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Intersections of \(\psi \)-classes on \(\overline{M}_{1,n}(m)\). (English) Zbl 1430.14065

Let \(\overline{M}_{g,n}\) be the moduli space of stable \(n\)-pointed genus \(g\) curves. For each \(i=1,\ldots, n\), the \(\psi_i\)-class is the cotangent line bundle class on \(\overline{M}_{g,n}\) associated with the \(i\)-th marked point. The study of top intersection numbers of \(\psi\)-classes has played an important role in enumerative geometry and Gromov-Witten theory. In particular, E. Witten made a conjecture [in: Surveys in differential geometry. Vol. I: Proceedings of the conference on geometry and topology, held at Harvard University, Cambridge, MA, USA, April 27-29, 1990. Providence, RI: American Mathematical Society; Bethlehem, PA: Lehigh University. 243–310 (1991; Zbl 0757.53049)] (first proved by M. Kontsevich [Commun. Math. Phys. 147, No. 1, 1–23 (1992; Zbl 0756.35081)]) that the generating function of intersection numbers of \(\psi\)-classes satisfies the KdV hierarchy of partial differential equations. Witten also showed two basic recursions for intersections of \(\psi\)-classes using the forgetful morphism \(\overline{M}_{1,n+1}\to \overline{M}_{1,n}\), which are called the string equation and the dilaton equation.
In the paper under review the author considers the case of genus one and an alternate modular compactification \(\overline{M}_{1,n}(m)\) parameterizing \(m\)-stable curves that are allowed to have nodes and elliptic \(k\)-fold points for \(k\leq m\) (previously constructed and studied by the author [Compos. Math. 147, No. 3, 877–913 (2011; Zbl 1223.14031); ibid. 147, No. 6, 1843–1884 (2011; Zbl 1260.14033)]). The author establishes a pair of recursions for intersections of \(\psi\)-classes on \(\overline{M}_{1,n}(m)\), which can be viewed as an analogue of the string equation and the dilaton equation. They differ from the original recursions by some “error” terms that arise from analyzing the behavior of \(\psi\)-classes under the rational forgetful map \(\overline{M}_{1,n+1}(m)\dashrightarrow \overline{M}_{1,n}(m)\).

MSC:

14H10 Families, moduli of curves (algebraic)
14H70 Relationships between algebraic curves and integrable systems
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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References:

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