Relative maps and tautological classes

By Carel Faber, R. Pandharipande. 2003

216 citations recorded by the source corpus.

The moduli space of stable relative maps to the projective line combines features of stable maps and admissible covers. We prove all standard Gromov-Witten classes on these moduli spaces of stable relative maps have tautological push-forwards to the moduli space of curves. In particular, the fundamental classes of all moduli spaces of admissible covers push-forward to tautological classes. Consequences for the tautological rings of the moduli spaces of curves include methods for generating new relations, uniform derivations of the socle and vanishing statements of the Gorenstein conjectures for the complete, compact type, and rational tail cases, tautological boundary terms for Ionel's, Looijenga's, and Getzler's vanishings, and applications to Gromov-Witten theory.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/98d04d55c094c773a6871165e735ea13fc6f9527

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Citations in the corpus

Algebraic cobordism revisited (2006, background): “Algebraic cobordism Ω∗ is also a universal theory in the less structured setting of oriented Borel-Moore functors of geometric type.”
Tautological and non-tautological cohomology of the moduli space of curves (2011, methodology): “2We follow here the definition of tautological classes in Chow given in [23].”
Relations in the tautological ring of the moduli space of curves (2013, background): “The tautological ring was originally defined by Mumford [26] in the case ofMg, but we follow Faber and Pandharipande [10] and first define the tautological ring of Mg,n: the tautological rings R(Mg,n) are simultaneously defined for all g, n to be the smallest Q-subalgebras of the Chow rings that are”
THE MODULI SPACE OF TWISTED CANONICAL DIVISORS (2015, unknown): “18See [11] for definitions and further references.”
BCOV theory via Givental group action on cohomological field theories (2008, unknown): “Also, there are general theorems proving the existence of some special families of tautological relations [16, 22, 12].”
Double ramification cycles on the moduli spaces of curves (2016, methodology): “The localization relations of [11] intertwine Hodge classes on the moduli spaces of curves with cycles obtained from maps”
Relative virtual localization and vanishing of tautological classes on moduli spaces of curves (2003, background): “Further applications are in [ FbPa2 ] (including different proofs of some of the results of Section 5), and [LLZ1, LLZ2].”
Curves on K3 surfaces and modular forms (2010, methodology): “The quasi-modularity of Theorem 4 is obtained in Section 7 from a boundary induction in the tautological ring of the moduli space of curves using the strong form of Getzler–Ionel vanishing proved in [13].”
A topological view of Gromov-Witten theory (2004, methodology): “(P 3 [3,4], S3) 2 −→ P 3 [3,4], S3, P 3 [3, 4 ] l −→ P 3 , S3, C3,4,”
Integrals of ψ-classes over double ramification cycles (2012, background): “It is known (see [1]) that the Poincaré dual cohomology class of DRg(a1, .”