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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.

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