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

By T. Graber, Ravi Vakil. 2003

237 citations recorded by the source corpus.

We prove a localization formula for the moduli space of stable relative maps. As an application, we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i − g + 1 genus 0 components. As consequences, we prove and generalize various conjectures and theorems about various moduli spaces of curves (due to Getzler, Ionel, Faber, Looijenga, Pandharipande, Diaz, and others). This theorem appears to be the geometric content behind these results; the rest is straightforward graph combinatorics. The theorem also suggests the importance of the stratification of the moduli space by number of rational components.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/85da4b36f2f170490bed7ae13cff84d84a6f34f5

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Citations in the corpus

Logarithmic Stable Maps (2008, background): “It is, however, not a condition friendly to moduli problems ([20, 21, 10]).”
Double ramification cycles on the moduli spaces of curves (2016, unknown): “The factor e(N∞) is 1 if the target (P1[r],∞) does not degenerate and e(N∞) = − t + ψ∞ ∏ e∈E( ) de if the target does degenerate [17].”
Polynomial recursion formula for linear Hodge integrals (2009, methodology): “It has become an effective tool of algebraic geometry for studying Hurwitz numbers and many related subjects [4, 13, 14, 15, 16, 17, 19, 24, 29, 33].”
K3 surfaces with Picard number one and infinitely many rational points (2005, background): “In [Graber and Vakil 2003b], admissible cover loci within the boundary of moduli spaces of stable maps play a key role in establishing the result that the degree 3g − 3 part of the tautological ring of Mg has dimension 1, providing further evidence for a conjecture by Faber, stating that R(Mg) is a ”
Recursion Formulae of Higher Weil–Petersson Volumes (2007, methodology): “[11] T. Graber and R. Vakil, Relative virtual localization and vanishing of tautological classes on moduli spaces of curves, Duke Math.”
The tropical vertex (2009, methodology): “A description of the obstruction space for relative maps can be found in [9].”
A topological view of Gromov-Witten theory (2004, methodology): “1 Rubber targets We will study the Gromov-Witten theory of the pair (Y,D0) via virtual localization [10, 11] with respect to the natural fiberwise C-action on Y discussed in Section 1.”
Orbifold techniques in degeneration formulas (2011, background): “able and discrete, the objects and arrows of T tw(S) agree with those given as in deﬁnition 2.1.5 also for non-reduced schemes. A proof is given in [ACFW]. Part (3) of the proposition was observed in [GV]. Part (4) applies for Xa curve and Da point, and Graber and Vakil identify T tw with a familiar”
Tropical Hurwitz numbers (2008, unknown): “Connections with the moduli spaces of curves, the most remarkable being the ELSV formula [2], are produced by virtual localization on these moduli spaces [7].”
Logarithmic Gromov–Witten theory with expansions (2019, background): “The virtual class of Li’s theory also satisfies a remarkable torus localization formula [28, 54].”