Special determinants in higher-rank Brill-Noether theory Brian Osserman University of California, Davis March 12, 2011 1
Classical Brill-Noether theory Let C be a smooth, projective curve of genus g over an algebraically closed field F. Classical Brill-Noether theory studies the moduli space G r d (C) of linear series pairs (L, V ) where L is a line bundle of degree d on C, and V Γ(C, L ) is an (r + 1)-dimensional space of global sections. The expected dimension of G r d (C) is ρ = g + (r + 1)(d r g). The study of G r d (C) is of basic importance due to the relationship between linear series and maps C P r, and also has applications ranging from the study of Weierstrass points to the geometry of the moduli space of curves. 2
The main results of classical Brill-Noether theory were proved over a number of papers by many people, with most of the progress occurring in the 1970 s. The Brill- Noether theorem was originally asserted by Brill and Noether in the 1870 s, and the proof was completed by Griffiths and Harris a century later. Theorem 1 If ρ 0, then for any C the space G r d (C) is nonempty, with every component of dimension at least ρ. If C is general, then G r d (C) is pure of dimension ρ, and in particular empty if ρ < 0. In addition, Fulton and Lazarsfeld and Gieseker (based on work of Petri) proved: Theorem 2 If ρ 1, then for any C the space G r d (C) is connected. If C is general, then for any ρ, the space G r d (C) is smooth of dimension ρ. We will focus our attention on the generalization of Theorem 1. 3
Higher rank A natural question is the following: Question 3 What happens when we replace line bundles with vector bundles of higher rank? The resulting objects can be interpreted in terms of maps to Grassmannians, and their study also has consequences for the geometry of the moduli space of curves (a special case arises in the work of Farkas and Popa giving a counterexample to the slope conjecture). Notation 4 Given r, k, d, let G k r,d (C) be the space of pairs (E, V ) with E a vector bundle of rank r and degree d on C, and V Γ(C, E ) a k-dimensional space of global sections. Here, if we do not wish to restrict to stable vector bundles, we should work systematically with stacks to get good dimensional behavior. 4
Even in rank 2, the situation is far more complicated than the classical case! There are many partial results, but we are nowhere near even a comprehensive conjecture. A natural generalization of the classical ρ is: ρ = r 2 (g 1) + 1 + k(d + r(1 g) k. In many cases, components of dimension ρ are known to exist. However, even on a general curve, one often has components of dimension larger than ρ. Given the likelihood of inductive structure on the problem, we will focus our attention on the case of rank 2. There are two known sources of large components: degeneracy, and special determinants. 5
Degeneracy We can always take M,M line bundles of degrees N + d and N, and for N >> 0 we will have Γ(C, M M ) k, so G k 2,d (C) no matter how negative ρ is. (Note on stacks: this locus has negative dimension, but decreasing linearly in k, while ρ decreases quadratically in k. Thus the dimension really is larger than ρ.) Because we are often interested only in semistable loci, we will not focus on this behavior. 6
Special determinants Notation 5 Given also L Pic d (C), let G k r,l (C) be the locus of Gk r,d (C) on which dete = L. The expected dimension of G k r,l (C) is ρ g. Bertram, Feinberg and Mukai discovered a special symmetry in the case L = ω is the canonical line bundle on C. They showed: Theorem 6 Every component of G k 2,ω(C) has dimension at least ( ) k ρ g +. 2 They conjectured that this should be sharp, and that one has nonemptiness (in the stable locus) if and only if ρ g + ( k 2) 0. Our focus is on generalizing these ideas. 7
Dimension results Our first theorem is: Theorem 7 Let L be a line bundle of degree d, and let δ be the minimal degree of an effective divisor such that h 1 (C, L ( )) 1. G k 2,L (C) has dimension at least ρ g + ( ) k δ 2. Corollary 8 If h 1 (C, L ) 1, then every component of G k 2,L least ρ g + ( k 2). This recovers the Bertram-Feinberg-Mukai result when L = ω. Then every component of Our second theorem, which begins to establish an expected pattern, is: (C) has dimension at Theorem 9 Let L be a line bundle of degree d, and suppose that h 1 (C, L ) 2. Then every component of G k 2,L (C) has dimension at least ρ g + 2( k 2). In the special case k = 2 we have the more general statement: Theorem 10 Let L be a line bundle of degree d, and suppose that h 1 (C, L ) = m. Then every component of G 2 2,L (C) has dimension at least ρ g + m. 8
Generalizations These results suggest generalizations, including to higher rank. The naive guess they would indicate is that in nice circumstances, we have that if h 1 (C, L ) = m, then every component of G k r,l (C) has dimension at least ( k ρ g + m r Note that this would have the bizarre consequence that for m 3 or r 3, and k sufficiently large, the modified expected dimension increases with k! More generally, one might expect to obtain a sequence of expected dimensions, determined by the sequence δ 1, δ 2,..., where δ i is the minimal degree of an effective divisor i such that h 1 (C, L ( )) i. This direction of generalization appears to be more subtle. ). 9
Degeneracy strikes back Our techniques in principle could be applied to prove a statement as above. However, generalizing either to r 3 or m 3, the analysis becomes much more delicate. Perhaps more interestingly, both directions of generalization naturally introduce certain nondegeneracy hypotheses on the pair (E, V ). Very degenerate examples show that some such hypotheses would be necessary for the statement to hold. Conclusion: degeneracy can not only cause the dimension to be higher than expected, but also lower! 10
