The image in $${\mathcal {M}}_g$$ M g of strata of meromorphic and quadratic differentials
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Mathematische Zeitschrift
The image in Mg of strata of meromorphic and quadratic differentials Andrei Bud1 Received: 28 April 2020 / Accepted: 3 September 2020 © The Author(s) 2020
Abstract We compute the dimension of the image of the map π Z : Z → Mg forgetting the markings, where Z is a connected component of the stratum Hkg (μ) of k-differentials with an assigned partition μ, for the cases when k = 1 with meromorphic partition and k = 2 when the quadratic differentials have at worst simple poles. Keywords Strata of differentials · Moduli of curves
1 Introduction For positive integers g and k, let μ = (m 1 , . . . , m n ) a partition of (2g − 2)k and define Hkg (μ) ⊆ Mg,n to be the moduli space of k-canonical divisors of type μ, parametrizing n m i pi ) ∼ pointed curves [C, p1 , . . . , pn ] satisfying OC ( i=1 = ωCk . It is natural to consider k a connected component Z of Hg (μ) and ask what is the dimension of the image of the forgetful map π Z : Z → Mg . We answer this question in the cases k = 1 with meromorphic partition and k = 2 when the quadratic differentials have at worst simple poles. Consequently, our results will provide new divisors along with higher codimension cycles on the moduli space Mg . To underline the importance of such cycles we point out that for k = 1 and μ = (2, 2, . . . , 2) the image of the even component is the divisor of curves with a vanishing theta null, see [15]. We start with the case k = 1 and drop the superscript k from the notation of the moduli of canonical divisors. It is obvious that if μ has a unique negative entry, equal to −1, the stratum Hg (μ) is empty. We will assume in what follows that the partition μ is not of this form and prove the following result: Theorem 1 For g ≥ 2, let μ be a strictly meromorphic partition of 2g − 2 of length n and Z a connected, non-hyperelliptic component of Hg (μ). Then the dimension of the image of the forgetful map π Z : Z → Mg is the expected one, that is, min {2g + n − 3, 3g − 3}.
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Andrei Bud [email protected] Institut für Mathematik, Humboldt Universität zu Berlin, Rudower Chausee 25, 12489 Berlin, Germany
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Together with the case of holomorphic differentials, treated by Gendron in [9], and the obvious case of hyperelliptic components, Theorem 1 completely answers the question for strata with k = 1. As a consequence, every stratum Hg (μ) has a connected component Z whose dimension of the image in Mg is equal to min {dim(Z ), 3g − 3}. We prove a similar theorem for the case k = 2 when the poles are at worst of order 1. When we have μ = (2m 1 , . . . , 2m n ) a positive partition of 4g − 4, the stratum H2g (μ) contains the components of Hg ( μ2 ). We will denote by Qg (μ) the union of connected components of H2g (μ) that are not components of Hg ( μ2 ). When μ has at least one odd entry we make the convention Qg (μ) = H2g (μ). If μ is a partition of 4g − 4 as above, there are four cases when Qg (μ) is empty; namely g = 1 and μ = (1, −1), or μ = (0) and the cases when g = 2 and μ = (3, 1) or μ = (4). We a
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