Du Val curves and the pointed Brill-Noether Theorem
aa r X i v : . [ m a t h . AG ] A p r DU VAL CURVES AND THE POINTED BRILL-NOETHER THEOREM
GAVRIL FARKAS AND NICOLA TARASCA
Abstract.
We show that a general curve in an explicit class of what we call Du Valpointed curves satisfies the Brill-Noether Theorem for pointed curves. Furthermore, weprove that a generic pencil of Du Val pointed curves is disjoint from all Brill-Noetherdivisors on the universal curve. This provides explicit examples of smooth pointed curvesof arbitrary genus defined over Q which are Brill-Noether general. A similar result isproved for 2-pointed curves as well using explicit curves on elliptic ruled surfaces. The pointed Brill-Noether Theorem concerns the study of linear series on a generalpointed algebraic curve [
C, p ] with prescribed ramification at the marked point p . Recallthat for a point p ∈ C and a linear series ℓ = ( L, V ) ∈ G rd ( C ), one denotes by α ℓ ( p ) : 0 ≤ α ℓ ( p ) ≤ . . . ≤ α ℓr ( p ) ≤ d − r the ramification sequence of ℓ at p . One says that p ∈ C is a ramification point of ℓ if α ℓr ( p ) >
0. For instance, the ramification points of the canonical linear series areprecisely the Weierstrass points of C . The total number of ramification points of ℓ , countedwith appropriate multiplicities, is given by the Pl¨ucker formula , see for instance [EH1]Proposition 1.1. Fixing a Schubert index α : 0 ≤ α ≤ . . . ≤ α r ≤ d − r , one can ask whena general pointed curve [ C, p ] of genus g carries a linear series ℓ ∈ G rd ( C ) with ramificationsequence α ℓ ( p ) ≥ α . The locus G rd ( C, p, α ) of linear series on C satisfying this conditionis a generalized determinantal variety of expected dimension ρ ( g, r, d, α ) := ρ ( g, r, d ) − w ( α ) , where ρ ( g, r, d ) := g − ( r + 1)( g − d + r ) and w ( α ) := α + · · · + α r is the weight of α .It is proved in [EH2] Theorem 1.1 that for a general pointed curve [ C, p ] ∈ M g, , eachcomponent of G rd ( C, p, α ), if nonempty, has dimension precisely ρ ( g, r, d, α ). Moreover,[EH2] Proposition 1.2 establishes that G rd ( C, p, α ) = ∅ if and only if r X i =0 max { α i + g − d + r, } ≤ g. The proofs in [EH2] rely on limit linear series and degeneration to the boundary of theuniversal curve C g := M g, . Up to now, no examples whatsoever of smooth pointed curves[ C, p ] ∈ C g verifying the pointed Brill-Noether Theorem have been known. This situationcontrasts the classical Brill-Noether Theorem; even though the original proof in [GH] useddegeneration to nodal curves, soon afterwards, in his well-known paper [Laz], Lazarsfeldshowed that sections of general polarized K Mathematics Subject Classification.
Key words and phrases.
Brill-Noether general smooth pointed curves, Du Val curves, rational and ruledsurfaces.
Since curves in the polarization class of a K explicit specialization of Lazarsfeld’s curves emerging from the paper [ABS]is worked out. It is shown that suitably general singular plane curves of degree 3 g havingmultiplicity g at eight points in P and multiplicity g − M g of thelocus of curves lying on K Du Val curves of genus g .One aim of this paper is to show that the Du Val curves introduced in [ABFS] lead toBrill-Noether general smooth pointed curves of any genus defined over Q . The essential ob-servation is that, unlike curves on general K S ′ be the blow-up of P at nine points p , . . . , p which are general in the sense of [ABFS] (see also Section 1 for the precisedefinition). Let E , . . . , E be the exceptional curves on S ′ . We denote by J ′ ∈ | − K S ′ | the unique smooth plane cubic passing through p , . . . , p and consider the linear systemon S ′ L g := (cid:12)(cid:12) gℓ − gE − · · · − gE − ( g − E (cid:12)(cid:12) , where ℓ ∈ Pic( S ′ ) is the proper transform of a line in P . The main result of [ABFS] isthat a general curve C ′ ∈ L g verifies the Brill-Noether-Petri Theorem. For each g ≥ p , . . . , p determine a 10-th point p = p ( g )10 which is the base point of L g . Infact, p ∈ C ′ · J ′ , for every C ′ ∈ L g . The point p is determined by the relation(1) p = p ( g )10 = − gp − · · · − gp − ( g − p ∈ J ′ , with respect to the group law of the elliptic curve. Under the genericity assumptions onthe points p , . . . , p we started with, the points p ( g )10 are distinct from one another, as wellas from p , . . . , p , see also Proposition 1. As in [ABFS], we set S := Bl p ( S ) and, byslight abuse of notation, we denote by E , . . . , E the corresponding exceptional curves.If C is the strict transform of C ′ , then |O S ( C ) | is a base point free linear system of curvesof genus g having a section induced by E (note that C · E = 1).A pointed Du Val curve is a smooth pointed curve [ C, p ] ∈ C g , where C ⊂ S is as aboveand { p } = C · E . Before stating our main results, we recall that for a linear system ℓ ∈ G rd ( C ) and points p , . . . , p n ∈ C , the pointed Brill-Noether number is defined as ρ ( ℓ, p , . . . , p n ) := ρ ( g, r, d ) − w (cid:0) α ℓ ( p ) (cid:1) − · · · − w (cid:0) α ℓ ( p n ) (cid:1) . Theorem 1.
