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Intersection numbers with Witten’s top Chern class. (English) Zbl 1141.14012
If \(C\) is a smooth projective curve of genus \(g\) with \(n\geq 1\) distinct marked points \(x_1,\dots,x_n\), an \(r\)-spin curve of type \((a_1,\dots,a_n)\) on \(C\) is a line bundle \(\mathcal T\) on \(C\) together with an identification between \(\mathcal T^{\otimes r}\) and \(\omega_C(-\sum a_ix_i)\). This definition can be extended to all Deligne–Mumford stable curves of genus \(g\) with \(n\) marked points, giving rise to the moduli space of stable curves with \(r\)-spin structure of type \((a_i)\), introduced by T. J. Jarvis [Int. J. Math. 11, No. 5, 637–663 (2000; Zbl 1094.14504)] and D. Abramovich and T. J. Jarvis [Proc. Am. Math. Soc. 131, No. 3, 685–699 (2003; Zbl 1037.14008)]. The structure of the moduli space of \(r\)-spin curves as a finite covering of the moduli space \(\overline{\mathcal M}_{g,n}\) of stable curves allows to define on the latter space a special cohomology class, called Witten’s top Chern class, which satisfies simple factorization rules.
The article under review deals with the computation of the intersection numbers of Witten’s class with powers of the \(\psi\)-classes. The main result is that these intersection numbers are entirely determined by genus \(0\) intersection numbers involving no \(\psi\)-classes, and the factorization rules for Witten’s class. This is proven by giving an algorithm for computing all these intersection numbers, completing the approach of S. V. Shadrin [Int. Math. Res. Not. 2003, No. 38, 2051–2094 (2003; Zbl 1070.14030)]. In particular, this involves giving a closed formula for the integrals of Witten’s class over certain divisors of \(\overline{\mathcal M}_{1,n}\), the so-called double ramification divisors.
The interest in the intersection numbers of Witten’s top Chern class and \(\psi\)-classes is motivated by the conjecture of E. Witten [in: Topological methods in modern mathematics, 235–269 (1993; Zbl 0812.14017)] that they can be arranged into a generating series which is a solution of the \(r\)-th Gelfand–Dikii hierarchy. This conjecture has been recently proved by the authors and C. Faber [Tautological relations and the \(r\)-spin Witten conjecture. Preprint 2006, arxiv:math/0612510].

MSC:
14H10 Families, moduli of curves (algebraic)
14H70 Relationships between algebraic curves and integrable systems
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References:
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