Non-Abelian Brill-Noether theory and Fano 3-folds

Abstract

A Brill-Noether locus is a subscheme of the moduli of bundles E over a curve C defined by requiring E to have a given number of sections, or homomorphisms from another bundle. There are a number of different types, that can be treated by determinantal methods, with symmetry or skewsymmetry arising from Serre duality. They have many beautiful applications to curves, K3 surfaces and Fano 3-folds.

arXiv commentFrom Sugaku 49:1 (1997), 1--24, translation by M. Reid to appear in the AMS series Sugaku Expositions, 1997

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• Extensions of line bundles and Brill--Noether loci of rank-two vector bundles on a general curve
• Stable maps and singular curves on K3 surfaces
• The non-abelian Brill-Noether divisor on ${\overline{\mathcal{M}}}_{13}$ and the Kodaira dimension of ${\overline{\mathcal{R}}}_{13}$
• Moduli spaces of stable sheaves on abelian surfaces
• Modular properties of nodal curves on K3 surfaces
• A Petri Theorem for Rank 2 Bundles with Canonical Determinant
• The Sp3-grassmannian and duality for prime Fano threefolds of genus 9