The Chow ring of the moduli space of curves of genus zero

doi:10.1016/j.jpaa.2016.07.016

Journal of Pure and Applied Algebra

Volume 221, Issue 4, April 2017, Pages 757-772

https://doi.org/10.1016/j.jpaa.2016.07.016 Get rights and content

Abstract

We give a new presentation of the intersection ring of the moduli space of curves of genus zero. Our description gives an explicit basis for the Chow groups and the intersection pairings between them.

Section snippets

Overview

The purpose of this note is to give a new presentation of the intersection ring of the moduli space ${\overline{M}}_{0, n}$ of stable n-pointed curves of genus zero. In the first section we recall general facts about the moduli space ${\overline{M}}_{0, n}$ . There are several constructions of this space. We recall four of them and present another construction of ${\overline{M}}_{0, n}$ as a blow-up of the variety ${(P^{1})}^{n - 3}$ . 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 ${\overline{M}}_{0, n}$

Recall that the moduli space $M_{0, n}$ parametrizes the isomorphism classes of pointed curves $(C; x_{1}, \dots, x_{n})$ , where C is a smooth curve of genus zero and the $x_{i}$ 's are distinct points on C. Every smooth curve of genus zero is isomorphic to the projective line $P^{1}$ and for every 3 distinct points $a, b, c$ on $P^{1}$ there exists a unique automorphism of the projective line sending these points to $0, 1, \infty$ respectively. This means that the space $M_{0, n}$ coincides with $U : = {(P^{1} - {0, 1, \infty})}^{n - 3} - Δ,$ where Δ stands for the big

The Chow ring of ${\overline{M}}_{0, n}$

We first review some general facts from [2] about the intersection ring of the blow-up $\tilde{Y}$ of a smooth variety Y along a smooth irreducible subvariety Z. When the restriction map from $A^{•} (Y)$ to $A^{•} (Z)$ is surjective, S. Keel has shown in [6] that the computations become simpler. We denote the kernel of the restriction map by $J_{Z / Y}$ so that $A^{•} (Z) = \frac{A^{•} (Y)}{J_{Z / Y}} .$ Define a Chern polynomial for $Z \subset Y$ , denoted by $P_{Z / Y} (t)$ , to be a polynomial $P_{Z / Y} (t) = t^{d} + a_{1} t^{d - 1} + \dots + a_{d - 1} t + a_{d} \in A^{•} (Y) [t],$ where d is the codimension of Z in Y

Examples

•
The moduli space ${\overline{M}}_{0, 3} = M_{0, 3}$ consists of a single point and its Chow ring is isomorphic to the ring of integers $Z$ .
•
${\overline{M}}_{0, 4} = P^{1}$ , and its intersection ring is $\frac{Z [a_{1}]}{(a_{1}^{2})}$ , where $a_{1}$ is the class of a point. Note that: $a_{1} = D_{1, 2} = D_{1, 3} = D_{1, 4} .$
•
${\overline{M}}_{0, 5}$ is $P^{1} \times P^{1}$ blown-up at 3 points. It is a del Pezzo surface of degree 5. The exceptional divisors of the blow-ups are $D_{1, 2, 3}, D_{1, 2, 4}, D_{1, 2, 5}$ . One has the following relations: $\begin{matrix} a_{1} = D_{1, 3} + D_{1, 2, 3} = D_{1, 4} + D_{1, 2, 4} = D_{1, 5} + D_{1, 2, 5}, \\ a_{2} = D_{2, 3} + D_{1, 2, 3} = D_{2, 4} + D_{1, 2, 4} = D_{2, 5} + D_{1, 2, 5}, \\ a_{1} + a_{2} = D_{1, 2} + D_{1, 2, 3} + D_{1,} \end{matrix}$

References (8)

B. Hassett
Moduli spaces of weighted pointed stable curves
Adv. Math.
(2003)
P. Deligne
Résumé des premiers exposés de A. Grothendieck
W. Fulton et al.
A compactification of configuration space
Ann. Math., Second Ser.
(1994)
R. Hartshorne
Algebraic Geometry, vol. 52
(1977)

There are more references available in the full text version of this article.

Cited by (6)

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The Chow ring of the moduli space of stable marked curves is generated by Keel's divisor classes. The top graded part of this Chow ring is isomorphic to the integers, generated by the class of a single point. In this paper, we give an equivalent graphical characterization on the monomials in this Chow ring, as well as the characterization on the algebraic reduction on such monomials. Moreover, we provide an algorithm for computing the intersection degree of tuples of Keel's divisor classes — we call it the forest algorithm; the complexity of which is $O (n^{3})$ in the worst case, where n refers to the number of marks on the stable curves in the ambient moduli space. Last but not least, we introduce three identities on multinomial coefficients which naturally came into play, showing that they are all equivalent to the correctness of the base case of the forest algorithm.
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We compute the Chow ring of an arbitrary heavy/light Hassett space ${\overline{M}}_{0, w}$ . These spaces are moduli spaces of weighted pointed stable rational curves, where the associated weight vector w consists of only heavy and light weights. Work of Cavalieri et al. [3] exhibits these spaces as tropical compactifications of hyperplane arrangement complements. The computation of the Chow ring then reduces to intersection theory on the toric variety of the Bergman fan of a graphic matroid. Keel [16] has calculated the Chow ring $A^{⁎} ({\overline{M}}_{0, n})$ of the moduli space ${\overline{M}}_{0, n}$ of stable nodal n-marked rational curves; his presentation is in terms of divisor classes of stable trees of $P^{1}$ 's having one nodal singularity. Our presentation of the ideal of relations for the Chow ring $A^{⁎} ({\overline{M}}_{0, w})$ is analogous. We show that pulling back under Hassett's birational reduction morphism $ρ_{w} : {\overline{M}}_{0, n} \to {\overline{M}}_{0, w}$ identifies the Chow ring $A^{⁎} ({\overline{M}}_{0, w})$ with the subring of $A^{⁎} ({\overline{M}}_{0, n})$ generated by divisors of w-stable trees, which are those trees which remain stable in ${\overline{M}}_{0, w}$ .
Tautological classes on the moduli space of hyperelliptic curves with rational tails
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We study tautological classes on the moduli space of stable n-pointed hyperelliptic curves of genus g with rational tails. The method is based on the approach of Yin in comparing tautological classes on the moduli of curves and the universal Jacobian. Our result gives a complete description of tautological relations. It is proven that all relations come from the Jacobian side. The intersection pairings are shown to be perfect in all degrees. We show that the tautological algebra coincides with its image in cohomology via the cycle class map. The latter is identified with monodromy invariant classes in cohomology.
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Chow rings of heavy/light hassett spaces via tropical geometry
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The Chow ring of the moduli space of curves of genus zero

Abstract

Section snippets

Overview

Construction of the moduli space M‾0,n

The Chow ring of M‾0,n

Examples

Adv. Math.

Résumé des premiers exposés de A. Grothendieck

A compactification of configuration space

Ann. Math., Second Ser.

Algebraic Geometry, vol. 52

Construction of the moduli space ${\overline{M}}_{0, n}$

The Chow ring of ${\overline{M}}_{0, n}$