A simplicial calculus for local intersection numbers at non-archimedian places on products of semi-stable curves

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 Zhang (Invent Math 179(1):1–73, 2010, §3). In our present work we achieve a generalisation to arbitrary d.