A general pointed Du Val curve [ C, p ] verifies the pointed Brill-NoetherTheorem, that is, dim G rd ( C, p, α ) = ρ ( g, r, d, α ) , when G rd ( C, p, α ) = ∅ . In particular, forevery linear system ℓ on C , one has ρ ( ℓ, p ) ≥ . Since the points p , . . . , p can be chosen to have rational coefficients, p = p ( g )10 ∈ P ( Q )and then [ C, p ] is also defined over Q . Hence, paralleling [ABFS] Corollary 1.3, our The-orem 1 provides examples of Brill-Noether general pointed curves of arbitrary genus g defined over Q . U VAL CURVES AND THE POINTED BRILL-NOETHER THEOREM 3 If W g denotes the locus of Weierstrass points in C g (known to be an irreducible divisor onthe universal curve), by direct calculation we show that the image of the family j : P → C g induced by a Lefschetz pencil of Du Val curves on S satisfies j ( P ) ∩ W g = ∅ , that is, for every pointed Du Val curve [ C, p ], the marked point p is not a Weierstrasspoint of C . As we point out in Corollary 1, this implies that j ( P ) is disjoint from allpointed Brill-Noether divisors on C g . We refer to Section 1 for detailed background onpointed Brill-Noether divisors on C g .0.1. Brill-Noether general -pointed curves on elliptic ruled surfaces. The Brill-Noether problem can be formulated for n -pointed curves [ C, p , . . . , p n ] and concernsthe variety of linear series ℓ ∈ G rd ( C ) having prescribed ramification α ℓ ( p i ) ≥ α i for i = 1 , . . . , n , given in terms of fixed Schubert indices α , . . . , α n . In Section 2, using decomposable elliptic ruled surfaces, we exhibit for the first time examples of smooth 2-pointed curves of arbitrary genus verifying the 2-pointed Brill-Noether Theorem. Theconstruction is inspired by a very nice note of Treibich [Tre].We start with an elliptic curve J and a non-torsion line bundle η ∈ Pic ( J ). Thedecomposable ruled surface φ : Y := P ( O J ⊕ η ) → J is endowed with two disjoint sections J and J respectively. Pick a point r ∈ J anddenote by f := φ − ( r ) the corresponding ruling of Y . We denote by s = s ( g ) ∈ J thepoint determined by the equation O J ( s − r ) = η ⊗ g . The linear system | gJ + f | consistsof curves of genus g and has two base points, namely { p } := φ − ( r ) · J and { q } := φ − ( s ) · J , respectively. We establish the following result: Theorem 2.
The -pointed curve [ C, p, q ] ∈ M g, , where C ∈ | gJ + f | is a general elementand p and q are as above, verifies the -pointed Brill-Noether Theorem. In particular, forevery linear series ℓ ∈ G rd ( C ) the inequality ρ ( ℓ, p, q ) ≥ holds. A Brill-Noether general 2-pointed curve supports a Brill-Noether general 1-pointedcurve obtained by dropping either marked point. In particular, both 1-pointed curves[
C, p ] and [
C, q ] in the statement of Theorem 2 verify the 1-pointed Brill-Noether Theo-rem as well. For details, we refer to Section 2. The proofs of both Theorems 1 and 2 areintimately related, and rely on a canonical degeneration within the corresponding linearsystem on the surface to a singular curve with an elliptic tail. This leads to an inductiveargument in the genus, which ultimately proves the desired Brill-Noether type theorems.Arguably, for many applications, the curves constructed in Theorem 2 are the simplestknown examples of Brill-Noether general smooth curves of arbitrary genus. They combinetwo desirable features: (i) The canonical elliptic tail degeneration in | gJ + f | providesa system of Brill-Noether general curves of any genus on the surface Y , which invitesinductive proofs and reduction to genus 1 curves and Schubert calculus problems in thespirit of limit linear series, and (ii) The general curve in | gJ + f | being smooth, one neednot build-up the degeneration set-up typical for limit linear series applications. A vivid G. FARKAS AND N. TARASCA instance of their use is the recent proof in [FK1] of the Prym-Green Conjecture concerningthe naturality of the resolution of a paracanonical curve ϕ K C ⊗ η : C ֒ → P g − , where C isa general curve of odd genus and η is an ℓ -torsion line bundles on C . The conjecture isproven for odd g and arbitrary ℓ using precisely the curves constructed in Theorem 2. Fora proof of the Prym-Green Conjecture using special K ℓ ≥ q g +22 — see [FK2].Both classes of curves constructed in Theorems 1 and 2 lie in the closure of the locusin M g of curves contained in a K Y is replaced by an indecomposable ruledsurface over an elliptic curve, provides examples of Brill-Noether general pointed curveswhich are not limits of K Acknowledgments:
Section 1 of this paper uses in an essential way the methods devel-oped in [ABFS]. The first author is grateful to Enrico Arbarello for interesting discussionsrelated to this circle of ideas. The presentation of the paper clearly benefitted from theinsightful remarks of two referees, whom we thank.1.
Pointed Du Val curves and Weierstrass points
We assume familiarity with the theory of limit linear series in the sense of [EH1]. Weneed a few facts concerning divisor classes on the universal curve C g := M g, . The rationalPicard group Pic( C g ) is generated by the Hodge class λ , the relative cotangent class ψ ,the boundary divisor class δ irr := [∆ irr ] of irreducible pointed stable curves of genus g andby the classes δ i := [∆ i ], where for each i = 1 , . . . , g −
1, the boundary divisor ∆ i ⊂ C g corresponds to a transverse union of two smooth curves of genus i and g − i respectively,meeting in one point, the marked points lying on the genus i component. If π : C g → M g is the morphism forgetting the marked point, the boundary divisors on M g and those on C g are related by the following formulas: π ∗ ( δ irr ) = δ irr , π ∗ ( δ i ) = δ i + δ g − i , for 1 ≤ i < g , and π ∗ ( δ g ) = δ g , for g even . If α : 0 ≤ α ≤ . . . ≤ α r ≤ d − r is a Schubert index of type ( r, d ), we introducethe complementary Schubert index α c : 0 ≤ d − r − α r ≤ . . . ≤ d − r − α ≤ d − r .When α i = 0 for i = 0 , . . . , r , we say that α is the trivial Schubert index. We recall thedefinition of pointed Brill-Noether divisors on C g . Fix integers r, d ≥ α : 0 ≤ α ≤ . . . ≤ α r ≤ d − r such that the expected dimension of the locus oflinear series of type g rd on a curve of genus g with prescribed ramification α at a givenpoint equals −
1. In other words, ρ ( g, r, d, α ) := g − ( r + 1)( g − d + r ) − w ( α ) = − . Let C rg,d ( α ) := (cid:8) [ C, p ] ∈ C g : G rd ( C, p, α ) = ∅ (cid:9) be the corresponding pointed Brill-Noetherlocus. For instance, W g := C g − g, g − (0 , . . . , ,
1) = n [ C, p ] ∈ C g : H ( C, ω C ( − gp )) = 0 o is the divisor of Weierstrass points. Since W g can be parametrized by the Hurwitz space of g -fold covers of P having a point of total ramification, it follows from [Arb] Theorem 2.5 U VAL CURVES AND THE POINTED BRILL-NOETHER THEOREM 5 that W g is an irreducible divisor. If ρ ( g, r, d ) = −
1, then C rg,d (0 , . . . ,
0) is the pull-back to C g of the Brill-Noether divisor M rg,d consisting of curves carrying a g rd .Cukierman [Cuk] computed the class of the closure W g of the Weierstrass divisor in C g :(2) [ W g ] = − λ + (cid:18) g + 12 (cid:19) ψ − g − X i =1 (cid:18) g − i + 12 (cid:19) δ i ∈ Pic( C g ) . We also recall [EH2] that the class of the pull-back to C g of the Brill-Noether divisors M rg,d is given by the formula(3) [ C rg,d (0 , . . . , c g,d,r · BN g , where c g,d,r ∈ Q > and BN g := ( g + 3) λ − g + 16 δ irr − g − X i =1 i ( g − i ) δ i ∈ Pic( C g ) . Remarkably, the pointed Brill-Noether divisors only span a 2-dimensional cone in Pic( C g ).It is shown in [EH3] Theorem 1.2 that C rg,d ( α ) is a proper subvariety of C g , having aunique divisorial component. The class of this component, which we shall denote by[ C rg,d ( α )] ∈ CH ( C g ), can be written as a linear combination[ C rg,d ( α )] = µ · [ W g ] + ν · BN g , for non-negative rational constants µ and ν , which are determined in [FT]. Definition 1.
We say that a pointed curve [
C, p ] ∈ C g is Brill-Noether general , if for everychoice of integers r, d and a corresponding Schubert index α of type ( r, d ), we havedim G rd ( C, p, α ) = ρ ( g, r, d, α ) or G rd ( C, p, α ) = ∅ . In particular, for every linear series ℓ ∈ G rd ( C ), the inequality ρ ( ℓ, p ) ≥ C, p ] is a Brill-Noether general pointed curve, by letting α be the trivial Schubertindex, we obtain that C is a Brill-Noether general (unpointed) curve. Lemma 1.
