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Alexander duality in intersection theory. (English) Zbl 0702.14002
For a scheme X of finite type over a field, write \(A_*(X)\) for its Chow group with rational coefficients. For a subscheme Y of X, or more generally a finite type map \(Y\to X\), write \(A_*(Y\to X)\) for the bivariant group in the sense of W. Fulton, [“Intersection theory” (1984; Zbl 0541.14005); chapter 7], again with rational coefficients. Following S. L. Kleiman (and A. Thorup) [in Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, part 2, Proc. Symp. Pure Math. 46, 321-370 (1987; Zbl 0664.14031)] the author considers the class of equidimensional schemes X as above, such that for any finite type map \(Y\to X\), the cap product with the fundamental class of X defines an isomorphism \(A_ k(Y\to X)\to A_{n-k}(Y)\), and such that a certain additional commutativity condition holds. From the analogy with Alexander duality for oriented manifolds, the author calls such schemes Alexander schemes. (Kleiman and Thorup called them \(C_{{\mathbb{Q}}}\)- orthocyclic.)
The main results of the paper show that Alexander schemes enjoy many intersection-theoretic properties of smooth schemes. For example, we have a natural commutative ring structure on \(A_*(id:X\to X)\), which yields an intersection product on \(A_*(X)\). Also, an Alexander scheme is geometrically unibranch. Over a perfect field, a curve is Alexander precisely if and only if it is geometrically unibranch, while a surface is Alexander if and only if it is geometrically unibranch, and if each component of any exceptional divisor in a resolution of its singularities is rational.
Further, the author presents evidence that the formal properties of Alexander schemes are sufficient to allow useful intersection calculations.
Reviewer: R.Speiser

14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14C05 Parametrization (Chow and Hilbert schemes)
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