is part of Proposition H in [Mus]. In the trigonal ase it is spanned by θ − x (this is easily extracted from the proof of Theorem 5 in [Kou]).These special families, together with smooth plane quintics, occupy important places in twodistinct classification schemes–namely the Martens-Mumford classification of Brill-Noetherloci and classification of curves by their Clifford index (which we will describe shortly).Our proof of Theorem I cannot be extended to the smooth plane quintic case. Thisis essentially due to the fact that every complete linear series of degree 5 and dimension1 on a smooth plane quintic has a base point. However, we have the following pleasantcharacterization of D : Theorem VI . Let C be a smooth plane quintic, and let Q = O C (1) . Then we have the equalityof cycles (0.3) D = 2 · Γ ( K C ⊗ M Q ) . The overt syzygetic content in this paper belongs to the proof of this theorem, whichemploys the minimal graded resolution of 4 points in P in a crucial way.We also obtain a strict outer bound for the effective cone of C . Theorem VII . If C is a smooth plane quintic, then there is no effective divisor on C whoseclass is proportional to θ − x. We conclude this introduction with a brief discussion of the Clifford index of a curve andits (possible) relation to our work. We define Cliff( C ) to bemin { deg L − | L | : L a line bundle on C ∋ dim | L | ≥ , dim | K C ⊗ L − | ≥ } . This is a rough measure of how general C is in the sense of moduli; one can check, forinstance, that Cliff( C ) = 0 precisely when C is hyperelliptic and that Cliff( C ) = 1 preciselywhen C is either trigonal or a smooth plane quintic. We mention in passing that the bundle M K C has been intensively studied with an eye towards the Green conjecture relating Cliff( C )to the syzygies of the canonical embedding of C (e.g. [PR],[Ein]) and that the tools developed y Voisin [V] in her proof of this conjecture for a general curve of even genus g have beenused by Pacienza [Pac] to compute the nef cone of C g for all such curves C. Given that the effective cones of C g − and C g − detect whether Cliff( C ) is 0 or greater, itis natural in light of the trigonal case and Theorem VII to ask if the effective cone of C g − detects whether Cliff( C ) is 1 or greater. Theorem VII does not preclude the possibility thatthe effective cone of C has an open boundary ray spanned by θ − x when C is a smoothplane quintic, and this points to two extreme scenarios: one where Theorem I continues tohold in the plane quintic case, and another where the closure of the part of the effective coneof C g − in the ( θ, x ) − plane is the same for all curves C satisfying Cliff( C ) = 1 . As of thiswriting, however, we have yet to find evidence that either of these is true.
Notation and Conventions:
We work over the field of complex numbers. C will alwaysdenote a smooth projective curve. The symbol g rd is used to refer to a linear series (completeor otherwise) on C of degree r and dimension d. All cycle classes on smooth varieties liein the algebraic cohomology ring with coefficients in R . We say that a property holds for a general curve of genus g if it holds on the complement of the union of countably manyproper subvarieties of the moduli space M g of smooth projective curves of genus g .1. Preliminaries on C d Intersection Theory . The following formula, which will be used freely, is a consequenceof the Poincar´e formula (p.25 of [ACGH]).
Lemma
For all ≤ k ≤ d ≤ g,x k θ d − k = g !( g − d + k )!1.2. Class Formulas . We begin with the basic result on classes of subordinate loci. This isLemma 3.2 on p.342 of loc. cit. emma Let ( L, V ) be a g rn on C, and assume r ≤ d ≤ n. Then Γ d ( L, V ) is an r − dimensional subvariety of C d whose class is γ d ( g rn ) := d − r X j =0 (cid:18) n − g − rj (cid:19) x j · θ d − r − j ( d − r − j )!While the following result is a corollary in the sense that it can be obtained in a straightfor-ward manner from Lemma 1.2, it can be obtained much more easily from the basic propertiesof the morphism a d . Corollary θ · γ d ( g d ) = 0 and x · γ d ( g d ) = 1 . (cid:3) The next two formulas are crucial to the proofs of Proposition III and Theorem I. Recallthat C d , which is the exceptional locus of the map a d , parametrizes effective divisors of degree d that move in a positive-dimensional linear series. The next formula follows immediatelyfrom the Theorem on p.326 of loc. cit.Proposition If the cycle C d is either empty or of the expected dimension d − g − , itsclass c d is equal to θ g − d +1 ( g − d + 1)! − xθ g − d ( g − d )! Proposition (Push-Pull formula) For each algebraic cycle Z in C d , let B k ( Z ) denote theclass of the cycle parametrizing the set { E ∈ C d − k : | D − E | 6 = ∅ for some D ∈ Z } . Then wehave the formula B k ( x α θ β ) = k X j =0 (cid:18) αk − j (cid:19)(cid:18) βj (cid:19)(cid:18) g − β + jj (cid:19) j ! x α − k + j θ β − j The derivation of the Push-Pull formula is outlined in exercise batch D of Chapter VIIIof loc. cit.
