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David McKinnon - An arithmetic Bézout theorem

DAVID MCKINNON, Department of Mathematics, University of California, Berkeley, California  94618, USA
An arithmetic Bézout theorem

In this paper, we prove two versions of an arithmetic analogue of Bézout's theorem, subject to some technical restrictions. The basic formula proven is $h(X\cap Y)=h(X)\deg(Y) + h(Y)\deg(X) + O(1)$. The theorems are inspired by the arithmetic Bézout theorem of Bost, Gillet, and Soulé, but improve upon them in two ways. First, we obtain an equality up to O(1) as the intersecting cycles vary in projective families. Second, we generalise this result to intersections of divisors on any regular, generically smooth, projective arithmetic variety. We present an application of these results to proving an analogue of Hilbert's Irreducibility Theorem for intersections of curves in ${\bf P}^2$.


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