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

Abstract

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 $n\le 4g+5$. In this paper we consider the complementary case. We calculate the canonical divisor of ${\overline{\mathcal{H}}}_{g,n}$ and show that it is effective for $n=4g+6$ and big for $n\le 4g+7$. This leads us to conjecture that ${\overline{\mathcal{H}}}_{g,n}$ has non-negative Kodaira dimension for $n=4g+6$ and is of general type for $n\ge 4g+7$.

Mayor revisions in v3. The main theorem, abstract and outline have been changed

Cited by in corpus (2)

• Pencils on surfaces with normal crossings and the Kodaira dimension of ${\overline{\mathcal{M}}}_{g,n}$
• Birational Geometry of moduli spaces of pointed curves