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Combinatorics of Mumford-Morita-Miller classes in low genus. (English) Zbl 1068.14028
The author shows how one can use elementary combinatorial arguments to obtain explicit formulae and relations for the universal cotangent classes \(\psi_p\) and the Mumford classes \(\kappa_m\) on the moduli spaces of stable curves of genus 0 and 1.
More precisely, in genus 0 it is known [S. Keel, Trans. Am. Math. Soc. 330, No.2, 545–574 (1992; Zbl 0768.14002)] that any cohomology class in \(H^*(\overline{\mathcal M}_{0,n};{\mathbb Q})\) can be written as a linear combination of monomials in Keel’s boundary classes \(\delta_{0,S}\). Using the formula \(\psi_p=\sum_{q_1,q_2\notin S\ni p}\delta_{0,S}\), and the recursive relations \(\kappa_m=\pi_p^*(\kappa_m)+\psi_p^m\) and \(\pi_p^*(\delta_{0,A})=\delta_{0,A}+\delta_{0,A\cup \{p\}}\) from E. Arbarello and M. Cornalba [Publ. Math., Inst. Hautes √Čtud. Sci. 88, 97–127 (1998; Zbl 0991.14012)], the author is able to determine explicit formulae expressing \(\psi_p^m\) and \(\kappa_m\) as linear combinations of monomials in the Keel classes. These formulae are then described in terms of the cohomology of the De Concini-Procesi model for the hyperplane arrangement of type \(A_{n-1}\).
In genus 1, explicit formulae for \(\psi_p^{m}\) and \(\kappa_m\) are found by means of the gluing morphisms \(\xi_{\text{irr.}}: \overline{\mathcal M}_{0,P\cup\{q_1,q_2\}}\to \overline{\mathcal M}_{1,P}\) and \(\xi_S: \overline{\mathcal M}_{0,S\cup\{r_1\}}\times\overline{\mathcal M}_{1,(P\setminus S)\cup\{r_2\}}\to \overline{\mathcal M}_{1,P}\), using the formula \(\psi_p={1\over {24}}\xi_{\text{irr.},*}(1)+ \sum_{S\ni p,| S| \geq 2}\xi_{S,*}(1)\) [E. Arbarello and M. Cornalba, loc. cit.] and the recursive relation \(\kappa_m={1\over {24}}\xi_{\text{irr.},*}(\kappa_{m-1})+ \sum_{| S| \geq 2}\xi_{S,*}(\kappa_{m-1}\otimes 1)\) from A. Kabanov and T. Kimura [Commun. Math. Phys. 194, No. 3, 651–674 (1998; Zbl 0974.14018)].
MSC:
14H10 Families, moduli of curves (algebraic)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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