Local invariants attached to Weierstrass points

TL;DR

This paper establishes exact formulas relating invariants of hyperelliptic curves, such as discriminants and Faltings height, to Weierstrass points and intersection theory, enhancing understanding of their arithmetic properties.

Contribution

It provides new explicit formulas connecting hyperelliptic invariants with Weierstrass point distributions and intersection theory, extending previous inequalities to exact relations.

Findings

Exact formula relating discriminants and Weierstrass points in semistable case.

Explicit formula for stable Faltings height involving Weierstrass points.

Validation of inequalities as equalities in specific cases.

Abstract

Let X/S be a hyperelliptic curve of genus g over the spectrum of a discrete valuation ring. Two fundamental numerical invariants are attached to X/S: the valuation of the hyperelliptic discriminant of X/S, and the valuation of the Mumford discriminant of X/S (equivalently, the Artin conductor). For a residue field of characteristic 0 as well as for X/S semistable these invariants are known to satisfy certain inequalities. We prove an exact formula relating the two invariants with intersection theoretic data determined by the distribution of Weierstrass points over the special fiber, in the semistable case. We also prove an exact formula for the stable Faltings height of an arbitrary curve over a number field, involving local contributions associated to its Weierstrass points.