# zbMATH — the first resource for mathematics

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
Full Text: