The Chow ring of the moduli space of curves of genus zero
Section snippets
Overview
The purpose of this note is to give a new presentation of the intersection ring of the moduli space of stable n-pointed curves of genus zero. In the first section we recall general facts about the moduli space . There are several constructions of this space. We recall four of them and present another construction of as a blow-up of the variety . In the next section we describe the intersection ring of the moduli space in terms of generators and relations. We give a
Construction of the moduli space
Recall that the moduli space parametrizes the isomorphism classes of pointed curves , where C is a smooth curve of genus zero and the 's are distinct points on C. Every smooth curve of genus zero is isomorphic to the projective line and for every 3 distinct points on there exists a unique automorphism of the projective line sending these points to respectively. This means that the space coincides with where Δ stands for the big
The Chow ring of
We first review some general facts from [2] about the intersection ring of the blow-up of a smooth variety Y along a smooth irreducible subvariety Z. When the restriction map from to is surjective, S. Keel has shown in [6] that the computations become simpler. We denote the kernel of the restriction map by so that Define a Chern polynomial for , denoted by , to be a polynomial where d is the codimension of Z in Y
Examples
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The moduli space consists of a single point and its Chow ring is isomorphic to the ring of integers .
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, and its intersection ring is , where is the class of a point. Note that:
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is blown-up at 3 points. It is a del Pezzo surface of degree 5. The exceptional divisors of the blow-ups are . One has the following relations:
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