Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a conjecture of Kazhdan and Polishchuk

Olivier Debarre

Abstract

We discuss a conjecture made by Alexander Polishchuk and David Kazhdan at the 2022 ICM about a variety naturally attached to any stable vector bundle of rank 2 and degree $2g- 1$ on a smooth projective complex curve of genus $g$.

On a conjecture of Kazhdan and Polishchuk

Abstract

We discuss a conjecture made by Alexander Polishchuk and David Kazhdan at the 2022 ICM about a variety naturally attached to any stable vector bundle of rank 2 and degree on a smooth projective complex curve of genus .

Paper Structure

This paper contains 6 sections, 8 theorems, 27 equations.

Table of Contents

  1. The conjecture
  2. Properties of the scheme $\mathscr{F}_E$ and the conjecture for general $E$
  3. General properties of the scheme $\mathscr{F}_E$
  4. Brill--Noether loci
  5. The conjecture for general vector bundles
  6. Proof of Theorem \ref{['mth']}

Key Result

Theorem 2

Let $C$ be a smooth projective curve of genus $g\ge 2$ and let $E$ be a stable vector bundle on $C$ of rank $2$ and degree $2g- 1$. Then $\mathscr{F}_E$ is irreducible, normal, and a local complete intersection of dimension $g$.

Theorems & Definitions (19)

  • Conjecture 1
  • Theorem 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Remark 5
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 9 more