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(Non-)Extendability of Abel-Jacobi Maps

Zev Rosengarten

TL;DR

This work analyzes when Abel-Jacobi maps on geometrically reduced projective curves extend from the smooth locus $C^{\rm{sm}}$ to larger separated curves. It introduces a universal contraction framework and an Albanese-type obstruction to extension, proving that, unless $C$ is modifiable, the Abel-Jacobi map cannot extend to any larger model, and that the image in the Jacobian is closed in general. For irreducible curves not isomorphic to $\mathbf{P}^1$, the Abel-Jacobi map is a closed embedding, with a reduction to nodal or cuspidal cubic covers when the normalization is $\mathbf{P}^1$. These results yield explicit criteria for extendability, closedness of the image, and embedding properties of Abel-Jacobi maps under degeneration, with modifiability by $\mathbf{P}^1$'s as the only known exception to extension in certain cases.

Abstract

We investigate the "natural" locus of definition of Abel-Jacobi maps. In particular, we show that, for a proper, geometrically reduced curve C -- not necessarily smooth -- the Abel-Jacobi map from the smooth locus C^{sm} into the Jacobian of C does not extend to any larger (separated, geometrically reduced) curve containing C^{sm} except under certain particular circumstances which we describe explicitly. As a consequence, we deduce that the Abel-Jacobi map has closed image except in certain explicitly described circumstances, and that it is always a closed embedding for irreducible curves not isomorphic to P^1.

(Non-)Extendability of Abel-Jacobi Maps

TL;DR

This work analyzes when Abel-Jacobi maps on geometrically reduced projective curves extend from the smooth locus to larger separated curves. It introduces a universal contraction framework and an Albanese-type obstruction to extension, proving that, unless is modifiable, the Abel-Jacobi map cannot extend to any larger model, and that the image in the Jacobian is closed in general. For irreducible curves not isomorphic to , the Abel-Jacobi map is a closed embedding, with a reduction to nodal or cuspidal cubic covers when the normalization is . These results yield explicit criteria for extendability, closedness of the image, and embedding properties of Abel-Jacobi maps under degeneration, with modifiability by 's as the only known exception to extension in certain cases.

Abstract

We investigate the "natural" locus of definition of Abel-Jacobi maps. In particular, we show that, for a proper, geometrically reduced curve C -- not necessarily smooth -- the Abel-Jacobi map from the smooth locus C^{sm} into the Jacobian of C does not extend to any larger (separated, geometrically reduced) curve containing C^{sm} except under certain particular circumstances which we describe explicitly. As a consequence, we deduce that the Abel-Jacobi map has closed image except in certain explicitly described circumstances, and that it is always a closed embedding for irreducible curves not isomorphic to P^1.
Paper Structure (5 sections, 14 theorems, 20 equations)

This paper contains 5 sections, 14 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Notation and terminology
  3. Preparations for the main results
  4. (Non-)extendability of Abel-Jacobi
  5. Abel-Jacobi is a closed embedding for irreducible curves

Key Result

Theorem 1.3

Let $C/K$ be a geometrically reduced projective curve, and let $\vec{c} := \{c_1, \dots, c_n\}$ be a collection of rational points in $C^{\rm{sm}}$, one in each irreducible component. Assume that $C$ is not modifiable. Then for any dominant open embedding $C^{\rm{sm}} \subsetneq C'$ with $C'/K$ a se

Theorems & Definitions (38)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • proof
  • Remark 1.5
  • Theorem 1.6
  • Example 1.7
  • Example 1.8
  • Example 1.9
  • ...and 28 more