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The Deligne-Mumford and the incidence variety compactifications of the strata of \(\Omega \mathcal{M}_g\). (Les compactifications de Deligne-Mumford et de la variété d’incidence des strates de \(\Omega \mathcal{M}_g\).) (English. French summary) Zbl 1403.14059
Let \(\mathcal{M}_g\) be the moduli space of algebraic curves of genus \(g\), and \(\Omega \mathcal{M}_g\) the Hodge bundle consisting of equivalence classes of pairs \((X, \omega)\) where \(X\) is a smooth curve of genus \(g\), and \(\omega\) is a non-zero holomorphic 1-form on \(X\). The Hodge bundle \(\Omega \mathcal{M}_g\) has a natural stratification given by the multiplicities of the zeros of \(\omega\). In the present paper the author studies two compactifications of the strata, namely the Deligne-Munford compactification and the incidence variety compactification. The goal of this study is to compute the Kodaira dimension of the strata.
The Deligne-Munford compactification of \(\mathcal{M}_g\) is the moduli space \(\overline{\mathcal{M}}_g\) of stable algebraic curves of arithmetic genus \(g\). Then, \(\Omega \overline{\mathcal{M}}_g\) is the moduli space of stable differentials, and the closure of a stratum inside \(\Omega \overline{\mathcal{M}}_g\) is the Deligne-Munford compactification of the stratum. On the other hand, the incidence variety compactification is a compactification of the projectivisation of the stratum, obtained as a quotient by the permutation of zeros.
Then, by using the compactifications, the author studies the Kodaira dimension of the strata. For a given \(n\)-tuple \((k_1, \dots, k_n)\) with \(\sum k_i = 2g-2\), Section 5 is devoted to study the Kodaira dimension of the stratum \(\mathbb{P} \Omega \mathcal{M}_g (k_1, \dots, k_n)\). First, (Prop. 5.6), the dimension of \(\mathbb{P} \Omega \mathcal{M}_g (1, \dots, 1)\) is \(- \infty\). And the main result is Theorem 5.10, according to which, for \(g \geq 2, (k_1, \dots, k_n) = (k_1, \dots, k_l, 1, \dots, 1)\), with \(k_i \geq 2\) for \(i \leq l\), \(\sum _{i=1} ^n k_i = 2g-2, \sum _{i=1} ^l k_i \leq g-2\), then the Kodaira dimension of the stratum \(\mathbb{P} \Omega \mathcal{M}_g (k_1, \dots, k_n)\) is \(- \infty\). Other strata are considered in this Section 5 of the paper, like \(\mathbb{P} \Omega \mathcal{M}_g (g-1, 1, \dots, 1)\), the hyperelliptic strata \(\mathbb{P} \Omega \mathcal{M}_g (g-1, g-1)\), or the odd and the even connected components of \(\mathbb{P} \Omega \mathcal{M}_g (2, \dots, 2)\).
The paper finishes in Sections 6 and 7 with the study of hyperelliptic minimal strata and their relationship with the hyperelliptic Weierstrass locus, and the boundary of the incidence variety compactification of the odd minimal stratum in genus 3.

14H15 Families, moduli of curves (analytic)
30F30 Differentials on Riemann surfaces
14E99 Birational geometry
14H45 Special algebraic curves and curves of low genus
Full Text: DOI
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