The image in $${\mathcal {M}}_g$$ of strata of meromorphic and quadratic differentials

Abstract

We compute the dimension of the image of the map $$\pi _Z :Z \rightarrow {\mathcal {M}}_g$$ π Z : Z → M g forgetting the markings, where Z is a connected component of the stratum $${\mathcal {H}}^k_g(\mu )$$ H g k ( μ ) of k-differentials with an assigned partition $$\mu $$ μ , for the cases when $$k=1$$ k = 1 with meromorphic partition and $$k=2$$ k = 2 when the quadratic differentials have at worst simple poles.

A Hurwitz divisor on the moduli of Prym curves

“…This follows immediately from Corollary 5 in [7]. Hence Im(ϕ α,β ) and C 1 d are transverse and the number of pairs (L, y 1 , .…”

“…We state without proof the following identities Proposition 2. 7 We consider the sums S k = i−1 s=0 s k 2i s and compute the first terms to be…”

Maximal gonality on strata of differentials and uniruledness of strata in low genus

“…Let Z be an irreducible component of H k g (μ) and consider the forgetful map π Z : Z → M g . Corollary 5 in [5] implies that a generic fibre of the map π Z has dimension h 0 (C, p 1 + • • • + p n ) − 1 where [C, p 1 , . .…”

“…Let be an irreducible component of and consider the forgetful map . Corollary 5 in [5] implies that a generic fibre of the map has dimension where is a generic point of . Remark Theorems 1.1 and 1.2 provide another proof of the fact that has generically finite fibers when or , the component is nonhyperelliptic and the partition is positive of length .…”

On the locus of genus $3$ curves that admit meromorphic differentials with a zero of order $6$ and a pole of order $2$

“…These loci are interesting subloci of M g , but only a few results on them are known. The dimension of (the irreducible components of) M g (µ) have been computed in [Gen18] in the holomorphic case and [Bud20] in the meromorphic case. Moreover, in the holomorphic case, when the locus M g (µ) is a divisor in M g , its class has been computed in [Mul17].…”