Evaluating tautological classes using only Hurwitz numbers. (English) Zbl 1160.14043
In the paper under review the authors use localization techniques to express the ratios of certain tautological classes in terms of simple Hurwitz numbers.
Let \(R^*(\mathcal M_g)\) be the tautological ring of \(\mathcal M_g\). A conjecture due to C. Faber predicts that \(R^*(\mathcal M_g)\) satisfies Poincaré duality with socle in degree \(g-2\) [see Faber, Carel (ed.) et al., Moduli of curves and abelian varieties. The Dutch intercity seminar on moduli. Braunschweig: Vieweg. Aspects Math. E33, 109–129 (1999; Zbl 0978.14029)] and, in fact, by results of C. Faber in [loc. cit.] and of E. Looijenga in [Invent. Math. 121, No. 2, 411–419 (1995; Zbl 0851.14017)], its degree \(g-2\) part is known to be one dimensional. A family of degree \(g-2\) tautological classes can be obtained by considering the \((2g-1)\)-dimensional moduli space \(\mathcal A_{dd}^g\) of connected genus \(g\) degree \(d\) covers of \(\mathbb P^1\) fully ramified over \(0\) and \(\infty\) and simply ramified over the other (\(2g\)) branch points, together with its natural (source) map \(\mathcal A_{dd}^g\to\mathcal M_g\) (for \(d=2\) this corresponds to the hyperelliptic locus of \(\mathcal M_g\)). This map can be extended to a map \(\overline{\mathcal A}_{dd}^g\to\overline{\mathcal M}_g\), where \(\overline{\mathcal A}_{dd}^g\) is the smooth and proper stack of admissible covers constructed by D. Abramovich, A. Corti and A. Vistoli in [Commun. Algebra 31, No. 8, 3547–3618 (2003; Zbl 1077.14034)], allowing to evaluate the classes \([\mathcal A_{dd}^g]\) in \(R^*(\mathcal M_g)\) by integrating with suitable elements of the Chow ring of \(\overline{\mathcal M}_g\).
Simple Hurwitz numbers are combinatorial objects counting ramified covers of \(\mathbb P^1\) with prescribed ramification data over \(0\) and simple ramification over the other (fixed) branch points. Even if, in general, Hurwitz numbers are hard to compute, in the case that we have full ramification over \(0\), there is a simple way of write them down in generating function form due to computations of B. Shapiro, M. Shapiro and A. Vainshtein in [Khovanskij, A. (ed.) et al., Topics in singularity theory. V. I. Arnold’s 60th anniversary collection. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 180(34), 219–227 (1997; Zbl 0883.05072)].
By localizing on spaces of admissible covers, the authors get a relation between generating functions for Hurwitz-Hodge integrals [see J. Bryan, T. Graber and R. Pandharipande, J. Algebr. Geom. 17, No. 1, 1–28 (2008; Zbl 1129.14075)] and for simple Hurwitz numbers fully ramified over \(0\). Combining this formula with computations due to the second author in [Algebra Number Theory 1, No. 1, 35–66 (2007; Zbl 1166.14036)], the ratios of several tautological classes can be expressed in a purely combinatorial way. In particular, this master relation gives back a formula due to J. Bryan and R. Pandharipande in [J. Am. Math. Soc. 21, No. 1, 101–136 (2008; Zbl 1126.14062)] computing all proportionalities of \([\mathcal A_{dd}^g]\) in \(R^*(\mathcal M_g)\).
