Relative virtual localization and vanishing of tautological classes of moduli spaces of curves.

*(English)*Zbl 1088.14007Let \(\overline{\mathcal M}_{g,n}\) denote the Deligne-Mumford compactification of the moduli space of curves of genus \(g\) with \(n\) marked points. The article under review deals with the study of the tautological ring of \(\overline{\mathcal M}_{g,n}\), which is defined as the minimal subring of the Chow ring of \(\overline{\mathcal M}_{g,n}\) containing the \(\psi\)-classes and any pushforward of them under the natural gluing and forgetful morphisms between moduli spaces of stable curves. The main result of the article is that all tautological classes of codimension \(i\) on \(\overline{\mathcal M}_{g,n}\) vanish on the open set of curves such that the number of rational components is at most \(i-g\). This was announced (under the name of Tautological Vanishing Theorem) in the review paper [R. Vakil, Notices Am. Math. Soc. 50, No. 6, 647–658 (2003; Zbl 1082.14033)], which contains a very readable introduction to the theory of tautological rings.

The tautological vanishing theorem extends and implies Getzler’s conjecture [E. Getzler, in: Integrable systems and algebraic geometry. Proc. 41st Taniguchi Symp., Kobe 1997, Kyoto 1997, World Scientific, 73–106 (1998; Zbl 1021.81056)] that all polynomials of degree \(g\) in the tautological classes \(\psi_i\) vanish on \({\mathcal M}_{g,n}\). Note that Getzler’s conjecture was proved in cohomology by E.-N. Ionel [Invent. Math. 148, No. 3, 627–658 (2002; Zbl 1056.14076)].

In order to prove the tautological vanishing theorem, the authors express the tautological classes as linear combination of “Hurwitz cycles”, which are, roughly speaking, cycle classes of subvarieties of \(\overline{\mathcal M}_{g,n}\) of curves which are multiple covers of degree \(d\) of the projective line with prescribed ramification at \(0\) and \(\infty\). Then Hurwitz cycles are shown to be contained in strata with many rational components, by deforming the target \(\mathbb P^1\). The technical tool on which the proof relies is a new virtual localization formula, obtained by applying the formalism of [T. Graber and R. Pandharipande, Invent. Math. 135, No. 2, 487–518 (1999; Zbl 0953.14035)] to the moduli space of stable maps from a variety \(X\), relative to a divisor \(D\). In particular, this requires to exhibit the moduli space of stable relative maps as a global quotient. The authors also review background material on the moduli space of stable relative maps, so as to make the article reasonably self-contained from this point of view.

In the last section of the paper, several applications of the tautological vanishing theorem are considered. A conjecture which has motivated much work in the theory of tautological rings is that the moduli space of stable curves (respectively curves of compact type, or curves with rational tails) behaves like the cohomology ring of a complex variety of a specific dimension. Analogous conjectures have been formulated for the moduli space of curves of compact type, and for curves with rational tails. In this paper, the tautological vanishing theorem is used to prove the socle statement of the three conjectures, i.e., that the tautological ring is one-dimensional in degree (respectively in the three cases) \(3g-3+n\), \(2g-3+n\), \(g-2+n\), and vanishes in all higher degrees. This extends results of E. Looijenga [Invent. Math. 121, No. 2, 411–419 (1995; Zbl 0851.14017)] and T. Graber and R. Vakil [Turk. J. Math. 25, No. 1, 237–243 (2001; Zbl 1040.14007)]. Furthermore, the theorem gives bounds on the possible dimension of complete subvarieties of the moduli spaces of the following types of curves: curves with compact type, curves with rational tails, curves with at most \(s\) components of genus \(0\). This is a generalization of the result of S. Diaz [Duke Math. J. 51, 405–408 (1984; Zbl 0581.14017)] for the moduli space of smooth curves of genus \(g\). Finally, a description of the tautological groups in dimension \(\leq 6\) is obtained.

The tautological vanishing theorem extends and implies Getzler’s conjecture [E. Getzler, in: Integrable systems and algebraic geometry. Proc. 41st Taniguchi Symp., Kobe 1997, Kyoto 1997, World Scientific, 73–106 (1998; Zbl 1021.81056)] that all polynomials of degree \(g\) in the tautological classes \(\psi_i\) vanish on \({\mathcal M}_{g,n}\). Note that Getzler’s conjecture was proved in cohomology by E.-N. Ionel [Invent. Math. 148, No. 3, 627–658 (2002; Zbl 1056.14076)].

In order to prove the tautological vanishing theorem, the authors express the tautological classes as linear combination of “Hurwitz cycles”, which are, roughly speaking, cycle classes of subvarieties of \(\overline{\mathcal M}_{g,n}\) of curves which are multiple covers of degree \(d\) of the projective line with prescribed ramification at \(0\) and \(\infty\). Then Hurwitz cycles are shown to be contained in strata with many rational components, by deforming the target \(\mathbb P^1\). The technical tool on which the proof relies is a new virtual localization formula, obtained by applying the formalism of [T. Graber and R. Pandharipande, Invent. Math. 135, No. 2, 487–518 (1999; Zbl 0953.14035)] to the moduli space of stable maps from a variety \(X\), relative to a divisor \(D\). In particular, this requires to exhibit the moduli space of stable relative maps as a global quotient. The authors also review background material on the moduli space of stable relative maps, so as to make the article reasonably self-contained from this point of view.

In the last section of the paper, several applications of the tautological vanishing theorem are considered. A conjecture which has motivated much work in the theory of tautological rings is that the moduli space of stable curves (respectively curves of compact type, or curves with rational tails) behaves like the cohomology ring of a complex variety of a specific dimension. Analogous conjectures have been formulated for the moduli space of curves of compact type, and for curves with rational tails. In this paper, the tautological vanishing theorem is used to prove the socle statement of the three conjectures, i.e., that the tautological ring is one-dimensional in degree (respectively in the three cases) \(3g-3+n\), \(2g-3+n\), \(g-2+n\), and vanishes in all higher degrees. This extends results of E. Looijenga [Invent. Math. 121, No. 2, 411–419 (1995; Zbl 0851.14017)] and T. Graber and R. Vakil [Turk. J. Math. 25, No. 1, 237–243 (2001; Zbl 1040.14007)]. Furthermore, the theorem gives bounds on the possible dimension of complete subvarieties of the moduli spaces of the following types of curves: curves with compact type, curves with rational tails, curves with at most \(s\) components of genus \(0\). This is a generalization of the result of S. Diaz [Duke Math. J. 51, 405–408 (1984; Zbl 0581.14017)] for the moduli space of smooth curves of genus \(g\). Finally, a description of the tautological groups in dimension \(\leq 6\) is obtained.

Reviewer: Orsola Tommasi (Mainz)

##### MSC:

14H10 | Families, moduli of curves (algebraic) |

14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |

14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |

14C15 | (Equivariant) Chow groups and rings; motives |

14F43 | Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) |

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\textit{T. Graber} and \textit{R. Vakil}, Duke Math. J. 130, No. 1, 1--37 (2005; Zbl 1088.14007)

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