A pointed curve [ C, p ] ∈ C g carries no linear series ℓ with ρ ( ℓ, p ) < if andonly if it does not belong to any locus C rg,d ( α ) , where ρ ( g, r, d, α ) = − .Proof. One implication being obvious, assume first there exists a linear series ℓ ∈ G rd ( C )with w (cid:0) α ℓ ( p ) (cid:1) > ρ ( g, r, d ) ≥
0. Then we can find a Schubert index α ′ : 0 ≤ α ′ ≤ . . . ≤ α ′ r ≤ d − r with w ( α ′ ) = ρ ( g, r, d ) + 1 ≤ w (cid:0) α ℓ ( p ) (cid:1) , such that α ′ ≤ α ℓ ( p ) (lexicographically). Hence ρ ( g, r, d, α ′ ) = − C, p ] ∈ C rg,d ( α ′ ). Finally, assume we are in the case when thereexists a linear series ℓ ∈ G rd ( C ) with ρ ( g, r, d ) < −
1. Then we can find d ′ > d and aSchubert index α ′ : 0 ≤ α ′ ≤ . . . ≤ α ′ r ≤ d ′ − r with α ′ i ≤ d ′ − d and w ( α ′ ) = ρ ( g, r, d ′ ) + 1.Hence [ C, p ] ∈ C rg,d ′ ( α ′ ), which finishes the proof. (cid:3) G. FARKAS AND N. TARASCA
We now turn to Du Val surfaces. In what follows, we denote by ≡ linear equivalence ofdivisors on varieties. Following [ABFS] Proposition 2.3, we recall that a set of nine distinctpoints p , . . . , p in P is said to be general if on the blown-up plane S ′ := Bl { p ,...,p } ( P ),every effective divisor D ′ ≡ dℓ − ν E − · · · − ν E with ν i ≥ D · J ′ = 0 is necessarily a multiple of J ′ . In particular, if p , . . . , p are general points, then the sum p + · · · + p ∈ J ′ is not torsion. Remark . Examples of sets of nine general points in P ( Q ) are easy to produce, if onestarts with a concrete elliptic curve defined over Q . For instance, it is shown in [ABFS] thatthe following points lying on the elliptic curve E : y = x + 17 are general: p = ( − , p = ( − , − , p = (2 , , p = (4 , p = (52 , p = (5234 , p = (8 , − p = (43 , p = (cid:16) , − (cid:17) . Recall the definition (1) of the points p ( g )10 ∈ J ′ , where g ≥ Proposition 1.
Assume that the points p , . . . , p are general. Then for k = 2 , . . . , g , thedifference p ( k )10 − p ( k − ∈ Pic ( J ′ ) is not torsion.Proof. Using (1), we obtain that p ( k − − p ( k )10 = p + · · · + p (with respect to the grouplaw of J ′ ), for each k ≥
2. As pointed out, this is not a torsion point on J ′ . (cid:3) We now introduce the pointed Du Val pencil in C g , which is a lift under the forgetfulmap π : C g → M g of the pencil of unpointed curves introduced in Section 4 of [ABFS].Recall that S := Bl p ( g )10 ( S ′ ) and we denote by L g the proper transform of the linear systemon S ′ denoted by the same symbol in the Introduction. The linear system of Du Val curvesof genus g − S , that is,Λ g − := (cid:12)(cid:12) g − ℓ − ( g − E − · · · − ( g − E − ( g − E (cid:12)(cid:12) appears as a hyperplane in the g -dimensional linear system L g . It consists precisely ofthe curves of the form D + J ∈ L g , where J ⊂ S denotes the proper transform of J ′ and D ∈ Λ g − . Since J ≡ ℓ − E − · · · − E , note that D · J = 1.We now choose a Lefschetz pencil in L g , which has 2 g − C base points. Let X := Bl g − ( S ) be the blow-up of S at those points. Since C · E = 1 for C ∈ L g , thecorresponding fibration f : X → P has a section induced by the proper transform of E on X . This induces a pencil in the universal curve j : P → C g . In what follows it will be convenient to use the notation C ∪ p C , for a stable curveconsisting of two irreducible components C and C respectively, meeting transversally ata point p . Proposition 2.
The intersection numbers of the pointed Du Val pencil with the generatorsof
Pic( C g ) are as follows: j ∗ ( λ ) = g, j ∗ ( ψ ) = 1 , j ∗ ( δ irr ) = 6( g + 1) , j ∗ ( δ ) = 1 , j ∗ ( δ i ) = 0 for i = 2 , . . . , g − . U VAL CURVES AND THE POINTED BRILL-NOETHER THEOREM 7
Proof.
One has j ∗ ( ψ ) = − E = 1. The restrictions of the classes λ, δ irr , δ , . . . , δ g − follow from [ABFS] Theorem 4.1 and are copied here for the sake of completeness. Thereexists precisely one element of the pencil f of the type D + J , for some D ∈ Λ g − . Since E · J = 1 while E · D = 0, the marked point lies on the elliptic component of thissingular element. The corresponding pointed stable curve is (cid:2) D ∪ p ( g − J ′ , p ( g )10 (cid:3) ∈ C g .Hence j ∗ ( δ ) = 1, and since π ∗ ( δ ) = δ + δ g − , it follows that j ∗ ( δ g − ) = 0. (cid:3) By direct computation, using (2) and (3), it follows that the pencil j ( P ) ⊂ C g hasintersection number zero with the Brill-Noether class BN g as well as with the Weierstrassdivisor W g , that is, j ∗ ( BN g ) = ( g + 3) g − g + 16 (6 g + 6) − ( g −
1) = 0 , and j ∗ ([ W g ]) = − g + (cid:18) g + 12 (cid:19) − (cid:18) g (cid:19) = 0 . Since the class of any pointed Brill-Noether divisor lies in the cone spanned by these classes[EH3] Theorem 1.2, it follows that the intersection number of j ( P ) with the closure ofany pointed Brill-Noether divisor is zero as well.We are now in a position to complete the proof of our main result. Proof of Theorem 1.