Special Subordinate Loci Associated To Basepoint-Free Pencils . The classical subordi-nate loci of primary interest to us are of the form Γ d ( L, V ) where (
L, V ) is a g d +1 . emma Let C be a curve of genus g ≥ , and let ( L, V ) be a basepoint-free g d +1 on C .Then Γ d ( L, V ) ∼ = C. Proof. (i) Let Y be the incidence variety supported on { ( p, D ) ∈ C × C d : p + D ∈ | V |} . Wewill proceed by showing that C and Γ d ( L, V ) are both isomorphic to Y . Since (
L, V ) is basepoint-free, we have well-defined bijective morphisms j : C → Y and j : Γ d ( L, V ) → Y given by j ( p ) = ( p, | V | ∩ | L ( − p ) | ) and j ( D ) = ( | L ( − D ) | , D ) . Thisimplies that g ≤ p a ( Y ) and p a (Γ( L, V )) ≤ p a ( Y ) . The morphisms j and j are sections of the projections ρ : Y → C and ρ : Y → Γ d ( L, V ) , respectively. Consequently ρ and ρ are bijective, so that p a ( Y ) ≤ g and p a ( Y ) ≤ p a (Γ d ( L, V )) . It follows that the arithmetic genera of the curves C, Γ d ( L, V ) , and Y are allequal; this implies that j and j are isomorphisms. (cid:3) Higher-Rank Subordinate Loci . Let d ≥ π , π be the projection mapsassociated to C × C d , and let U be the universal divisor on C × C d , i.e. the incidence varietysupported on { ( p, D ) ∈ C × C d : p ∈ supp( D ) } . For any vector bundle F of rank r on C, wedefine E ( F ) := π ∗ ( π ∗ F ⊗ O U )which is a vector bundle of rank rd on C d whose fibre over D ∈ C d is H ( D, F | D ) . If (
F, V )is a coherent system, the restriction maps V → H ( D, F | D ) globalize to a vector bundlemorphism α ( F,V ) : V ⊗ O C d → E ( F ) . Definition: Γ d ( F, V ) is the degeneracy locus associated to α ( F,V ) . If V = H ( C, F ) , we willdenote Γ d ( F, V ) by Γ d ( F ) . Proposition
Let ( F, V ) be a coherent system on C of rank r ≥ and degree f such that rd = dim V. Then the class of the virtual divisor Γ d ( F, V ) on C d is rθ − ( rd + rg − f − r ) x. Proof.
By Porteous’ Formula, the class we seek is c ( E ( F )) , which we will extract from theChern character of E ( F ) . he calculation is almost identical to the proof of Lemma 2.5 on p.340 of [ACGH]. Inwhat follows, η denotes the class of the pullback via π of a point on C and the pullbacksvia π of the classes x and θ are also denoted by x and θ, respectively. By Grothendieck-Riemann-Roch, we have thattd( C d ) · ch( E ( F )) = ( π ) ∗ (td( C × C d ) · ch( π ∗ F ⊗ O U ))Canceling out the td( C d ) term and using the short exact sequence0 → π ∗ F ( −U ) → π ∗ F → π ∗ F ⊗ O U → U from p.338 of loc. cit. , we have thatch( E ( F )) = ( π ) ∗ ((1 + (1 − g ) η ) · (( r + f η ) − ( r + ( f − rd ) η − rηθ ) e − x ))= ( π ) ∗ ( r + ( f + r (1 − g )) η − ( r + ( f − rd + r − rg ) η − rηθ ) e − x )= ( f + r (1 − g )) + ( rd + rg − f − r + rθ ) e − x . (cid:3) Proofs of Proposition III and Theorem I
We will require two major results of Brill-Noether theory. The first of these is Gieseker’sTheorem (e.g. Theorem 1.6 on p. 214 of [ACGH]):
Theorem
Let C be a general curve of genus g. Let d, r be integers satisfying d ≥ and r ≥ . Then G rd ( C ) is smooth of dimension g − ( r + 1)( g − d + r ) . We will also use the following theorem of Fulton and Lazarsfeld (e.g. Theorem 1.4 onp.212 of loc. cit. ): Theorem
Let C be a smooth curve of genus g. Let d, r be integers such that d ≥ , r ≥ . Assume that g − ( r + 1)( g − d + r ) ≥ . hen G rd ( C ) is connected. The following corollaries are immediate applications of these theorems; the first followsfrom Theorem 2.1, while the second follows from combining Theorems 2.1 and 2.2.