Let \(R^*(\mathcal M_g)\) be the tautological ring of \(\mathcal M_g\). A conjecture due to C. Faber predicts that \(R^*(\mathcal M_g)\) satisfies Poincaré duality with socle in degree \(g-2\) [see Faber, Carel (ed.) et al., Moduli of curves and abelian varieties. The Dutch intercity seminar on moduli. Braunschweig: Vieweg. Aspects Math. E33, 109–129 (1999; Zbl 0978.14029)] and, in fact, by results of C. Faber in [loc. cit.] and of E. Looijenga in [Invent. Math. 121, No. 2, 411–419 (1995; Zbl 0851.14017)], its degree \(g-2\) part is known to be one dimensional. A family of degree \(g-2\) tautological classes can be obtained by considering the \((2g-1)\)-dimensional moduli space \(\mathcal A_{dd}^g\) of connected genus \(g\) degree \(d\) covers of \(\mathbb P^1\) fully ramified over \(0\) and \(\infty\) and simply ramified over the other (\(2g\)) branch points, together with its natural (source) map \(\mathcal A_{dd}^g\to\mathcal M_g\) (for \(d=2\) this corresponds to the hyperelliptic locus of \(\mathcal M_g\)). This map can be extended to a map \(\overline{\mathcal A}_{dd}^g\to\overline{\mathcal M}_g\), where \(\overline{\mathcal A}_{dd}^g\) is the smooth and proper stack of admissible covers constructed by D. Abramovich, A. Corti and A. Vistoli in [Commun. Algebra 31, No. 8, 3547–3618 (2003; Zbl 1077.14034)], allowing to evaluate the classes \([\mathcal A_{dd}^g]\) in \(R^*(\mathcal M_g)\) by integrating with suitable elements of the Chow ring of \(\overline{\mathcal M}_g\).
Simple Hurwitz numbers are combinatorial objects counting ramified covers of \(\mathbb P^1\) with prescribed ramification data over \(0\) and simple ramification over the other (fixed) branch points. Even if, in general, Hurwitz numbers are hard to compute, in the case that we have full ramification over \(0\), there is a simple way of write them down in generating function form due to computations of B. Shapiro, M. Shapiro and A. Vainshtein in [Khovanskij, A. (ed.) et al., Topics in singularity theory. V. I. Arnold’s 60th anniversary collection. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 180(34), 219–227 (1997; Zbl 0883.05072)].
By localizing on spaces of admissible covers, the authors get a relation between generating functions for Hurwitz-Hodge integrals [see J. Bryan, T. Graber and R. Pandharipande, J. Algebr. Geom. 17, No. 1, 1–28 (2008; Zbl 1129.14075)] and for simple Hurwitz numbers fully ramified over \(0\). Combining this formula with computations due to the second author in [Algebra Number Theory 1, No. 1, 35–66 (2007; Zbl 1166.14036)], the ratios of several tautological classes can be expressed in a purely combinatorial way. In particular, this master relation gives back a formula due to J. Bryan and R. Pandharipande in [J. Am. Math. Soc. 21, No. 1, 101–136 (2008; Zbl 1126.14062)] computing all proportionalities of \([\mathcal A_{dd}^g]\) in \(R^*(\mathcal M_g)\).
Reviewer: Margarida Melo (Roma)
MSC:
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 14H10 | Families, moduli of curves (algebraic) |
Citations:
Zbl 0978.14029; Zbl 0851.14017; Zbl 1077.14034; Zbl 0883.05072; Zbl 1129.14075; Zbl 1126.14062; Zbl 1166.14036References:
| [1] | Dan Abramovich, Alessio Corti, and Angelo Vistoli. Twisted bundles and admissible covers. Comm in Algebra, 31(8):3547-3618, 2001. · Zbl 1077.14034 |
| [2] | Dan Abramovich, Tom Graber, and Angelo Vistoli. Gromov-Witten theory of Deligne-Mumford stacks. Preprint: math.AG/0603151, 2006. · Zbl 1193.14070 |
