Affine Geometry of Strata of Differentials

Abstract

Affine varieties among all algebraic varieties have simple structures. For example, an affine variety does not contain any complete algebraic curve. In this paper we study affine related properties of strata of k-differentials on smooth curves which parameterize sections of the k-th power of the canonical line bundle with prescribed orders of zeros and poles. We show that if there is a prescribed pole of order at least k, then the corresponding stratum does not contain any complete curve. Moreover, we explore the amusing question whether affine invariant manifolds arising from Teichmüller dynamics are affine varieties, and confirm the answer for Teichmüller curves, Hurwitz spaces of torus coverings, hyperelliptic strata as well as some low genus strata.

Tautological ring of strata of differentials

“…On the other hand, the hyperelliptic strata components P(2g−2) hyp and P(g−1, g−1) hyp are isomorphic to certain moduli spaces of pointed smooth rational curves, hence their rational Picard groups are trivial, and so is η. Similarly some low genus strata can be realized as images of finite morphisms from affine varieties with trivial rational Picard group (see [C3,Section 4]), hence η is trivial for those strata.…”

Positivity of Divisor Classes on the Strata of Differentials

“…In addition, knowing ample divisor classes can help determine the birational type of the underlying variety, e.g., the canonical class of a variety of general type can be written as the sum of an ample divisor class and an effective divisor class. We hope to compare the canonical class of the strata with the ample divisor classes produced in this paper in order to determine the birational type of the strata (see [G,Ba,C2] for some partial results). Finally, the second part of Theorem 1.1 yields an approach to analyze the base loci of divisor classes outside of the ample cone of the strata, hence it can help us better understand the effective cone decomposition of the strata as well as alternate birational models in the sense of Mori's program.…”

Les strates ne possèdent pas de variétés complètes

“…Ce problème est encore peu exploré dans le cas des strates de différentielles abéliennes. Chen à montré dans [3] qu'il n'existe pas de variétés complètes dans les strates de différentielles abéliennes méromorphes. Récemment Hamenstädt [6] a affirmé avoir résolu cette question dans le cas holomorphe en montrant que les strates sont affines.…”