We shall establish by induction on g that the general member ofthe Du Val pencil satisfies the pointed Brill-Noether Theorem. For g = 1, we have that[ C, p ] ∈ C and it is well-known that each smooth pointed elliptic curve is Brill-Noethergeneral, see e.g. [EH2] Theorem 1.1. Assuming the statement for Du Val curves of genus g −
1, suppose by contradiction that there exist r, d ≥ α such thatdim G rd ( C, p, α ) > ρ ( g, r, d, α ), for each C ∈ L g , where { p } = C ∩ E .Let j : P → C g be a Lefschetz pencil of Du Val curves on S . As explained in Proposition2, the pencil contains a unique elliptic tail degeneration (cid:2) D ∪ p ( g − J ′ , p ( g )10 (cid:3) , where D is anelement of Λ g − . Then the variety G rd (cid:16) D ∪ J ′ , p ( g )10 , α (cid:17) of limit linear series ℓ = ( ℓ D , ℓ J ′ ) ∈ G rd ( D ) × G rd ( J ′ ) on D ∪ p ( g − J ′ satisfying the ramifica-tion condition α ℓ (cid:0) p ( g )10 (cid:1) ≥ α is of dimension at least ρ ( g, r, d, α ) + 1. Note that [ D, p ( g − ]can be assumed to be a general Du Val curve of genus g −
1, for every curve from Λ g − appears as an elliptic tail degeneration in a genus g Du Val pencil.Let ℓ be a general point of an irreducible component Z of G rd (cid:16) D ∪ J ′ , p ( g )10 , α (cid:17) of maximaldimension, and set β := α ℓ D ( p ( g − ). By the additivity of the Brill-Noether number withrespect to marked points, we write ρ ( g, r, d, α ) = ρ (cid:16) ℓ, p ( g )10 (cid:17) ≥ ρ (cid:16) ℓ D , p ( g − (cid:17) + ρ (cid:16) ℓ J ′ , p ( g )10 , p ( g − (cid:17) . G. FARKAS AND N. TARASCA p ( g )10 p ( g − p ( g − k )10 kJ ′ D = D g − k Figure 1.
The k -th step of the elliptic tail specialization in a Du Val pencil.By the construction in [EH1] Theorem 3.3 of the variety of limit linear series, Z is birationalto an irreducible component of the product G rd (cid:16) D, p ( g − , β (cid:17) × G rd (cid:16) J ′ , ( p ( g − , β c ) , ( p ( g )10 , α ) (cid:17) . By assumption, each component of G rd (cid:16) D, p ( g − , β (cid:17) has dimension ρ ( g − , r, d, β ).Moving to J ′ , first observe that ρ (cid:16) ℓ J ′ , p ( g )10 , p ( g − (cid:17) ≥
0. Indeed, assuming otherwise, wedenote by ( a , . . . , a r ) and ( b , . . . , b r ) the vanishing sequences of ℓ J ′ at the points p ( g − and p ( g )10 respectively, and obtain that there exist indices 0 ≤ i < j ≤ r such that a i + b r − i = a j + b r − j = d. In particular, the underlying line bundle of the linear series ℓ J ′ corresponds to the divisors a i · p ( g − + b r − i · p ( g )10 ≡ a j · p ( g − + b r − j · p ( g )10 , from which it follows that p ( g − − p ( g )10 is atorsion class in Pic ( J ′ ), which contradicts Proposition 1.Furthermore, it implicitly follows from [EH1], and it is spelled-out explicitly in [Oss]Lemma 2.1, that every 2-pointed elliptic curve [ E, x, y ] ∈ M , , where the difference O E ( x − y ) is not a torsion class, is Brill-Noether general. This follows from the observationthat for a line bundle L ∈ Pic d ( E ) which is not given by a divisor on E supported onlyat x and y , the flags in H ( E, L ) ∼ = C d given by the vanishing of sections at x and y respectively, are transversal. In particular, Schubert cycles in G (cid:0) r + 1 , H ( E, L ) (cid:1) definedin terms of these flags intersect in the expected dimension. Applying this fact to the caseat hand, we finddim G rd (cid:16) J ′ , ( p ( g − , β c ) , ( p ( g )10 , α ) (cid:17) = ρ (1 , r, d, β c , α ) := ρ (1 , r, d ) − w ( β c ) − w ( α ) . Putting all together, we obtain that ρ ( g, r, d, α ) < dim Z = dim G rd (cid:16) D, p ( g − , β (cid:17) + dim G rd (cid:16) J ′ , ( p ( g − , β c ) , ( p ( g )10 , α ) (cid:17) = ρ ( g − , r, d, β ) + ρ (1 , r, d, β c , α ) ≤ ρ ( g, r, d, α ) , which is a contradiction. Therefore, the singular pointed curve (cid:16) D ∪ J ′ , p ( g )10 (cid:17) is Brill-Noether general. (cid:3) U VAL CURVES AND THE POINTED BRILL-NOETHER THEOREM 9
Corollary 1.
The image of a Du Val pencil j : P → C g is disjoint from all pointedBrill-Noether divisors C rg,d ( α ) .Proof. As noted in Proposition 2, we have j ( P ) · C rg,d ( α ) = 0. Either j ( P ) ∩ C rg,d ( α ) = ∅ ,or else, j ( P ) ⊂ C rg,d ( α ). The proof of Theorem 1 rules out the second possibility. (cid:3) In general it is not known whether C rg,d ( α ) is pure of codimension 1. However, whenthis happens, for instance in the case of the Weierstrass divisor W g , Corollary 1 showsthat every pointed Du Val curve is Brill-Noether general with respect to linear series ofthat type.1.1. Towards the effective cone of C g . The Slope Conjecture [HM] on effective divisorson M g used to predict that the Brill-Noether divisors M rg,d of curves with a linear series g rd where ρ ( g, r, d ) = − R ⊂ M g induced by a Lefschetz pencilof genus g curves on a general polarized K X, H ), with H = 2 g − nef , thatis, it intersects every effective divisor on M g non-negatively. Note that the intersectionnumbers of R with the generators of Pic( M g ) are given as follows, see for instance [FP]: R · λ = g + 1 , R · δ irr = 6 g + 18 and R · δ i = 0 , for i = 1 , . . . , j g k . Although the Slope Conjecture is false for high g , see [FP] and [Far], it is known to hold for g ≤ g = 11. The statement played an important role in Mukai’s work on alternativebirational models of M g for g = 7 , , D on M g having small slope, that is, satisfying R · D <
0, which necessarily contain thelocus in M g of curves that lie on K C g . Problem 1.