Corollary If C is a general curve of genus g ≥ , then the basepoint-free g g − ’s are anonempty Zariski open subset of G g − ( C ) . (cid:3) Corollary If C is a general curve of genus g, and d, r are integers such that d ≥ , r ≥ , and g − ( r + 1)( g − d + r ) ≥ , then G rd ( C ) is irreducible. (cid:3) Proof of Proposition III:
Let C be a general curve of genus g ≥ , and let m be a positiveinteger satisfying m ≤ g − . Consider the diagram C g − m × C mπ (cid:15) (cid:15) σ / / C g − m C g − m where σ is the addition map and π is projection onto the first factor. Since C is general, theBrill-Noether Theorem implies that C g − m is of pure dimension g − m − , and since σ is afinite map of smooth varieties, the same is true of σ − ( C g − m ) . Therefore D m := π ∗ σ ∗ ( C g − m )is a virtual divisor on C g − m .We now compute the class of D m . We have from Proposition 1.4 that c g − m = θ m +1 ( m + 1)! − xθ m m !Combining this with Proposition 1.5, we have that the class of D m is(2.1) B m ( c g − m ) = (cid:18) gm (cid:19)(cid:18) g − mg · θ − x (cid:19) . The cycle D m is an effective divisor on C g − m precisely when the intersection σ − ( C g − m ) ∩ π − ( D ) is at most finite for general D ∈ C g − m . Our next and final task is to verify this ondition for m = 1 and m = 2 . Note that since σ − ( C g − m ) ∩ π − ( D ) = { D } × C m forall D ∈ C g − m , we must necessarily consider a general element of C g − m − C g − m . (TheBrill-Noether Theorem implies that C g − m is a proper Zariski-closed subset of C g − m for all m ≥ . )If m = 1 , then σ − ( C g − ) ∩ π − ( D ) parametrizes the points q ∈ C for which dim | D + q | ≥ , or equivalently, the base points of | K C ( − D ) | . If D ∈ C g − − C g − , then | K C ( − D ) | is1-dimensional, and thus has at most finitely many base points.On our way to settling the case m = 2 , we justify the claim (stated in the Introduc-tion) that for all m ≥ D ∈ C g − m , the linear series | K C ( − D ) | is basepoint-free of dimension 2 m − φ K C ( − D ) is birational onto its im-age. We assume throughout that D ∈ C g − m − C g − m . By Riemann-Roch, we have thatdim | K C ( − D ) | = 2 m − , and by Brill-Noether, the set of all D ∈ C g − m for which | K C ( − D ) | admits a base point is a proper Zariski-closed subset of C g − m ; this establishes the first partof the claim. If φ K C ( − D ) is not birational onto its image, its image is necessarily a nondegen-erate rational curve of degree at least 2 m − P m − since a general curve of genus g cannotbe a multiple cover of an irrational curve (cf. Exercise C-6 in Chapter VIII of [ACGH]).The morphism φ K C ( − D ) then factors through a branched cover of P by C having degree atmost g +2 m − m − . But this is impossible, since any branched cover of P by C must have degreeat least g + 1 by Brill-Noether.The characterization of D m in terms of secant planes stated in the Introduction impliesthat D parametrizes D ∈ C g − for which φ K C ( − D ) fails to be an embedding. Since φ K C ( − D ) is birational onto its image for general D ∈ C g − and the image of φ K C ( − D ) has at mostfinitely many singular points, it follows that σ − ( C g − ) ∩ π − ( D ) is at most finite for general D ∈ C g − . This concludes the proof that D is an effective divisor. (cid:3) Proof of Theorem I:
