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The Gauss Map on Theta Divisors with Transversal $\mathrm{A}_1$ Singularities

Constantin Podelski

Abstract

We use Lagrangian specialization to compute the degree of the Gauss map on Theta divisors with transversal $\mathrm{A}_1$ singularities. This computes the Gauss degree for a general abelian variety in the loci $\mathcal{A}^δ_{t,g-t}$ that form some of the irreducible components of the Andreotti-Mayer loci. We also prove that the first coefficient of the Lagrangian specialization is the Samuel multiplicity of the singular locus.

The Gauss Map on Theta Divisors with Transversal $\mathrm{A}_1$ Singularities

Abstract

We use Lagrangian specialization to compute the degree of the Gauss map on Theta divisors with transversal singularities. This computes the Gauss degree for a general abelian variety in the loci that form some of the irreducible components of the Andreotti-Mayer loci. We also prove that the first coefficient of the Lagrangian specialization is the Samuel multiplicity of the singular locus.
Paper Structure (12 sections, 22 theorems, 146 equations)

This paper contains 12 sections, 22 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Lagrangian Specialization
  3. Generalities on Lagrangian Specialization
  4. First Coefficients in the Lagrangian Specialization
  5. Application to Theta Divisors
  6. Theta Divisors with Transversal A1 Singularities
  7. Deformation of PPAV's
  8. Computation of the Gauss Degree
  9. The Family Agt
  10. Definition of the Family
  11. Gauss Degree on Agt
  12. Numerical Analysis of the Degree

Key Result

Theorem 1

Let $(A,\Theta)\in \mathcal{A}_g$ such that $\Theta$ has transversal $\mathrm{A}_1$ singularities, then where $B=\mathrm{Sing}(\Theta)$ and $C\in| L\raisebox{-.5ex}{$|$}_{B}|$ is a general divisor in the linear system.

Theorems & Definitions (44)

  • Theorem 1: \ref{['Theorem: Gauss degree theta divisor with smooth singular locus']}
  • Theorem 2: \ref{['Thm: Gauss Degree on A^d_g_1,g_2 ']}
  • Proposition 1: Leading term of the Lagrangian specialization
  • Theorem 3: Second term of the Lagrangian specialization
  • Example 1
  • Corollary 1
  • Remark
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • ...and 34 more