Cycles on arithmetic schemes and Euler characteristics of curves. (English) Zbl 0654.14004

Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, part 2, Proc. Symp. Pure Math. 46, No. 2, 421-450 (1987).
[For the entire collection see Zbl 0626.00011.]
This paper develops a theory of algebraic cycles and intersection numbers on regular arithmetic schemes and uses it to establish a formula describing the behaviour of the Euler characteristic in a degenerating family of curves in mixed characteristic case. To explain this formula, let \(f: X\to S=Spec(A)\) be a flat and proper morphism, with A a complete discrete valuation ring, X a regular scheme and the generic fibre \(X_ g\) smooth. If \(X_ s\) is the closed fibre of f, then one defines the Euler number of \(X_ z\) (with \(z=s\) or \(z=\) geometric generic point \(\bar g\)) \(\chi(X_ z)\) using the étale cohomology. The formula one is interested in asks to compute the difference \(\chi(X_ s)-\chi(X_{\bar g})\). In characteristic zero such a formula is well known. For example, if \(X_ s\) has only isolated singularities, then \((- 1)^{\dim(X)}(\chi(X_ s)-\chi(X_{\bar g}))\) is just the sum of the Milnor numbers of the singularities of X. If S is of pure characteristic \(p>0\) such a formula is in general wrong, but it can be “corrected” using the socalled Swan conductor. In the arithmetic case the author is able to define the intersection number \((\Delta_ X\cdot \Delta_ X)_ s=(-1)^{\dim (X)}c_{\dim (X)}(\Omega\) \(1_{X/S})\) as a 0-cycle of \(X_ s\) and then he conjectures that \((\Delta_ X\cdot \Delta_ X)=- sw(X/S)+\chi(X_ s)-\chi (X_{\bar g})\), where \(\Delta_ X\) is the diagonal of X and \(sw(X/S)\) is the Swan correction. Finally, he shows that this formula is true if the fibre dimension of X/S is one.
[See also the author’s paper in Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 85-90 (1987; see review 14016)].
Reviewer: L.Bădescu


14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14H10 Families, moduli of curves (algebraic)
14A15 Schemes and morphisms
14C99 Cycles and subschemes
57R20 Characteristic classes and numbers in differential topology
14F45 Topological properties in algebraic geometry