Connected Components of H_(r,g)^A (G)

The Hurwitz space is the space of genus g covers of the Riemann sphere with branch points and the monodromy group . In this paper, we enumerate the connected components of the Hurwitz spaces for a finite primitive group of degree 7 and genus zero except . We achieve this with the aid of the computer algebra system GAP and the MAPCLASS package.

AFFINE GEOMETRY OF STRATA OF DIFFERENTIALS

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.