A Simplicial Calculus for Local Intersection Numbers at Nonarchimedian Places on Products of Semi-stable Curves

TL;DR

This paper develops a combinatorial simplicial calculus to explicitly compute local intersection numbers in the special fibre of products of semi-stable curves over discrete valuation rings, generalizing previous work to arbitrary dimensions.

Contribution

It introduces a new simplicial calculus and localization formula for calculating intersection numbers in the special fibre of higher-dimensional semi-stable curve products.

Findings

Explicit intersection number calculations in dimensions three and four.

A generalized localization formula applicable to arbitrary dimensions.

Practical procedure for computing self-intersections in the special fibre.

Abstract

We analyse the subring of the Chow ring with support generated by the irreducible components of the special fibre of the Gross-Schoen desingularization of a d-fold self product of a semi-stable curve over the spectrum of a discrete valuation ring. For this purpose we develop a calculus which allows to determine intersection numbers in the special fibre explicitly. As input our simplicial calculus needs only combinatorial data of the special fibre. It yields a practical procedure for calculating even self intersections in the special fibre. The first ingredient of our simplicial calculus is a localization formula, which reduces the problem of calculating intersection numbers to a special situation. In order to illustrate how our simplicial calculus works, we calculate all intersection numbers between divisors with support in the special fibre in dimension three and four. The localization formula and the general idea were already presented for d=2 in a paper of Shou-Wu Zhang. In our present work we achieve a generalisation to arbitrary d.