GIT versus Baily-Borel compactification for $K3$'s which are double covers of $\mathbb P^1\times\mathbb P^1$
Abstract
In previous work, we have introduced a program aimed at studying the birational geometry of locally symmetric varieties of Type IV associated to moduli of certain projective varieties of K3 type. In particular, a concrete goal of our program is to understand the relationship between GIT and Baily-Borel compactifications for quartic K3 surfaces, K3's which are double covers of a smooth quadric surface, and double EPW sextics. In our first paper (arXiv:1607.01324), based on arithmetic considerations, we have given conjectural decompositions into simple birational transformations of the period maps from the GIT moduli spaces mentioned above to the corresponding Baily-Borel compactifications. In our second paper (arXiv:1612.07432) we studied the case of quartic K3's; we have given geometric meaning to this decomposition and we have partially verified our conjectures. Here, we give a full proof of our conjectures for the moduli space of K3's which are double covers of a smooth quadric surface. The main new tool here is VGIT for (2,4) complete intersection curves.
Keywords
Cite
@article{arxiv.1801.04845,
title = {GIT versus Baily-Borel compactification for $K3$'s which are double covers of $\mathbb P^1\times\mathbb P^1$},
author = {Radu Laza and Kieran O'Grady},
journal= {arXiv preprint arXiv:1801.04845},
year = {2022}
}
Comments
61 pages. Version 2 supersedes Version 1 (somewhat preliminary). Full details and arguments are provided. This gives a complete, highly non-trivial illustration of our Hassett-Keel-Looijenga program (related "easier" known examples include degree 2 K3s and cubic fourfolds)