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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.

##### MSC:
 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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