We begin by showing the irreducibility of D , which we know to be ofcodimension 1 in C g − by Proposition III. There is a natural rational map φ : C × G g − ( C ) D , ( p, L )
7→ |L ( − p ) | hich is well-defined (outside a closed set) and dominant thanks to Corollary 2.3. By Corol-lary 2.4, C × G g − ( C ) is irreducible, so D is irreducible as well.Let us now consider the curves Γ g − ( L, V ) , where ( L, V ) is a g g − on C. By Lemma 1.2,we have that their common numerical class is γ g − ( g g − ) = g − X j =0 (cid:18) − j (cid:19) x j θ g − − j ( g − − j )!It then follows that θ · γ g − ( g g − ) = g − X j =0 ( − j ( j + 1) g !( j + 2)!( g − − j )!= g · g − X j =0 ( − j ( j + 1) (cid:18) g − j + 2 (cid:19) = g · (cid:18) g − X j =0 ( − j ( j + 2) (cid:18) g − j + 2 (cid:19) − g − X j =0 ( − j (cid:18) g − j + 2 (cid:19)(cid:19) = g · (cid:18) g − X j =2 ( − j j (cid:18) g − j (cid:19) − g − X j =2 ( − j (cid:18) g − j (cid:19)(cid:19) = g · (( g − − ( g − g. We need one other intersection number: x · γ g − ( g g − ) = g − X j =0 ( − j ( j + 1) g !( j + 3)!( g − − j )! = g − X j =0 ( − j ( j + 1) (cid:18) gj + 3 (cid:19) = − g X j =3 ( − j ( j − (cid:18) gj (cid:19) = 2 g X j =3 ( − j (cid:18) gj (cid:19) − g X j =3 ( − j j (cid:18) gj (cid:19) = g − . It then follows from setting m = 1 in (2.1) that D · γ g − ( g g − ) = (( g − θ − gx ) · γ g − ( g g − ) = 0 . Suppose there exists an irreducible effective divisor E on C g − whose class is proportionalto θ − tx for some t > g − . Then E · γ g − ( g g − ) < . Since Γ g − ( L, V ) is irreducible forgeneral (
L, V ) by Lemma 1.6 and Corollary 2.3, we have that the curves Γ g − ( L, V ) cover E . We may then conclude that D = E , which is absurd. (cid:3) . The Virtual Divisor Γ d ( K C ⊗ M L )In preparation for the proofs of Theorems II and VI, we present two results that yield twodifferent ways of showing that Γ d ( K C ⊗ M L ) is a virtual divisor in the cases of interest to us.The first result, which we state without proof, is a straightforward consequence of applyingGreen’s “ K p, − Theorem” (Theorem 3.c.1 in [Gr]) and Lazarsfeld’s description of Koszul co-homology in terms of kernel bundles (e.g. Theorem 2.6 in [ApFa]) to the Koszul cohomologygroup K r − ,r ( C, L ). It can also be viewed as a slight modification of Proposition 3.4 in [Tei].
Proposition
Let C be an irrational curve and let k ≥ be an integer such that thegeneral complete pencil of degree k on C is basepoint-free. Let L be a line bundle of degree k − such that h ( C, L ) = k − . Then h ( C, M ∗ L ) = k − . (cid:3) For the statement and proof of the second result, we reuse the notation from Section 1.4.
Proposition
Let C be a nonhyperelliptic curve, let d be an integer satisfying ≤ d ≤ g − , and define C ∗ d to be the quasiprojective variety C d − C d . Furthermore, define U ∗ to be therestriction of U to C × C ∗ d , let r be an integer satisfying r ≥ dg − d , and let L be a line bundleon C of degree r ( g − d ) + 1 . Then the degeneracy locus of the multiplication map µ L : H ( C, L ) ⊗ π ∗ ( π ∗ K C ⊗ O ( −U ∗ )) → π ∗ ( π ∗ ( K C ⊗ L ) ⊗ O ( −U ∗ )) on C ∗ d extends to a virtual divisor on C d whose class is rθ − ( r + 1) x. Proof.
