Syzygies using vector bundles

This paper studies syzygies of curves that have been embedded in projective space by line bundles of large degree. The proofs take advantage of the relationship between syzygies and spaces of section of vector bundles associated to the given line bundles.Comment: To appear in Transactions of the AM

Existence of vector bundles of rank two with fixed determinant and sections

Consider the scheme B_{2,L}^k of stable vector bundles of rank two and fixed determinant L which have at least k sections. Under suitable numerical conditions and for generic L, we show the existence of a component of the expected dimension of B_{2,L}^k.Comment: To appear in Proceedings of the Japanese Academy of Scienc

Green's Conjecture for the generic canonical curve

Green's Conjecture states the following : syzygies of the canonical model of a curve are simple up to the p^th stage if and only if the Clifford index of C is greater than p. We prove that the generic curve of genus g satisfies Green's conjecture.Comment: 13 pages Late

Rank two vector bundles with canonical determinant

Denote by B^k_{2,K} the locus of vector bundles of rank two and canonical determinant. We show that for a generic curve of genus g, B^k_{2,K} is non-empty if g is sufficiently large.Comment: To appear in Mathematische Nachrichte

Injectivity of the Petri map for twisted Brill-Noether loci

Let C be a generic curve, E a generic vector bundle on C. Then, for every line bundle on C the twisted Petri map P:H^0(C,L\otimes E)\otimes H^0(C, K\otimes L^*\otimes E^{*})--> H^0(C, K) is injective.Comment: To appear in manuscripta mathematic

Injectivity of the symmetric map for line bundles

Let C be a generic non-singular curve of genus g defined over a field of characteristic different from 2. We show that for every line bundle on C of degree at most g+1, the natural product map S^2(H^0(L))\to H^0(C,L^2) is injective. We also show that the bound on the degree of L is sharp.Comment: To appear in Manuscripta Mathematica. No changes in the paper. Word "generic" added to the abstrac

Existence of coherent systems

A coherent system of type (r,d,k) on a curve C is a pair (E,V) where E is a vector bundle of rank r and degree d and V is a space of sections of E of dimension k. There is a condition of stability on coherent systems that depends on a parameter \alpha and allows to construct moduli spaces for such pairs. This paper shows non-emptiness of these moduli spaces when k>r and some mild conditions on the degree and genus are satisfied.Comment: To appear in International Journal of Mathematic

Stable extensions by line bundles

Let C be an algebraic curve of genus g. Consider extensions E of a vector bundle F'' of rank n'' by a vector bundle F' of rank n'. The following statement was conjectured by Lange: If 0<n'deg F''-n''degF'\le n'n''(g-1), then there exist extensions like this with E stable. We prove this result for the generic curve when F' is a line bundle. Our method uses a degeneration argument to a reducible curve.Comment: plain te

On Lange's Conjecture

Let C be an algebraic curve of genus g. Let E be a vector bundle of rank n and degree d. Consider among all subbundles F' of E of rank n' those of maximal degree d'. Then s_n'(E)= n'd-nd'\le n'(n-n')g. If E is stable s_n'(E)>0 while if E is generic s_n'(E)\ge n'(n-n')(g-1) . The following statement was conjectured by Lange: If 0<s\le n'(n-n')(g-1), then there exist stable vector bundles with s_n'(E)=s. We prove this result for the generic curve. We also clarify what happens in the interval n'(n-n')(g-1)<s\le n'(n-n')g Our method uses a degeneration argument to a reducible curve. A similar result has been obtained by L.Bambrila-Paz and H.Lange using a different method.Comment: plain te

Maps between moduli spaces of vector bundles and the base locus of the theta divisor

Given a vector bundle $E$ of rank $r$ and degree $d$ on a curve $C$ of genus $g$, one can associate to $E$ in a natural way several other vector bundles. For example, one can take wedge powers of $E$. If $E$ is generated by global sections, the kernel of the evaluation map of sections is again a vector bundle. Also, new vector bundles can be produced by taking elementary transformations centered at a fixed point. Under suitable conditions on degree and rank, these constructions can be carried out globally. While all this processes seem quite elementary, very little is known about the resulting maps. The purpose of this paper is to fill in this gap.Comment: 6 page