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

TL;DR

This paper investigates the Kodaira dimension of the moduli space of hyperelliptic curves with marked points, calculating the canonical divisor and exploring its positivity to determine the space's geometric type.

Contribution

It computes the canonical divisor of the moduli space and establishes its effectiveness and bigness for specific numbers of marked points, advancing understanding of its Kodaira dimension.

Findings

Canonical divisor is effective for n=4g+6

Canonical divisor is big for n ≤ 4g+7

Conjecture on the space being of general type for n ≥ 4g+7

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 $n \leq 4g+5$. 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 $n\leq 4g+7$. 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 $n \geq 4g+7$.