For simplicity we define F := π ∗ ( π ∗ K C ⊗ O ( −U ∗ )) and G L := π ∗ ( π ∗ ( K C ⊗ L ) ⊗O ( −U ∗ )) . By Riemann-Roch and Grauert’s Theorem, F and G L are locally free sheaves on C ∗ d of respective ranks g − d and ( r + 1)( g − d ) . As D varies over C ∗ d , the exact sequence0 → H ( C, K C ( − D )) → H ( C, K C ) → H ( D, K C | D ) → C ∗ d :0 → F → H ( C, K C ) ⊗ O C ∗ d → Ω C d ⊗ O C ∗ d → t follows from taking determinants that det F = K − C d ⊗ O C ∗ d . Since C is nonhyperelliptic,we have by Martens’ Theorem that C d is of codimension 2 or greater. Consequently det F extends over C d to K − C d by Hartogs’ Theorem, and c ( K − C d ) = − θ − ( g − d − x. A similar argument shows that det G L extends over C d to det E ( K C ⊗ L ) ∗ , so that c ( G L ) = − c (Γ d ( K C ⊗ L )) = − θ − ( r + 1)( g − d ) x by Lemma 1.2 or Proposition 1.7. Therefore the class of our degeneracy locus is c ( G L ) − ( r + 1) c ( K − C d ) = rθ − ( r + 1) x. (cid:3) Proof of Theorem II
The following is an immediate consequence of either Proposition 3.2 or a combination ofPropositions 1.7 and 3.1.
Lemma If C is nonhyperelliptic of genus g ≥ , then for all p ∈ C we have that Γ g − ( K C ⊗ M K C ( − p ) ) is a virtual divisor on C g − with class ( g − θ − ( g − x. (cid:3) Proof of Theorem II:
We begin by simultaneously showing that the virtual divisor Γ g − ( K C ⊗ M K C ( − p ) ) is an honest divisor and that it contains X p . Let D ∈ C g − − D be given. Thenthe degree- g pencil | K C ( − D ) | is basepoint-free, and we have the short exact sequence0 → T C ( D ) → H ( C, K C ( − D )) ⊗ O C → K C ( − D ) → . Twisting by K C ( − p ) and taking cohomology, we see that the kernel of µ K C ( − p ) ,K C ( − D ) isisomorphic to H ( C, O C ( D − p )) . We then have a natural isomorphism H ( C, K C ⊗ M K C ( − p ) ( − D )) ∼ = H ( C, O C ( D − p )) . ecall from the Introduction that Γ g − ( K C ⊗ M K C ( − p ) ) is supported on the set of all D ∈ C g − for which µ K C ( − p ) ,K C ( − D ) fails to be injective. It then follows that D ∈ C g − − D iscontained in Γ g − ( K C ⊗ E ∗ K C ( − p ) ) precisely when D ∈ X p . Since D is irreducible and Γ g − ( K C ⊗ M K C ( − p ) ) is a determinantal subscheme of C g − having the expected dimension, either Γ g − ( K C ⊗ M K C ( − p ) ) contains all of D or it onlycontains D ∩ X p . If the latter was true, Γ g − ( K C ⊗ M K C ( − p ) ) would be supported on X p , and therefore would have class proportional to x. But this contradicts Lemma 4.1.The same Lemma, together with (2.1), shows that D and X p each occur with multiplicity1 in Γ g − ( K C ⊗ M K C ( − p ) ), thereby establishing (0.1). (cid:3) Recall that the movable cone of a projective variety X is the closure of the convex cone in N R ( X ) spanned by classes of divisors whose stable base locus has no divisorial component.Theorem C of [Mus] implies that for a general curve of genus g ≥ , the class θ − x of thesubordinate locus Γ g − ( K C ( − p )) lies in the interior of the movable cone of C g − and hasstable base locus C g − . In light of this result, the present characterization of D + X p as ahigher-rank subordinate locus naturally associated to K C ( − p ) seems to suggest that theclass ( g − θ − ( g − x lies outside of the movable cone of C g − and has stable base locus D , but we currently have no proof of this.5. The Case Of A Smooth Plane Quintic
This final section contains the proofs of Theorems VI and VII. From this point forward, C always denotes a smooth plane quintic, and Q denotes O C (1) . Note that Q ⊗ ∼ = K C bythe adjunction formula.Given that the general curve of genus 6 cannot be realized as a smooth plane quintic,it is worth reminding the reader that the classes x and θ exist in N R ( C d ) and are linearlyindependent for all curves C and all d ≥ . It can be shown using Proposition 3.9 in [Pir]that N R ( C d ) is generated by x and θ when C is a general smooth plane quintic, but we willnot need this fact. ur first task in this section is to show that D is an irreducible effective divisor on C . Theargument for the case of a general curve (which is spread out over the proofs of PropositionIII and Theorem I) fails in the plane quintic case for two reasons. One is that we cannot usethe Brill-Noether Theorem to show that D is an effective divisor. Another is that G ( C ) isno longer irreducible; this is a consequence of the next result. Lemma
Let ( L, V ) be a g on C. (i) If ( L, V ) is incomplete, then it is a sub-pencil of |Q| . In particular, the general in-complete g on C is basepoint-free. (ii) If ( L, V ) is complete, then it is of the form |Q ( r − s ) | , where r, s are distinct pointsin C. In particular, every complete g on C has a base point.Proof. (i) Let ( L, V ) be an incomplete g on C. By Clifford’s Theorem, the dimension of | L | is exactly 2, and since there is exactly one g on C, we must have that L ∼ = Q . (ii) Thisfollows from Exercise F-1 in Chapter VIII of [ACGH]. (cid:3) The following characterization of C will also be helpful. Lemma
We have the set-theoretic equality C = Γ ( Q ) . Proof.