| [3] | Jim Bryan, Tom Graber, and Rahul Pandharipande. The orbifold quantum cohomology of \(C^2/Z_3\) and Hurwitz-Hodge integrals. Preprint:math.AG/0510335, 2005. · Zbl 1129.14075 |
| [4] | Jim Bryan and Rahul Pandharipande. The local Gromov-Witten theory of curves. Preprint: math.AG/0411037, 2004. · Zbl 1126.14062 |
| [5] | Jim Bryan and Rahul Pandharipande. Curves in Calabi-Yau threefolds and topological quantum field theory. Duke Math. J., 126(2):369-396, 2005. · Zbl 1084.14053 |
| [6] | Renzo Cavalieri. Phd thesis. http://www.math.lsa.umich.edu/\( \sim\) crenzo/ thesis.pdf, 2005. |
| [7] | Renzo Cavalieri. A TQFT for intersection numbers on moduli spaces of admissible covers. Preprint: mathAG/0512225, 2005. · Zbl 1166.14036 |
| [8] | Renzo Cavalieri. Generating functions for Hurwitz-Hodge integrals. Preprint:mathAG/0608590, 2006. · Zbl 1186.14058 |
| [9] | Renzo Cavalieri. Hodge-type integrals on moduli spaces of admissible covers. In Dave Auckly and Jim Bryan, editors, The interaction of finite type and Gromov-Witten invariants (BIRS 2003), volume 8. Geometry and Topology monographs, 2006. · Zbl 1258.14012 |
| [10] | Carel Faber. A conjectural description of the tautological ring of the moduli space of curves. In Moduli of curves and abelian varieties, Aspects Math., E33, pages 109-129. Vieweg, Braunschweig, 1999. · Zbl 0978.14029 |
| [11] | Carel Faber and Rahul Pandharipande. Hodge integrals and Gromov-Witten theory. Invent. Math., 139(1):173-199, 2000. · Zbl 0960.14031 |
| [12] | C. Faber and R. Pandharipande. Relative maps and tautological classes. J. Eur. Math. Soc. (JEMS), 7(1):13-49, 2005. · Zbl 1084.14054 |
| [13] | I. P. Goulden, D. M. Jackson, and R. Vakil. The Gromov-Witten potential of a point, Hurwitz numbers, and Hodge integrals. Proc. London Math. Soc. (3), 83(3):563-581, 2001. · Zbl 1074.14520 |
| [14] | Ian Goulden, David Jackson, and Ravi Vakil. A short proof of the \(\lambda_g\)-conjecture without Gromov-Witten theory: Hurwitz theory and the moduli of curves. Preprint:mathAG/0604297, 2006. · Zbl 1205.14068 |
| [15] | Tom Graber and Ravi Vakil. On the tautological ring of \(M_g,n\). Turkish J. Math., 25(1):237-243, 2001. · Zbl 1040.14007 |
| [16] | Tom Graber and Ravi Vakil. Hodge integrals, Hurwitz numbers, and virtual localization. Compositio Math., 135:25-36, 2003. · Zbl 1063.14032 |
| [17] | Tom Graber and Ravi Vakil. Relative virtual localization and vanishing of tautological classes on moduli spaces of curves. Duke Math J., 130(1):1-37, 2005. · Zbl 1088.14007 |
| [18] | Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipande, Richard Thomas, Cumrun Vafa, Ravi Vakil, and Eric Zaslow. Mirror Symmetry. AMS CMI, 2003. · Zbl 1044.14018 |
| [19] | Eleny-Nicoleta Ionel. Relations in the tautological ring of \(M_g\). Duke Math. J., 129(1):157-186, 2005. · Zbl 1086.14023 |
| [20] | Eduard Looijenga. On the tautological ring of \(M_g\). Invent. Math., 121(2):411-419, 1995. · Zbl 0851.14017 |
| [21] | David Mumford. Toward an enumerative geometry of the moduli space of curves. Arithmetic and Geometry, II(36):271-326, 1983. · Zbl 0554.14008 |
| [22] | B. Shapiro, M. Shapiro, and A. Vainshtein. Ramified coverings of \(S\sp 2\) with one degenerate branch point and enumeration of edge-ordered graphs. In Topics in singularity theory, volume 180 of Amer. Math. Soc. Transl. Ser. 2, pages 219-227. Amer. Math. Soc., Providence, RI, 1997. · Zbl 0883.05072 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