For what values of g is the Du Val pencil j : P → C g nef, that is, j ∗ ( D ) ≥ ,for every effective divisor D on C g ? For which g does this inequality hold for all effectivedivisors D on C g such that π ( D ) = M g ? In light of Corollary 1, a closely related question is whether the Weierstrass divisor W g is extremal in the effective cone Eff( C g ). The hypothesis that W g is extremal has recentlyreceived further credence due to [Pol]. Note that for the pull-backs to C g of the effectivedivisors on M i +10 constructed in [Far], Problem 1 has a negative answer. For instance,when g = 10, the divisor in question is Z := (cid:8) [ C, p ] ∈ C : C lies on a K (cid:9) , and [ Z ] = 7 λ − δ irr − δ − δ − δ − δ − · · · ∈ Pic( C ), see [FP] Theorem 1.6. Byapplying Proposition 2, we compute j ∗ ([ Z ]) = − <
0. We are unaware of any exampleof an effective divisor D on C g that is not a pull-back of an effective divisor from M g andwhich satisfies j ∗ ( D ) < Brill-Noether general two-pointed curves via elliptic surfaces
In this section we construct explicit smooth 2-pointed curves of arbitrary genus verifyingthe Brill-Noether Theorem. Given a smooth curve C , distinct points p, q ∈ C and twoSchubert indices α : 0 ≤ α ≤ . . . ≤ α r ≤ d − r and β : 0 ≤ β ≤ . . . ≤ β r ≤ d − r, we consider the variety G rd (cid:16) C, ( p, α ) , ( q, β ) (cid:17) of linear series ℓ ∈ G rd ( C ) verifying ramifica-tion conditions at two points: α ℓ ( p ) ≥ α and α ℓ ( q ) ≥ β. We say that [
C, p, q ] satisfies the 2-pointed Brill-Noether Theorem if for any α and β ,dim G rd (cid:16) C, ( p, α ) , ( q, β ) (cid:17) = ρ ( g, r, d, α, β ) := ρ ( g, r, d ) − w ( α ) − w ( β ) , unless G rd (cid:16) C, ( p, α ) , ( q, β ) (cid:17) = ∅ . Eisenbud and Harris [EH2] Theorem 1.1 established the2-pointed Brill-Noether Theorem for general 2-pointed curves by use of degeneration. Asin the case of 1-pointed curves, up to now no explicit example of a smooth Brill-Noethergeneral 2-pointed curve has been known. We construct such curves using decomposableelliptic ruled surfaces.We start with an elliptic curve J and consider a non-torsion line bundle η ∈ Pic ( J ).Let φ : Y := P ( O J ⊕ η ) → J be the ruled surface corresponding to a decomposable rank 2 vector bundle. We denote by J and J the disjoint sections of Y such that N J /Y = η and N J /Y = η ∨ . In particular, J = J = 0. Observe that J ≡ J − φ ∗ ( η ). We fix a point r ∈ J and let f = f r := φ − ( r ) be the corresponding ruling. For each g ≥
1, we denote by s = s ( g ) thepoint on the base elliptic curve J determined by O J ( s ( g ) − r ) = η ⊗ g . Since η is not a torsion class, we have s ( g ) = r , for all g ≥
1. Furthermore, the difference s ( g ) − s ( g − ∈ Pic ( J ) is not a torsion class. As explained in the Introduction, we set { p } = J · f r and { q ( g ) } := J · f s ( g ) . Lemma 2.
We have that h ( Y, O Y ( gJ + f r )) = g + 1 . The general point of the linearsystem | gJ + f r | is a smooth curve of genus g passing through the points p and q ( g ) .Proof. By direct calculation, using Riemann-Roch, we find that h ( Y, O Y ( gJ + f r )) = h (cid:16) O J ( r ) ⊗ Sym g ( O J ⊕ η ) (cid:17) = deg (cid:16) O J ( r ) ⊗ Sym g ( O J ⊕ η ) (cid:17) = g + 1 . Furthermore, since K Y ≡ − J + φ ∗ ( η ) ≡ − J + φ ∗ ( η ∨ ), from the adjunction formula weobtain that a smooth curve C ∈ | gJ + f r | has genus g .From [FGP] Proposition 11, since η is non-torsion, the base points of | gJ + f r | lie on J + J = |− K Y | . Since O J ( gJ + f r ) = O J ( p ), the point p must lie in the base locus of U VAL CURVES AND THE POINTED BRILL-NOETHER THEOREM 11 | gJ + f r | . Finally, since O J ( gJ + f r ) = η ⊗ g ⊗ O J ( f r ) = O J ( q ( g ) ), it follows that q ( g ) belongs to each curve C ∈ | gJ + f r | . Hence, the base locus of | gJ + f r | consists of thepoints p and q ( g ) . (cid:3) Therefore, on each curve from the linear system | gJ + f r | we can single out two markedpoints, p and q = q ( g ) . These are precisely the points for which the Brill-Noether Theoremwill be proved. Theorem 3.
