Let X be a non-singular projective curve of genus g ≥ 2 over an algebraically closed field of characteristic zero. Let M(n, d) denote the moduli space of stable bundles of rank n and degree d on X and W k−1 n,d the Brill-Noether loci in M(n, d). We prove that, if 0 ≤ d ≤ n and W k−1 n,d is non-empty, then it is irreducible of the expected dimension and smooth outside W k n,d . We prove further that in this range W k−1 n,d is non-empty if and only if d > 0, n ≤ d+(n−k)g and (n, d, k) 6 (n, n, n). We also prove irreducibility and non-emptiness for the semistable BrillNoether loci.

Geography of brill-noether loci for small slopes

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Generalised parabolic bundles and applications to torsionfree sheaves on nodal curves

Let X be a curve of genus g A coherent system on X consists of a pair (E,V), where E is an algebraic vector bundle over X of rank n and degree d and V is a subspace of dimension k of the space of sections of E The stability of the coherent system depends on a parameter α We study the variation of the moduli space of coherent systems when we move the parameter As an application, we analyze the cases k=1,2,3 and n=2 explicitly For small values of α, the moduli spaces of coherent systems are related to the Brill–Noether loci, the subschemes of the moduli spaces of stable bundles consisting of those bundles with at least a prescribed number of independent sections The study of coherent systems is applied to find the dimension, prove the irreducibility, and in some cases calculate the Picard groups of the Brill–Noether loci with k≤3

Coherent systems and Brill-Noether theory

Brill-Noether theory for stable vector bundles

Let $C$ be a curve of genus $g\geq 2$. A coherent system on $C$ consists of a pair $(E,V)$ where $E$ is an algebraic vector bundle of rank $n$ and degree $d$ and $V$ is a subspace of dimension $k$ of sections of $E$. The stability of the coherent systems depend on a parameter $\tau$. We study the variation of the moduli space of coherent systems when we move the parameter. As an application, we analyse the cases $k=1,2,3$ and $n=2$ explicitly. For small values of $\tau$, the moduli space of coherent systems is related to the Brill-Noether loci, the subspaces of the moduli space of stable bundles consisting of those bundles with a prescribed number of sections. The study of coherent systems is applied to find the dimension, irreducibility, and in some cases, the Picard group, of the Brill-Noether loci with $k\leq 3$.

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Narasimhan Sundaram

Papers

Special divisors and vector bundles