Any complete linear series of degree 4 on C is necessarily a pencil by Clifford’s The-orem. By Exercise A-12 in Chapter V of loc. cit. we have that every g on C is of the form |Q ( − s ) | for some s ∈ C ; the desired equality follows at once. (cid:3) Proposition D is an irreducible effective divisor on C . Proof.
We begin by showing that D is an effective divisor on C . The argument from theproof of Proposition III will suffice once we know that C is a proper Zariski-closed subsetof C and that C is irreducible of the expected dimension. The first statement followsfrom Lemma 5.2. As for the second statement, Lemma 5.1 implies that the Brill-Noetherlocus W ( C ) ⊆ Pic ( C ) is a translate of the difference surface C − C ; therefore W ( C ) is rreducible of dimension 2. Since C = a − ( W ( C )) and a − ( L ) ∼ = P for general L ∈ W ( C ) , we have that C is irreducible of dimension 3.We now proceed to show that D is irreducible. Define the rational map e φ : C × C × C C by ( q, r, s ) q + |Q ( r − s − q ) | . The degree-4 linear series |Q ( r − s − q ) | has nonnegativedimension for all r, s, q ∈ C since Q is very ample, and it is zero-dimensional precisely when r = s and r = q. Therefore e φ is well-defined on a nonempty Zariski-open subset of C × C × C. It remains to show that e φ dominates D . By part (i) of Lemma 5.1 we know that if D ∈ C is subordinate to an incomplete g on C, it is necessarily subordinate to |Q| . Since Γ ( Q ) is of codimension 2 in C , it follows thatthe general element of D is subordinate to a complete g . Lemma 5.2 and part (ii) of Lemma5.1 imply that if D ∈ C − C is subordinate to a complete g on C, then D + q ∈ |Q ( r − s ) | for some r, s, q ∈ C such that r = s and r = q. In particular, D = e φ ( q, r, s ) . (cid:3) For the proof of Theorem VI, we will need the following Lemma, whose derivation isanalogous to that of Lemma 4.1.
Lemma
The locus Γ ( K C ⊗ M Q ) is a virtual divisor on C whose class is θ − x. (cid:3) Proof of Theorem VI . By (2.1) and Lemma 5.4, the equality of cycles (0.3) will beestablished once we check equality of the underlying sets. Since Γ ( K C ⊗ M Q ) is a virtualdivisor and D is an irreducible effective divisor by Proposition 5.3, it suffices to show thatΓ ( K C ⊗ M Q ) is a nonempty subset of D . This will be accomplished by a useful characteri-zation of Γ ( K C ⊗ M Q ) , which we now describe.Let ι Q : C ֒ → P be the embedding induced by Q . Recall that for a closed subscheme X of P N , the space H ( P N , Ω P N (1) ⊗ I X ( k )) is the kernel of the multiplication map H ( P N , O P N (1)) ⊗ H ( P N , I X ( k )) → H ( P N , I X ( k + 1)) .e. the space of linear syzygies among the degree- k elements of the saturated homogenousideal of X. As mentioned in the Introduction, for each D ∈ C the space H ( C, K C ⊗ M Q ( − D )) is the kernel of the multiplication map H ( C, Q ) ⊗ H ( C, K C ( − D )) → H ( C, Q ⊗ K C ( − D )) . Restriction from P to C yields the following commutative diagram with exact rows. −−−−−→ H ( P , Ω P (1) ⊗ I ι Q ( D ) (2)) −−−−−→ H ( P , O P (1)) ⊗ H ( P , I ι Q ( D ) (2)) −−−−−→ H ( P , I ι Q ( D ) (3)) y y y −−−−−→ H ( C, E ∗Q ⊗ K C ( − D )) −−−−−→ H ( C, Q ) ⊗ H ( K C ( − D )) −−−−−→ H ( C, K C ⊗ Q ( − D )) Since h i ( P , O P ( − h i ( P , O P ( − i = 0 , , the right two vertical arrowsare isomorphisms, and therefore so is the leftmost vertical arrow. It follows that D ∈ Γ ( K C ⊗ M Q ) precisely when the quadrics in P which vanish along the subscheme ι Q ( D )admit linear syzygies.For any D ∈ C , there are three distinct possibilities for ι Q ( D ) :(a) ι Q ( D ) is collinear.