https://www.mdu.se/

The moduli of principally polarized abelian varieties of dimension g ≤ 5 can be studied using Prym varieties and their deformations. This thesis studies the ℓ-adic cohomology of the Prym moduli space R_₂, which parametrizes smooth curves of genus two equipped with a nonzero two-torsion point on their Jacobian. Our approach uses the geometry of singular plane quartics to give a combinatorial description of this moduli space. This perspective also suggests possible generalizations of the combinatorial construction to Prym varieties of higher dimension.

Specifically, our description identifies R₂ with configurations of points and lines in P^², considered up to projective equivalence. We then identify this space with the quotient of M_₀,₆ by S_₂ × S_₄. We use the Grothendieck-Lefschetz trace formula, together with purity and the symmetric-group action on the cohomology of M_₀,₆, to obtain information about the ℓ-adic cohomology of R_₂. This provides an explicit approach to studying the cohomology of the moduli space through the geometry of nodal plane quartics.

Along the way, we investigate geometric properties of the bitangents and theta lines of singular plane quartics. These extend aspects of the classical geometry of the 28 bitangents to a smooth plane quartic.