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Weierstrass cycles in moduli spaces and the Krichever map. (English) Zbl 1393.14022

Let \(\widehat{\mathcal C \mathcal M}_g\) be the moduli space of triples \((C,p,z)\) where \(C\) is a projective integral curve of genus \(g\), \(p\) is a nonsingular point, and \(z\) is a local coordinate system at \(p\) sending a closed set \(D\) containing \(p\) onto the unit disk with \(z(p) = 0\). Let \(k: \widehat{\mathcal C \mathcal M}_g \to \mathrm{Gr}(\mathcal H)\) be the extended Krichever map that embeds \(\widehat{\mathcal C \mathcal M}_g\) into the Sato Grassmannian \(\mathrm{Gr}(\mathcal H)\) in the sense of G. Segal and G. Wilson [in: Surveys in differential geometry. Vol. IV: Integral systems [Integrable systems]. Lectures on geometry and topology. Cambridge, MA: International Press. 403–466 (1998; Zbl 0939.35163)]. The authors study the \(S^1\)-equivariant cohomology induced by the Krichever map \(k\) and express the answers in terms of tautological classes \(\lambda\) and \(\psi\), where \(\lambda\) are the Chern classes of the Hodge bundle and \(\psi\) is the cotangent line class associated to the marked point \(p\). As applications, the authors compute the Weierstrass cycle class when it has expected dimension and obtains some relations for the tautological ring of the moduli space of curves.

MSC:

14H10 Families, moduli of curves (algebraic)

Citations:

Zbl 0939.35163
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