Abstract
We show that the K-moduli spaces of log Fano pairs
$\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$
, where C is a
$(4,4)$
curve and their wall crossings coincide with the VGIT quotients of
$(2,4)$
, complete intersection curves in
$\mathbb {P}^3$
. This, together with recent results by Laza and O’Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of
$(4,4)$
curves on
$\mathbb {P}^1\times \mathbb {P}^1$
and the Baily–Borel compactification of moduli of quartic hyperelliptic K3 surfaces.
Birational geometry of moduli of curves with an S3-cover
