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A few remarks on sections of the Picard bundle of family of curves

Lorenzo Fassina, Gian Pietro Pirola

Abstract

We study sections of the relative Picard bundle of a family of curves of genus $g \geq 2$ through the rank of the associated normal function. Using Griffiths' formula for the infinitesimal invariant and higher Schiffer variations, we establish a numerical inequality relating the rank, the minimal support of a representing divisor and the modular dimension of the family. When the modular map is dominant, we obtain a sharp classification: equality occurs only for multiples of odd theta characteristics or of the canonical section. As applications, we derive geometric consequences for plane curves, obtaining results on intersections with very general quartic curves, in the spirit of the work of Chen-Riedl-Yeong, and with quintic curves.

A few remarks on sections of the Picard bundle of family of curves

Abstract

We study sections of the relative Picard bundle of a family of curves of genus through the rank of the associated normal function. Using Griffiths' formula for the infinitesimal invariant and higher Schiffer variations, we establish a numerical inequality relating the rank, the minimal support of a representing divisor and the modular dimension of the family. When the modular map is dominant, we obtain a sharp classification: equality occurs only for multiples of odd theta characteristics or of the canonical section. As applications, we derive geometric consequences for plane curves, obtaining results on intersections with very general quartic curves, in the spirit of the work of Chen-Riedl-Yeong, and with quintic curves.
Paper Structure (15 sections, 16 theorems, 158 equations)

This paper contains 15 sections, 16 theorems, 158 equations.

Table of Contents

  1. The Relative Picard Bundle
  2. Normal functions
  3. Basic theory of normal functions
  4. Rank of a normal function
  5. The Griffiths infinitesimal invariant
  6. Definition of the infinitesimal invariant
  7. Normal function of rank 0
  8. Cohomology and higher-Schiffer variations
  9. Griffiths' formula for decomposable tensors and applications
  10. Griffiths formula
  11. First results
  12. Proofs of the Main Results
  13. Plane Curves
  14. The Case of Quartic Plane Curves
  15. The Case of Quintic Plane Curves

Key Result

Theorem 1

Let $\pi : \mathcal{C} \to Y$ be a family of complex curves of genus $g \ge 2$. Assume that the image of the modular map $m: Y \to \mathcal{M}_g$ contains an analytic open subset. Let $\psi$ be a section of the Picard bundle $\mathrm{Pic}^n(\pi)$, and consider the associated normal function If $\nu$ is locally constant, then $d_S(\psi) = g-1$ if and only if, for a general point $y \in Y$ and ever

Theorems & Definitions (42)

  • Theorem 1: = Theorem \ref{['theo-spin-section']}
  • Theorem 2
  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 3.1
  • ...and 32 more