On the Kodaira dimension of the moduli space of hyperelliptic curves with marked points

Irene Schwarz

Published 2020-02-09Version 1

It is known that the moduli space $\overline{\mathcal{H}}_{g,n}$ of genus g stable hyperelliptic curves with $n$ marked points is uniruled for $n \geq 4g+5$. In this paper we consider the complementary case and show that $\overline{\mathcal{H}}_{g,n}$ has non-negative Kodaira dimension for $n = 4g+6$ and is of general type for $n \geq 4g+7$. Important parts of our proof are the calculation of the canonical divisor and establishing that the singularities of $\overline{\mathcal{H}}_{g,n}$ do not establish adjunction conditions.