The -pointed curve [ C, p, q ] ∈ M g, , where C ∈ | gJ + f r | is general and p and q := q ( g ) are as above, verifies the -pointed Brill-Noether Theorem, that is, dim G rd (cid:16) C, ( p, α ) , ( q, β ) (cid:17) = ρ ( g, r, d, α, β ) or G rd (cid:16) C, ( p, α ) , ( q, β ) (cid:17) = ∅ , for all Schubert indices α and β .Proof. Assume by contradiction that for a 2-pointed curve [
C, p, q ( g ) ], where C ∈ | gJ + f | is a general element, the Brill-Noether Theorem fails for certain Schubert indices α and β , that is, there exists a component of G rd (cid:16) C, ( p, α ) , ( q, β ) (cid:17) whose dimension exceeds ρ ( g, r, d, α, β ). Then, similarly to the proof of Theorem 1, we consider a specialization of C to the sublinear system { J } + | ( g − J + f r | ∼ = P g − , which appears as a hyperplanein | gJ + f r | ∼ = P g . The 2-pointed curve corresponding to the general element of thissubsystem is a curve of the form[ D ∪ J , p ∈ D, q ( g ) ∈ J ] ∈ M g, , where D ∈ | ( g − J + f r | is a smooth curve of genus g − p and thepoint q ( g − ∈ J · f s ( g − . Note that D ∩ J = { q ( g − } . Observe moreover that under theisomorphism φ = φ | J : J ∼ = → J , we have q ( g ) − q ( g − = φ ∗ ( s ( g ) ) − φ ∗ ( s ( g − ) = φ ∗ ( η ) ∈ Pic ( J ) , that is, the difference q ( g ) − q ( g − is not torsion on J .The proof now follows by induction. By semicontinuity, the variety of limit linear series ℓ of type g rd on D ∪ J verifying the ramification conditions α ℓ ( p ) ≥ α and α ℓ ( q ( g ) ) ≥ β must have a component Z of dimension strictly greater than ρ ( g, r, d, α, β ). Denote by ℓ = ( ℓ D , ℓ J ) a general point of Z . We may assume that ℓ is a refined limit linear series.Set γ := α ℓ D ( q ( g − ). Then Z is birationally isomorphic to the product G rd (cid:16) D, ( p, α ) , ( q ( g − , γ ) (cid:17) × G rd (cid:16) J , ( q ( g − , γ c ) , ( q ( g ) , β ) (cid:17) . By induction on the genus, we may assume that [
D, p, q ( g − ] ∈ M g − , satisfies the2-pointed Brill-Noether Theorem, in particulardim G rd (cid:16) D, ( p, α ) , ( q ( g − , γ ) (cid:17) = ρ ( g − , r, d, α, γ ) . Since q ( g ) − q ( g − ∈ Pic ( J ) is not torsion, as we have observed [ J , q ( g − , q ( g ) ] ∈ M , is a Brill-Noether general 2-pointed curve, hencedim G rd (cid:16) J , ( q ( g − , γ c ) , ( q ( g ) , β ) (cid:17) = ρ (1 , r, d, γ c , β ) . Using the additivity of the Brill-Noether number, we havedim Z = ρ ( g − , r, d, α, γ ) + ρ (1 , r, d, γ c , β ) = ρ ( g, r, d, α, β ) , a contradiction. (cid:3) Remark . Since a Brill-Noether general n -pointed curve supports a Brill-Noether general m -pointed curve for all m < n obtained by dropping n − m of the marked points, it followsthat the curve C ∈ | gJ + f r | satisfies the (unpointed) Brill-Noether Theorem as well.2.1. Brill-Noether general pointed curves which are not limits of K sections. The Du Val curves considered in [ABFS] and in Section 1 of this paper are known to liein the closure in M g of the locus of curves of genus g lying on a K S ⊂ P g having canonical hyperplane sections have been classified by Epema[Epe]. All such surfaces are potentially limits in P g of smooth polarized K g −
2. A criterion for when such surfaces smooth to K K indecomposable elliptic ruled surfacesand is inspired by [Tre].We fix again an elliptic curve J and denote by E the unique indecomposable vectorbundle on J defined by the exact sequence0 −→ O J → E −→ O J −→ . Let ϕ : X ′ := P ( E ) → J be the induced ruled surface. We fix a point r ∈ J andset f := ϕ − ( r ), therefore f = 0. Let J ⊂ X ′ be the unique section of ϕ having N J /Y ′ = O J and set { q } := J · f r . In a way similar to the proof of Lemma 2, one canshow that the general element of the linear system | gJ + f | is a curve of genus g passingthrough the point q .Each curve C ∈ | gJ + f | has a distinguished marked point, namely q ∈ C · J . In [Tre],Treibich considers curves in the linear system | gJ + f | and sketches an argument usingFay’s trisecant formula for showing that a general curve C ∈ | gJ + f | is Brill-Noethergeneral. Reasoning in a way very similar to the proof of Theorem 3, we prove the strongerfact that the general curve [ C, q ] satisfies the pointed Brill-Noether Theorem.Since the linear system | gJ + f | has a base point, we denote by ǫ : X := Bl q ( X ′ ) → X ′ the blow-up of X at q and by E the exceptional divisor. We keep denoting by J and f , the strict transforms of the curves denoted by the same symbols on X ′ . Finally, let C ⊂ X be the strict transform of a curve in the linear system | gJ + f | . Then C · E = 1and C = 2 g −
1. Since | C | is base point free, a Lefschetz pencil in this linear systeminduces a family of pointed curves ι : P → C g . U VAL CURVES AND THE POINTED BRILL-NOETHER THEOREM 13
Proposition 3.