Degeneration techniques Degeneration techniques played a central role in the development of classical Brill- Noether theory, and appear if anything more important in the higher-rank case. Montserrat Teixidor i Bigas has developed a generalization to higher rank of the Eisenbud-Harris theory of limit linear series, and has successfully applied it to prove many results. However, dimension bounds play a central role in limit linear series theory, and the point of view of the construction makes it difficult to reconcile with the Bertram-Feinberg- Mukai modified expected dimension. This is necessary in particular if one wants to prove the Bertram-Feinberg-Mukai existence conjecture via degenerations. Indeed, Teixidor has proved some existence results in the case of canonical determinant which are conditional on addressing this issue. 11
Teixidor and I have been working together to address this question. The idea is based on an alternate limit linear series construction from my thesis, the construction of which has only been carried out for curves with two smooth components glued at a node. Building on this construction, we prove: Theorem 11 For degenerations to curves with two components, if we restrict to the case of canonical determinant, the Bertram-Feinberg-Mukai modified expected dimension is compatible with limit linear series constructions. It appears that our techniques will generalize without too much difficulty to handle chains of smooth curves, which appears to be the crucial case. Of course, we hope to generalize to arbitrary curves of compact type. 12
Construction of G k r,d (C) Let M r,d (C) be the moduli space of vector bundles of rank r and degree d on C. Let U be the universal bundle on C M r,d (C). Let D be a sufficiently ample divisor on C, pulled back to C M r,d (C). Let π : G = G(k, p 2 U (D)) M r,d (C) be the relative Grassmannian, with universal subbundle V π p 2 (U (D)). Then G k r,d (C) G is the locus on which the composed map vanishes. V π p 2 (U (D) D ) 13
This explains where ρ comes from: dim M r,d (C) = r 2 (g 1) + 1; by Riemann-Roch, the relative dimension of G is k(d + r deg D + r(1 g) k); and the codimension of G k r,d (C) is at most rkv rkπ p 2 (U (D) D ) = kr deg D. Thus the dimension of every component of G k r,d (C) is at least which is ρ. r 2 (g 1) + 1 + k(d + r deg D + r(1 g) k) kr deg D, 14
Symmetries from special determinants We can modify the above construction slightly as follows: with D sufficiently ample, p 2 U (D) is a subbundle of p 2 (U (D)/U ( D)), which has rank 2r deg D. This induces a morphism G(k, p 2 U (D)) G 1 := G(k, p 2 (U (D)/U ( D))). On the other hand, pulling back p 2 (U /U ( D)) to G(k, p 2 U (D)) induces a morphism G(k, p 2 U (D)) G 2 := G(r deg D, p 2 (U (D)/U ( D))), so we obtain a morphism ϕ : G(k, p 2 U (D)) G 1 Mr,d (C) G 2. 15
A point of G(k, p 2 U (D)) corresponds to a point of G k r,d (C) if and only if the corresponding subbundle, considering inside p 2 U (D)/U ( D), is in fact contained in p 2 U /U ( D). That is, G k r,d (C) is precisely the preimage under the map ϕ of the incidence correspondence I G 1 Mr,d (C) G 2. We can rederive ρ from this description. Given a line bundle L Pic d (C), the same construction works with M r,l (C) in place of M r,d (C) to construct G k r,l (C). 16
Now, consider the case of canonical determinant, and r = 2. Then we have an isomorphism ψ : 2 U p 1ω. By taking local representatives near each point of D, applying ψ, and summing residues, we produce an alternating form on p 2 (U (D)/U ( D)) which is in fact easily checked to be symplectic. Moreover, p 2 U (D) is isotropic by the residue theorem, and p 2 (U /U ( D)) is isotropic because the resulting local differential forms have no poles. Thus, in the above construction, the map ϕ factors through the product of symplectic Grassmannians SG(k, p 2 (U (D)/U ( D))) M2,d (C) SG(2 deg D, p 2 (U (D)/U ( D))). 17
This changes the codimension of the incidence corresponse I : indeed, it may be realized as a relative Grassmannian of relative dimension k(2 deg D k) over SG(2 deg D, p 2 (U (D)/U ( D))), and since SG(k, p 2 (U (D)/U ( D))) has relative dimension k(4 deg D k) ( k 2), the codimension of I is 2k deg D ( k 2). Thus the dimension of G k 2,ω(C) is at least 3(g 1) + k(d + 2 deg D + 2 2g k) 2k deg D + giving the modified dimension bound of Bertram-Feinberg-Mukai. ( ) k 2 = ρ g + ( ) k, 2 18
Techniques The variations on this used to obtain Theorems 7, 9, and 10 all stem from the observation that the canonical determinant hypothesis is not the crucial part of the argument. Rather, it is the existence of a map 2 U p 1ω which allows us to construct the symplectic form. We can immediately obtain Corollary 8 from this observation. Theorem 7 follows via the same idea, using a more detailed analysis involving a Grassmannian parametrizing isotropic subspaces for a partially degenerate alternating form. Theorem 9 is the most difficult of the three. For both it and Theorem 10, the point is that if h 1 (C, L ) = m, then we obtain an m-dimensional space of maps to p 1ω, and thus an m-dimensional space of symplectic forms, for which our bundles are simultaneously isotropic. We thus analyze the resulting multiply symplectic Grassmannians (a non-general intersection of m symplectic Grassmannians) to obtain the theorems. 19
Finally, to prove Theorem 11, the idea is as follows. The alternate limit linear series moduli space we use involves spaces of linked Grassmannians, which take the place of the relative Grassmannian used in the construction for smooth curves. We thus develop a theory of symplectic linked Grassmannians, and are able to use it to combine the limit linear series construction with the canonical determinant construction to show that the necessary dimension bounds hold. 20