Rational Picard Group of Moduli of Pointed Hyperelliptic Curves

Abstract

We determine the rational Picard group of the moduli spaces of smooth pointed hyperelliptic curves and of their Deligne-Mumford compactification, over the field of complex numbers.

The rational Chow rings of moduli spaces of hyperelliptic curves with marked points

“…In [22,Theorem 1.1], Scavia shows that, over the complex numbers, the ψ classes and boundary divisors form a basis for the Picard group of H g,n . Our I g,n is the complement of all boundary divisors in H g,n besides the divisor ∆ of irreducible nodal curves.…”

“…Indeed, to prove rationality, one must understand a dense open subset of the space, whereas to determine the Chow ring, one must understand a full stratification and how the pieces fit together. Previous work on the intersection theory of H g,n has determined the Chow group in codimension 1 (Scavia [22]) and the full (integral) Chow ring in the case of 1 marked point (Pernice [21]). Here, we determine the full rational Chow ring A * (H g,n ) for n ≤ 2g + 6.…”

The Kodaira classification of the moduli of hyperelliptic curves

“…The inclusion O C (−x) → O C canonically induces a map of the corresponding exact sequences (2.7). The middle vertical arrow [Sca,Prop. 5.4] that H g,n is smooth on the locus of n-pointed hyperelliptic curves that remain stable after forgetting the markings.…”