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Degenerations of abelian differentials. (English) Zbl 1388.14080
D. Eisenbud and J. Harris introduced the concept of limit linear series for degenerations of linear series on algebraic curves [Invent. Math. 85, 337–371 (1986; Zbl 0598.14003)]. In the present article, the author pursues an analogous theory for moduli spaces of abelian differentials on compact Riemann surfaces.
For precise statements, let \(\mathcal{M}_g\) be the moduli space of genus \(g\) curves and \(\overline{\mathcal{M}}_g\) its Deligne-Mumford compactification. The Hodge bundle \(\mathcal{H} \rightarrow \mathcal{M}_g\), extends to a rank \(g\) vector bundle \(\overline{\mathcal{H}} \rightarrow\overline{\mathcal{M}}_g\). Inside of \(\mathcal{H}\), let \(\mathcal{H}(\mu) \subset \mathcal{H}\) be the stratum of holomorphic abelian differentials with signature \(\mu\), for \(\mu = (m_1,\dots,m_n)\) a partition of \(2g-2\). In particular, \(\mathcal{H}(\mu)\) parametrizes pairs \((C,\omega)\), where \(C\) is a smooth, connected, compact, complex curve of genus \(g\), and \(\omega\) is a holomorphic abelian differential on \(C\) with divisor of zeros: \[ (\omega)_0 = m_1 p_1 + \dots + m_n p_n, \] for distinct points \(p_1,\dots,p_n \in C\). Furthermore, let \(\overline{\mathcal{H}}(\mu) \subset \overline{\mathcal{H}}\) be the closure of \(\mathcal{H}(\mu) \subset \mathcal{H}\).
In this context, the author’s first result pertains to those strata closures \(\overline{\mathcal{H}}(\mu)\) which contain a given fixed \((C,\omega) \in \overline{\mathcal{H}}\). The author then refines this result by marking the zeros of differentials in \(\mathcal{H}(\mu)\). For example, inside of \(\mathcal{M}_{g,n}\), the moduli space of genus \(g\) curves with \(n\)-marked points, consider \(\mathcal{P}(\mu) \subseteq \mathcal{M}_{g,n}\) the stratum of holomorphic canonical divisors with signature \(\mu\) and its closure \(\overline{\mathcal{P}}(\mu) \subseteq \overline{\mathcal{M}}_{g,n}\) in the Deligne-Mumford compactification of \(\mathcal{M}_{g,n}\). In this setting, the author gives necessary and sufficient conditions for a given marked curve, of compact type and with a single node, to lie in \(\overline{\mathcal{P}}(\mu)\). The author also obtains a version of this result which applies to the strata of meromorphic canonical divisors with a given signature.
The author then studies the connected components for the strata of holomorphic differentials and builds on work of M. Kontsevich and A. Zorich [Invent. Math. 153, No. 3, 631–678 (2003; Zbl 1087.32010)]. Indeed, the author obtains a result which distinguishes the boundary points of the spin components in the locus of curves of pseudocompact type. Finally, the author describes Weierstrass point behaviour for general differentials in \(\mathcal{H}(\mu)\). These results contribute to the study of the nonhyperelliptic components of the strata of holomorphic differentials.

14H10 Families, moduli of curves (algebraic)
14H15 Families, moduli of curves (analytic)
14K20 Analytic theory of abelian varieties; abelian integrals and differentials
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