A SIMPLICIAL CALCULUS FOR LOCAL INTERSECTION NUMBERS AT NONARCHIMEDIAN 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 desin-gularization 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 al-lows 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 sit-uation. 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 wer