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AFFINE GEOMETRY OF STRATA OF DIFFERENTIALS

Part of: Birational geometry Curves

Published online by Cambridge University Press:  26 October 2017

Dawei Chen
Dawei Chen*
Affiliation:
Department of Mathematics, Boston College, Chestnut Hill, MA 02467, USA ([email protected])
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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.

Keywords

strata of differentialsmoduli space of curvescomplete curvesaffine varietiesaffine invariant manifolds

MSC classification

Primary: 14H10: Families, moduli (algebraic) 14H15: Families, moduli (analytic) 14E99: None of the above, but in this section
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 18 , Issue 6 , November 2019 , pp. 1331 - 1340
Copyright
© Cambridge University Press 2017 

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Footnotes

The author is partially supported by NSF CAREER Award DMS-1350396.

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