1,722,383 research outputs found
The tautological ring of the moduli space of curves
No full textThe tautological ring of the moduli space of curves M_g is a subring R^*(M_g) of the Chow ring A^*(M_g). The tautological ring can also be defined for other moduli spaces of curves, such as the moduli space of curves of compact type M^c_g or the moduli space of Deligne-Mumford stable pointed curves Mbar_{g,n}. We conjecture and prove various results about the structure of the tautological ring.
In particular, we give two proofs of the Faber-Zagier relations, a large family of relations between the kappa classes in R^*(M_g) that contains all known relations. The first proof (joint work with R. Pandharipande) uses the virtual geometry of the moduli space of stable quotients developed by Marian, Oprea, and Pandharipande. The second proof (joint work with R. Pandharipande and D. Zvonkine) uses Witten's class on the moduli space of 3-spin curves and the classification of semisimple cohomological field theories by Givental and Teleman. The second proof has the disadvantage that it only proves the image of the Faber-Zagier relations in cohomology, but the advantage that it also proves an extension of the relations to Mbar_{g,n} that was conjectured by the author. These relations on Mbar_{g,n} and their restrictions to smaller moduli spaces of curves seem to describe all known relations in the tautological ring.
We also prove several combinatorial results about the structure of the Gorenstein quotient rings of R^*(M_g) and R^*(M^c_g). This includes several new families of relations that are similar to the Faber-Zagier relations, as well as joint work with F. Janda giving formulas for ranks of restricted socle pairings in R^*(M^c_g).
The appendix presents data obtained by computer calculations of the tautological relations on Mbar_{g,n} and their restrictions to M^c_{g,n} and M^{rt}_{g,n} for small values of g and n. The data suggests several new locations in which the tautological ring might not be a Gorenstein ring
The 4-fold Pandharipande--Thomas vertex
Full text linkWe give a conjectural but full and explicit description of the (K-theoretic)
equivariant vertex for Pandharipande--Thomas stable pairs on toric Calabi--Yau
4-folds, by identifying torus-fixed loci as certain quiver Grassmannians and
prescribing a canonical half of the tangent-obstruction theory. For any number
of non-trivial legs, the DT/PT vertex correspondence can then be verified by
computer in low degrees.Comment: 37 page
Stable maps and branch divisors
Full text linkWe construct a natural branch divisor for equidimensional projective morphisms where the domain has lci singularities and the target is nonsingular. The method involves generalizing a divisor construction of Mumford from sheaves to complexes. The construction is valid in flat families. The generalized branch divisor of a stable map to a nonsingular curve X yields a canonical morphism from the space of stable maps to a symmetric product of X. This branch morphism (together with virtual localization) is used to compute the Hurwitz numbers of covers of the projective line for all genera and degrees in terms of Hodge integrals
The Generic Nontriviality of the Faber-Pandharipande Cycle
No full textWe present a simple and characteristic-free proof of a result of Green and Griffiths, which states that for the generic curve C of genus g≥4, the Faber-Pandharipande cycle K×K−(2g−2)KΔ is nontorsion in CH2(C×C)
The Remodeling Conjecture and the Faber-Pandharipande Formula
No full textInternational audienceIn this note, we prove that the free energies F g constructed from the Eynard-Orantin topological recursion applied to the curve mirror to C3 reproduce the Faber-Pandharipande formula for genus g Gromov-Witten invariants of C3 . This completes the proof of the remodeling conjecture for C3
Gromov--Witten/Pandharipande--Thomas correspondence via conifold transitions
Full text linkGiven a (projective) conifold transition of smooth projective threefolds from
to , we show that if the Gromov--Witten/Pandharipande--Thomas descendent
correspondence holds for the resolution , then it also holds for the
smoothing with stationary descendent insertions. As applications, we show
the correspondence in new cases.Comment: We add the reference to John Pardon's work and point out our results
are not covered by hi
ALMOST CLOSED 1-FORMS
Full text linkWe construct an algebraic almost closed 1-form with zero scheme not expressible (even locally) as the critical locus of a holomorphic function on a non-singular variety. The result answers a question of Behrend-Fantechi. We correct here an error in our paper (D. Maulik, R Pandharipande and R. P. Thomas, Curves on K3 surfaces and modular forms, J. Topol. 3 (2010) 937-996. arXiv:1001.2719v3), where an incorrect construction with the same claimed properties was propose
The 4-fold Pandharipande--Thomas vertex and Jeffrey--Kirwan residue
No full textInternational audienceWe present a contour integral formalism for computing the K-theoretic equivariant Pandharipande--Thomas (PT) 4-vertex. Within the Jeffrey--Kirwan (JK) residue framework, we show that the PT 4-vertex can be obtained from the same integrand as the Donaldson--Thomas (DT) 4-vertex by choosing a different reference vector. We illustrate the formalism through examples involving curves and surfaces on the 4-fold. Furthermore, we investigate the DT/PT correspondence for the 4-fold setting together with its higher rank and supergroup-like generalizations
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