Cycles in Jacobians: infinitesimal results

TL;DR

This paper investigates the properties of Ceresa cycles in Jacobians of generic curves, providing explicit calculations of infinitesimal invariants and extending results to hyperelliptic cases with K-theoretic analogues.

Contribution

It introduces a method to derive invariants of Ceresa cycles from basic cycles and explicitly determines the zero locus for genus g=4, also extending to hyperelliptic curves with K-theoretic constructions.

Findings

Explicit zero locus for infinitesimal invariants in genus 4

Derivation of invariants from basic cycles

K-theoretic analogue for hyperelliptic genus 3

Abstract

Let C be a generic smooth curve of genus g\geqslant 4. We study normal functions and infinitesimal invariants associated to Ceresa cycles W_{k}-W_{k}^{-}, k=2,...,g-2. We show how they can be obtained from the normal function associated to the basic cycle C-C^{-} and, for k=2, we also explicitely determine the zero locus of the infinitesimal invariant. For C hyperelliptic of genus g=3, we define the K-theoretic counterpart of W_{2}-W_{2}^{-}, generalizing a construction of A. Collino, and show that it is indecomposable.