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Strata of abelian differentials and the Teichmüller dynamics. (English) Zbl 1273.14054
Let \(\mathbb P \mathcal H\) be the projectivization of the total space of the Hodge bundle over \(M_g\). So \(\mathbb P \mathcal H\) parametrizes pairs \((C,\omega)\) with \(C\) a smooth genus \(g\) curve and \(\omega\) a nonzero holomorphic differential on \(C\) defined up to a scalar. To each point of \(\mathbb P\mathcal H\) one can associate a partition of \(2g-2\), given by the multiplicities of all zeroes of \(\omega\); we get a stratification of \(\mathbb P\mathcal H\) indexed by partitions. A finite cover of each stratum, given by ordering the zeroes of \(\omega\), maps naturally to \(M_{g,n}\) (where \(n\) is the number of zeroes). The paper under review studies the classes of these images in the Chow ring of \(M_{g,n}\).
Section 2 shows how Porteous’s formula can be used to express these classes in \(A^\bullet(M_{g,n})\) in terms of tautological classes. Section 3 works instead on the compactification \(\overline M_{g,n}\) but treats only divisors, and the author writes down test curves to produce a formula for the closure of the unique codimension one stratum corresponding to differentials with at least one multiple zero. This formula had previously been found by different methods in [D. Korotkin and P. Zograf, Math. Res. Lett. 18, No. 3, 447–458 (2011; Zbl 1246.32014)]. Section 4 proves that certain Brill–Noether divisors on \(\overline M_{g,n}\) (for \(g\) and \(n\) small) are extremal in the effective cone, using that these Brill–Noether divisors are swept out by images of Teichmüller curves.

14H10 Families, moduli of curves (algebraic)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
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