2021/06/25 by Ignacio Barros, Barros, Ignacio, Scott Mullane +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2106.13774
We study the birational geometry of the moduli spaces of hyperelliptic curves with marked points. We show that these moduli spaces have non ℚ-factorial singularities. We complete the Kodaira classification by proving that these spaces have Kodaira dimension 4g+3 when the number of markings is 4g+6 and are of general type when the number of markings is n≥4g+7. Similarly, we consider the natural finite cover given by ordering the Weierstrass points. In this case, we provide a full Kodaira classification showing that the Kodaira dimension is negative when n≤3, one when n=4, and of general type when n≥ 5. For this, we carry out a singularity analysis of ordered and unordered pointed Hurwitz spaces. We show that the ordered space has canonical singularities and the unordered space has non-canonical singularities. We describe all non-canonical points and show that pluricanonical forms defined on the full regular locus extend to any resolution. Further, we provide a full classification of the structure of the pseudo-effective cone of Cartier divisors for the moduli space of hyperelliptic curves with marked points. We show the cone is non-polyhedral when the number of markings is at least two and polyhedral in the remaining cases.