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A conjectural description of the tautological ring of the moduli space of curves. (English) Zbl 0978.14029

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).
Let \({\mathcal M}_g\) denote the moduli space of smooth, genus \(g\geq 2\) curves and let \(A^*({\mathcal M}_g)\) its (rational) Chow ring. Mumford defined natural classes in \(A^*({\mathcal M}_g)\) and conjectured that they generated the stable cohomology of \({\mathcal M}_g\) [see D. Mumford in: Arithmetic and geometry, Vol. II, Progr. Math. 36, 271-328 (1983; Zbl 0554.14006)]. This is true for \(g\leq 5\) (due to Mumford for \(g=2\), same reference as above, to the author for \(g=3,4\) [C. Faber, Ann. Math. (2) 132, 331-419 (1990; Zbl 0721.14013); 421-449 (1990; Zbl 0735.14021)], and to E. Izadi for \(g=5\) [The moduli space of curves, Proc. Conf., Texel 1994, Prog. Math. 129, 267-304 (1995; Zbl 0862.14017)]. Let us denote by \(R^*({\mathcal M}_g)\) the tautological ring.
In the paper under review the author formulates a number of conjectures giving a rather complete description of \(R^*({\mathcal M}_g)\). These conjectures, beside of giving explicit proportionality relation between the tautological classes, assert that \(R^* ({\mathcal M}_g)\) “behaves like” the algebraic cohomology ring of a nonsingular projective variety of dimension \(g-2\); i.e. it satisfies the hard Lefschetz property and Hodge positivity property with respect to the ample generator of the Picard group of \({\mathcal M}_g\). Moreover the author gives a survey of known results on \(R^*({\mathcal M}_g)\) (all supporting his conjectures) and \(A^*({\mathcal M}_g)\), and explains techniques and methods used by many authors over the years.
For the entire collection see [Zbl 0933.00030].

MSC:

14H10 Families, moduli of curves (algebraic)
14C05 Parametrization (Chow and Hilbert schemes)
14C15 (Equivariant) Chow groups and rings; motives
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
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