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Local structure of theta divisors and related loci of generic curves

Nero Budur

Abstract

For a generic compact Riemann surface the theta function is at every point on the Jacobian equal to its first Taylor term, up to a holomorphic change of local coordinates and multiplication by a local holomorphic unit. More generally, any Brill-Noether locus of twisted stable vector bundles on a smooth projective curve is at every point L locally étale isomorphic with its tangent cone if the Petri map at L is injective. This assumption has various consequences for Brill-Noether loci: positive answers to the monodromy conjecture for generalized theta divisors and to questions of Schnell-Yang on log resolutions and Whitney stratifications, and formulas for local b-functions, log canonical thresholds, topological zeta functions, and minimal discrepancies.

Local structure of theta divisors and related loci of generic curves

Abstract

For a generic compact Riemann surface the theta function is at every point on the Jacobian equal to its first Taylor term, up to a holomorphic change of local coordinates and multiplication by a local holomorphic unit. More generally, any Brill-Noether locus of twisted stable vector bundles on a smooth projective curve is at every point L locally étale isomorphic with its tangent cone if the Petri map at L is injective. This assumption has various consequences for Brill-Noether loci: positive answers to the monodromy conjecture for generalized theta divisors and to questions of Schnell-Yang on log resolutions and Whitney stratifications, and formulas for local b-functions, log canonical thresholds, topological zeta functions, and minimal discrepancies.
Paper Structure (5 sections, 14 theorems, 39 equations)

This paper contains 5 sections, 14 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Deformations with cohomology constraints
  3. From formal to étale neighborhoods
  4. Generic matrices
  5. Proof of Theorem \ref{['thrmLCThrk']}

Key Result

Theorem 1.1

(Ke, A+) Let $L\in\mathop{\mathrm{Pic}}\nolimits^d(C)$ with $0\neq h^0(L)h^1(L)$. If $\pi_L$ is injective, the tangent cone $TC_LW_d^r$ is the closed subscheme defined by the ideal generated by the minors of size $h^0(L)-r$ of the $h^1(L)\times h^0(L)$ matrix of linear forms on $\mathop{\mathrm{H}}\

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.2
  • Proposition 2.3
  • Remark 2.4
  • Theorem 2.6
  • ...and 13 more