On coherent systems of type (n,d,n+1) on Petri curves

We study coherent systems of type $(n,d,n+1)$ on a Petri curve $X$ of genus $g\ge2$. We describe the geometry of the moduli space of such coherent systems for large values of the parameter $\alpha$. We determine the top critical value of $\alpha$ and show that the corresponding ``flip'' has positive codimension. We investigate also the non-emptiness of the moduli space for smaller values of $\alpha$, proving in many cases that the condition for non-emptiness is the same as for large $\alpha$. We give some detailed results for $g\le5$ and applications to higher rank Brill-Noether theory and the stability of kernels of evaluation maps, thus proving Butler's conjecture in some cases in which it was not previously known.Comment: 33 page

New examples of twisted Brill-Noether loci I

Our purpose in this paper is to construct new examples of twisted Brill-Noether loci on curves of genus $g\ge2$. Many of these examples have negative expected dimension. We deduce also the existence of a new region in the Brill-Noether map, whose points support non-empty standard Brill-Noether loci.Comment: 20 pages, 2 figures. Comments welcom

Stability of projective Poincare and Picard bundles

Let $X$ be an irreducible smooth projective curve of genus $g\ge3$ defined over the complex numbers and let ${\mathcal M}_\xi$ denote the moduli space of stable vector bundles on $X$ of rank $n$ and determinant $\xi$, where $\xi$ is a fixed line bundle of degree $d$. If $n$ and $d$ have a common divisor, there is no universal vector bundle on $X\times {\mathcal M}_\xi$. We prove that there is a projective bundle on $X\times {\mathcal M}_\xi$ with the property that its restriction to $X\times\{E\}$ is isomorphic to $P(E)$ for all $E\in\mathcal{M}_\xi$ and that this bundle (called the projective Poincar\'e bundle) is stable with respect to any polarization; moreover its restriction to $\{x\}\times\mathcal{M}_\xi$ is also stable for any $x\in X$. We prove also stability results for bundles induced from the projective Poincar\'e bundle by homomorphisms $\text{PGL}(n)\to H$ for any reductive $H$. We show further that there is a projective Picard bundle on a certain open subset $\mathcal{M}'$ of $\mathcal{M}_\xi$ for any $d>n(g-1)$ and that this bundle is also stable. We obtain new results on the stability of the Picard bundle even when $n$ and $d$ are coprime.Comment: One typo corrected; final version accepted for publication in Bull. London Math. So

Stability of generalised Picard sheaves

Let $C$ be a smooth irreducible complex projective curve of genus $g \geq 2$ and $M_1$ a moduli space of stable vector bundles over $C$. A (generalised) Picard sheaf is the direct image on $M_1$ of the tensor product of the Poincar\'e or universal bundle on $M_1\times C$ by the pullback of a vector bundle $E_0$ on $C$; when the degree of $E_0$ is sufficiently large, this sheaf is a bundle and coincides with the Fourier-Mukai transform of $E_0$. In this paper we include all results known to us and many new ones on the stability of the Picard sheaves when $M_1$ is one of the Picard variety of line bundles of degree $d$ on $C$, the moduli space of stable vector bundles of rank $n$ and degree $d$ on $C$ with $n,d$ coprime or the moduli space of stable bundles of rank $n$ and fixed determinant of degree $d$. We prove in particular that, if $E_0$ is a stable bundle of rank $n_0$ and degree $d_0$ with $nd_0 + n_0d > n_0n(2g-1)$, then the pullbacks of the Picard bundle on the moduli space of stable bundles by morphisms analogous to the Abel-Jacobi map are stable; moreover, if $nd_0 + n_0d > n_0n(n + 1)(g-1) + n_0$, then the Picard bundle itself is stable with respect to a theta divisor.Comment: 14 page

Hodge polynomials of some moduli spaces of Coherent Systems

When $k<n$, we study the coherent systems that come from a BGN extension in which the quotient bundle is strictly semistable. In this case we describe a stratification of the moduli space of coherent systems. We also describe the strata as complements of determinantal varieties and we prove that these are irreducible and smooth. These descriptions allow us to compute the Hodge polynomials of this moduli space in some cases. In particular, we give explicit computations for the cases in which $(n,d,k)=(3,d,1)$ and $d$ is even, obtaining from them the usual Poincar\'e polynomials.Comment: Formerly entitled: "A stratification of some moduli spaces of coherent systems on algebraic curves and their Hodge--Poincar\'e polynomials". The paper has been substantially shorten. Theorem 8.20 has been revised and corrected. Final version accepted for publication in International Journal of Mathematics. arXiv admin note: text overlap with arXiv:math/0407523 by other author