Normal functions and the height of Gross-Schoen cycles

TL;DR

This paper establishes a new relation between the height of Gross-Schoen cycles and the self-intersection of dualizing sheaves on pointed curves, using normal functions and biextensions.

Contribution

It proves a variant of Zhang's formula connecting heights and intersection theory, introducing a new approach via Bloch line bundles and their Chern forms.

Findings

The Chern form of the Bloch line bundle is non-negative.

Calculated the class of the line bundle in the Picard group.

Established a new formula relating heights and intersection numbers.

Abstract

We prove a variant of a formula due to S. Zhang relating the Beilinson-Bloch height of the Gross-Schoen cycle on a pointed curve with the self-intersection of its relative dualizing sheaf. In our approach the height of the Gross-Schoen cycle occurs as the degree of a suitable Bloch line bundle. We show that the Chern form of this line bundle is non-negative, and we calculate its class in the Picard group of the moduli space of pointed stable curves of compact type. The basic tools are normal functions and biextensions associated to the cohomology of the universal jacobian.