The numerical features of the pencil ι : P → C g are as follows: ι ∗ ( λ ) = g − , ι ∗ ( ψ ) = 1 , ι ∗ ( δ irr ) = 6( g − , ι ∗ ( δ ) = 1 , ι ∗ ( δ g − ) = 1 , and ι ∗ ( δ i ) = 0 for i = 2 , . . . , g − . Proof.
We blow-up X at the 2 g − | C | and denote by h : e X → P the induced fibration. Clearly h ( e X, O e X ) = 0 and H ( e X, O e X ) ∼ = H ( J, O J )is 1-dimensional, therefore χ ( e X, O e X ) = 0. Accordingly, ι ∗ ( λ ) = χ ( e X, O e X ) + g − g − . By the Noether formula, the total number of singular fibres in the pencil ι is given by ι ∗ ( δ ) = c ( e X ) + 4 g − χ ( e X , O e X ) − K e X + 4 g − g − . In the pencil ι there exists a unique curve from the linear system | ( g − J + f | + J ,which is viewed as a hyperplane inside | gJ + f | . This singular curve is of the type(4) t = (cid:2) D ∪ E ∪ J , e q := f · E ∈ E (cid:3) ∈ C g , where D ∈ | ( g − J + f | is a smooth curve of genus g − D ∩ J = ∅ (on X ).Note that the rational curve E intersects both D and J at one point. Forgetting themarked point e q , the stable model of this curve is [ D ∪ q J ] ∈ M g . The point t lies onboth boundary divisors ∆ and ∆ g − , which implies ι ∗ ( δ ) = ι ∗ ( δ g − ) = 1, therefore ι ∗ ( δ irr ) = 6( g − (cid:3) Corollary 2.
The numerical features of the pencil ¯ ι := π ◦ ι : P → M g obtained byforgetting the marked point, are given by: ¯ ι ∗ ( λ ) = g − , ¯ ι ∗ ( δ irr ) = 6( g − , ¯ ι ∗ ( δ ) = 2 , ¯ ι ∗ ( δ i ) = 0 , for i = 2 , . . . , j g k . Proof.
The only thing which has to be observed is that ¯ ι ∗ ( δ ) = ι ∗ ( δ ) + ι ∗ ( δ g − ) = 2. (cid:3) Using Proposition 3 it is now immediate to check that the pencil ι , just like the Du Valpencil, satisfies the relations ι ∗ ( BN g ) = 0 and ι ∗ ([ W g ]) = 0 . Theorem 4.
The general pointed curve [ C, q ] , where C ∈ | gJ + f | and { q } = J · f ,verifies the pointed Brill-Noether Theorem.Proof. The proof proceeds by induction on g in a way mirroring the proofs of Theorems1 and 3. Assume by contradiction that the pointed Brill-Noether Theorem fails for everysmooth curve [ C, q ], where C ∈ | gJ + f | . By choosing a Lefschetz pencil ι in | gJ + f | as above, the same conclusion holds for the degenerate pointed curve t = [ D ∪ E ∪ J , e q ].That is, the variety of limit linear series ℓ of type g rd on D ∪ E ∪ J such that a ℓ ( e q ) ≥ α has a component Z of dimension strictly greater than ρ ( g, r, d, α ), for some r, d, and α .For ℓ = ( ℓ D , ℓ E , ℓ J ) a general point of Z , let γ D := α ℓ D ( D · E ) and γ J := α ℓ J ( J · E ).Then Z is birationally isomorphic to G rd (cid:16) D, ( D · E, γ D ) (cid:17) × G rd (cid:16) E, ( E · D, γ cD ) , ( E · J , γ cJ ) , ( e q, α ) (cid:17) × G rd (cid:16) J , ( J · E, γ J ) (cid:17) . Both the 3-pointed rational curve [
E, E · D, E · J , e q ] ∈ M , , as well as the 1-pointedelliptic curve [ J , J · E ] ∈ M , verify the pointed Brill-Noether Theorem. By inductionthe same can be assumed for [ D, D · E ] ∈ C g − . It follows thatdim Z = ρ ( g − , r, d, γ D ) + ρ (0 , r, d, γ cD , γ cJ , α ) + ρ (1 , r, d, γ J ) = ρ ( g, r, d, α ) , a contradiction. (cid:3) Remark . Using [Epe] page 34, we observe that the indecomposable elliptic surface P ( E )is not the desingularization of a surface in P g with canonical curve sections. This canalso be seen from the self-intersection formula C = 2 g − g −
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U VAL CURVES AND THE POINTED BRILL-NOETHER THEOREM 15
Humboldt-Universit¨at zu Berlin Institut f¨ur MathematikUnter den Linden 6, 10099 Berlin, Germany
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