(b) ι Q ( D ) is noncollinear, but contains a collinear subscheme of length 3.(c) ι Q ( D ) contains no collinear subscheme of length 3.It follows from the discussion in Example 3.12 of [Eis] and the upper-semicontinuity of Bettinumbers that the quadrics in P which vanish along ι Q ( D ) admit linear syzygies preciselywhen (a) or (b) holds. In particular, Γ ( Q ) = C is a subset of Γ ( K C ⊗ M Q ) , so thatΓ ( K C ⊗ M Q ) is nonempty.It remains to show that Γ ( K C ⊗ M Q ) ⊆ D . If (a) holds, and ℓ ⊆ P is the line spannedby ι Q ( D ) , then D is subordinate to the (incomplete) g on C obtained by projecting awayfrom a point on ℓ which does not lie on C. If (b) holds, then D = D ′ + r, where ι Q ( D ′ ) iscollinear of length 3 and r is not on the line spanned by ι Q ( D ′ ) . If s ∈ C is one of the pointsfor which D ′ ∈ Γ ( Q ( − s )) , we have that D ∈ Γ ( Q ( r − s )) . Since |Q ( r − s ) | is a g , we aredone. (cid:3) .2. Proof of Theorem VII . We begin by determining the components of the subordinatelocus associated to a complete g on C. Proposition
For p, s ∈ C, p = s, define Z p,s := p + Γ ( Q ( − s )) . Then we have the equalityof cycles (5.1) Γ ( Q ( p − s )) = Z p,s + Γ ( Q ( − s )) Proof.
We first establish the equality at the level of sets and then use a specialization ar-gument to show that each component occurs with multiplicity 1. Let D ∈ Γ ( Q ( p − s )) begiven. Then there exists r ′ ∈ C such that D + s + r ′ ∈ |Q ( p ) | , and since |Q ( p ) | is a g withbase point p by Clifford’s Theorem, we have that either r ′ = p or D − p is effective. In thefirst case, we have that D ∈ Γ ( Q ( − s )) and in the second case we have that D ∈ Z p,s . Thereverse inclusion Z p,s ∪ Γ ( Q ( − s )) ⊆ Γ ( Q ( p − s )) follows immediately from the definitions.By Lemma 1.6, Z p,s ∼ = C. Furthermore, Γ ( Q ( − s )) ∼ = P , and since Z p,s is a smoothsubvariety of the divisor X p which meets Γ ( Q ( − s )) transversally in one point by Corollary1.3, we have that Z p,s meets Γ ( Q ( − s )) properly in one point. Therefore the reduciblecurve Z p,s + Γ ( Q ( − s )) has arithmetic genus 6. Another application of Lemma 1.6 showsthat Γ ( Q , V ) ∼ = C whenever ( Q , V ) is a basepoint-free pencil of degree 5. Since G ( C ) isconnected (this follows, for instance, from Theorem 2.2) we have that the arithmetic genusof Γ ( Q ( p − s )) is also 6, so that (5.1) holds. (cid:3) The next two propositions conclude the proof of Theorem VII. We denote by Z the nu-merical class of the curves Z p,s . Proposition ( θ − x ) · Z = 0 . roof. Combining Proposition 5.5 and Corollary 1.3 with our calculations from the proof ofTheorem I, we have that θ · Z = θ · γ ( g ) − θ · γ ( g ) = 6 x · Z = x · γ ( g ) − x · γ ( g ) = 3and the result follows. (cid:3) Proposition If E is an effective divisor on C whose class is proportional to θ − tx forsome t > , then E · Z ≥ . Proof.
