Degeneration of the modified diagonal cycle

TL;DR

This paper studies how the modified diagonal cycle behaves under degenerations of algebraic curves, revealing connections to indecomposable higher Chow cycles in specific degenerations.

Contribution

It provides a detailed analysis of the degeneration process of the modified diagonal cycle and links it to higher Chow cycles on degenerated curves.

Findings

Degeneration induces a higher Chow cycle on the central fiber.

For genus three non-hyperelliptic curves degenerating to nodal curves, the cycle corresponds to an indecomposable higher Chow cycle.

The specialization map captures the cycle's behavior during degeneration.

Abstract

In this note, we revisit the modified diagonal cycle of Gross and Schoen. We look at degenerations of this cycle, induced by a degeneration of the curve C, and explain how the specialization map with respect to the central fiber produces a higher Chow cycle. When C is non-hyperelliptic of genus three, and degenerates to a nodal curve, the degeneration of the cycle corresponds to an indecomposable higher Chow cycle on a curve of genus two.