Intersection theory of moduli space of stable \(n\)-pointed curves of genus zero.

*(English)*Zbl 0768.14002From the introduction: “This paper concerns the intersection theory of the moduli space of \(n\)-pointed stable curves of genus 0. F. F. Knudsen [Math. Scand. 52, 161-199 (1983; Zbl 0506.14019)] constructs the space, which we call \(X_ n\), and shows it is a smooth complete variety. We give an alternative construction of \(X_ n\), via a sequence of blow- ups of smooth varieties along smooth codimension two subvarieties, and using our construction:

(1) We show that the canonical map from the Chow groups to homology (in characteristic zero) \(A_ *(X_ n)@>cl>>H_ *(X_ n)\) is an isomorphism.

(2) We give a recursive formula for the Betti numbers of \(X_ n\).

(3) We give an inductive recipe for determining dual bases in the Chow ring \(A^*(X_ n)\).

(4) We calculate the Chow ring. It is generated by divisors, and we express it as a quotient of a polynomial ring by giving generators for the ideal of relations.

Once we have described \(X_ n\) via blow-ups, our results follow from application of some general results on the Chow rings of regular blow-ups which we develop in an appendix”.

(1) We show that the canonical map from the Chow groups to homology (in characteristic zero) \(A_ *(X_ n)@>cl>>H_ *(X_ n)\) is an isomorphism.

(2) We give a recursive formula for the Betti numbers of \(X_ n\).

(3) We give an inductive recipe for determining dual bases in the Chow ring \(A^*(X_ n)\).

(4) We calculate the Chow ring. It is generated by divisors, and we express it as a quotient of a polynomial ring by giving generators for the ideal of relations.

Once we have described \(X_ n\) via blow-ups, our results follow from application of some general results on the Chow rings of regular blow-ups which we develop in an appendix”.

Reviewer: E.Stagnaro (Padova)