We may suppose without loss of generality that E is irreducible. The conclusionfollows immediately from the calculations in the previous proof if t = , so we will assumeotherwise from now on. In particular, we are assuming that E = Γ ( K C ⊗ E ∗Q ) . Since thecurves Z p,s cover Γ ( K C ⊗ E ∗Q ) , there necessarily exist distinct p ′ , s ′ ∈ C such that Z p ′ ,s ′ * E . Therefore we have that E · Z ≥ , and it remains to check that this intersection is positive.By Nakamaye’s theorem on stable base loci of nef and big divisors (e.g. Theorem 10.3.5in [Laz]) applied to the class θ, the numerical hypothesis on E implies that C ⊂ E . Since Z p ′ ,s ′ ∩ C = ∅ , the result follows. (cid:3) Funding:
This work was partially supported by the National Science Foundation [RTGDMS-0502170].
Acknowledgments:
The author would like to thank S. Casalaina-Martin, E. Cotterill, J.Kass, R. Lazarsfeld, M. Nakamaye, G. Pacienza, J. Ross, and J. Starr for valuable discussionsand correspondence. He would also like to thank the anonymous referees for numerous helpfulsuggestions and for pointing towards a gap in the initial statement of Proposition III.Department of Mathematics, University of MichiganEast Hall, 530 Church Street nn Arbor, MI 48109-1043email: ymustopa @ umich . edu References [ACGH] Arbarello, Enrico, Maurizio Cornalba, Phillip Griffiths and Joe Harris.
Geometry of AlgebraicCurves, Volume I , New York: Springer-Verlag, 1985.[ApFa] Aprodu, Marian, and Gavril Farkas. “Koszul cohomology and applications to moduli”, to appear in
Aspects of vector bundles and moduli, Clay Mathematical Institute Volume 10 [Chan] Chan, Kungho. “A characterization of double covers of curves in terms of the ample cone of secondsymmetric product”,
J. Pure. App. Alg.
212 no. 12 (2008): 2623-32.[Cot] Cotterill, Ethan. “Geometry of curves with exceptional secant planes: linear series along the generalcurve”, to appear in
Math. Zeit. [Deb] Debarre, Olivier. “Seshadri Constants Of Abelian Varieties”, in
The Fano Conference , Univ. Torino,Turin, 2004: 379-94.[Ein] Ein, Lawrence. “A remark on the syzygies of the generic canonical curves”,
J. Differential Geom.
The Geometry of Syzygies , New York: Springer-Verlag, 2005.[Fa] Farkas, Gavril. “Higher ramification and varieties of secant divisors on the generic curve”,
J. LondonMath. Soc.
78 (2008): 418-40.[vdGK] van der Geer, Gerard, and Alexis Kouvidakis. “Cycle Relations On Jacobian Varieties”,
CompositioMath
143 (2007): 900-08.[Gr] Green, Mark. “Koszul Cohomology And The Geometry Of Projective Varieties”,
J. Differential Geom.
19 (1984): 125-71.[Herb] Herbaut, Fabien. “Algebraic Cycles On The Jacobian Of A Curve With A Linear System Of GivenDimension”,
Compositio Math.
143 (2007): 883-99.[Iz1] Izadi, Elham. “Deforming Curves In Jacobians To Non-Jacobians I: Curves in C (2) ”, Geom. Dedicata
116 (2005): 87-109.[Iz2] Izadi, Elham. “Deforming curves In Jacobians To Non-Jacobians II. Curves in C ( e ) , ≤ e ≤ g − Geom. Dedicata
115 (2005): 33-63. Kou] Kouvidakis, Alexis. “Divisors on Symmetric Products of Curves”,
Trans. Amer. Math. Soc.
337 (1993):117-28.[Laz] Lazarsfeld, Robert.
Positivity in Algebraic Geometry II, New York: Springer-Verlag, 2004. [Mus] Mustopa, Yusuf. “Residuation of Linear Series and the Effective Cone of C d ”, to appear in Amer. J.Math. [Pac] Pacienza, Gianluca. “On the nef cone of symmetric products of a generic curve”,
Amer. J. Math.
Algebraic Ge-ometry and Commutative Algebra II , Kinokuniya, Tokyo, 1984.[Pir] Pirola, Gian Pietro. “Base Number Theorem for Abelian Varieties”,
Math. Ann.
282 (1988): 361-368[Tei] Teixidor i Bigas, Montserrat. “Syzygies using vector bundles”,
Trans. Amer. Math. Soc.
359 (2007):897-908.[V] Voisin, Claire. “Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface”,
J.Eur. Math. Soc.2 (2004): 363-404.