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The Gauss map for bielliptic Prym varieties

Constantin Podelski

Abstract

We completely describe the degree of the Gauss map of the theta divisor of bielliptic Prym varieties. We characterize bielliptic Prym varieties whose Gauss degree is the same as Jacobians. We also construct bielliptic Prym varieties with a very low Gauss degree. In dimension $5$, we obtain a complete description of the Gauss degree on the Andreotti-Mayer locus.

The Gauss map for bielliptic Prym varieties

Abstract

We completely describe the degree of the Gauss map of the theta divisor of bielliptic Prym varieties. We characterize bielliptic Prym varieties whose Gauss degree is the same as Jacobians. We also construct bielliptic Prym varieties with a very low Gauss degree. In dimension , we obtain a complete description of the Gauss degree on the Andreotti-Mayer locus.
Paper Structure (17 sections, 44 theorems, 264 equations, 1 figure, 1 table)

This paper contains 17 sections, 44 theorems, 264 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Summary of the results in dimension 5
  3. The family Egt
  4. The construction
  5. The evaluation bundle
  6. The Gauss degree on Egt
  7. Numerical analysis
  8. The Gauss locus
  9. The boundary of Egt
  10. Preliminaries
  11. Degenerations in Egt
  12. The Gauss degree on the boundary of Egt
  13. Computation of the coefficient mu
  14. Additional isolated singularities
  15. A combinatorial question on groups
  16. ...and 2 more sections

Key Result

Theorem 1

For $0\leq t \leq g/2$ and $(P,\Xi)=\mathrm{Prym}(\tilde{C}/C)\in \mathscr{E}'_{g,t}$, we have where $b_n=\binom{2n}{n}$ denotes the middle binomial coefficient and $N_\mathrm{ad}(\tilde{C}/C)$ is a combinatorial invariant of double covers of bielliptic curves (see Def: eta).

Figures (1)

  • Figure 1.3: Possible Gauss degrees on $\mathcal{N}^{(5)}_1$

Theorems & Definitions (88)

  • Theorem 1: \ref{['Thm: degree Gauss Map on Egt, in general']} and \ref{["Prop: Gauss degree on Egt'"]}
  • Remark
  • Corollary 1
  • Corollary 2: \ref{['Cor: EEg0 irreducible component of Gauss locus']}
  • Theorem 2: \ref{["Thm: Degree Egt with E cycle of P1's"]}
  • Example 1
  • Proposition 1: \ref{['Sec: App: A special ppav in arbitrary dimension']}
  • Corollary 3
  • Corollary 4
  • Corollary 5
  • ...and 78 more