Tautological algebras of moduli spaces of curves (Q2848316)

This survey article contains the lecture notes for a course that the author taught at the Park City Mathematics Institute on moduli spaces of Riemann surfaces in July, 2011. The material is organized in two parts titled ``Lecture 1'' and ``Lecture 2'', respectively, which are to reflect roughly the first and second half of the course.NEWLINENEWLINE More precisely, let \(g\geq 2\) be an integer and denote by \(M_g\) the moduli space of nonsingular projective curves of genus \(g\). The subject of the first lecture is the so-called tautological ring \(R^*(M_g)\) of \(M_g\). This graded ring is defined as the \(\mathbb{Q}\)-subalgebra of the rational Chow ring \(A^*(M_g)\) of \(M_g\) generated by the tautological classes as constructed in \textit{D. Mumford's} seminal paper ``Towards an enumerative geometry of the moduli space of curves'' [Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 271--328 (1983; Zbl 0554.14008)].NEWLINENEWLINE In the course of the first lecture, the author recalls Mumford's definition of these so-called kappa-classes as well as different methods for producing relations between them. Next he discusses his own conjecture on a precise description of \(R^*(M_g)\) published several years ago in [the author, in: Moduli of curves and abelian varieties. The Dutch intercity seminar on moduli. Braunschweig: Vieweg. Aspects Math. E33, 109--129 (1999; Zbl 0978.14029)], together with the various results on it as obtained by himself and others in the meantime.NEWLINENEWLINE Finally, he surveys some very recent developments indicating that the relations that suffice to prove the conjecture for \(g\leq 23\) may not be sufficient for larger values of \(g\).NEWLINENEWLINE The second lecture deals with the moduli space \(M_{g,n}\) of \(n\)-pointed nonsingular curves of genus \(g\) where the inequality \(2g-2+ n>0\) holds. More precisely, the Deligne-Mumford compactification \(\overline M_{g,n}\) of \(M_{g,n}\) as well as some natural partial compactifications of \(M_{g,n}\) sitting in between \(M_{g,n}\) and \(\overline M_{g,n}\) are discussed, and various developments concerning the tautological rings of these (partially) compactified moduli spaces are briefly touched upon, mainly through references to the ample current research literatur in this context. As for more information concerning these topics, the survey article ``Tautological and non-tautological cohomology of the moduli space of curves'' by the author and \textit{R. Pandharipande} [Preprint, \url{arxiv:1101.5489}, to appear in ``Handbook of Moduli (G. Farkas et al. (eds.), 2013)] is highly recommended.NEWLINENEWLINE As these lectures are directed to graduate students and non-specialists,each of them ends with a set of related exercises and problems stimulating both further reading and self-reliant research.NEWLINENEWLINEFor the entire collection see [Zbl 1272.30002].