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affine invariant manifolds

Journal of the Institute of Mathematics of Jussieu (1)

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AFFINE GEOMETRY OF STRATA OF DIFFERENTIALS
Part of
- Curves
- Birational geometry
Dawei Chen
Journal:

Journal of the Institute of Mathematics of Jussieu / Volume 18 / Issue 6 / November 2019

Published online by Cambridge University Press:

26 October 2017, pp. 1331-1340

Print publication:

November 2019
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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.