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Kieran O'Grady

Publications and source records attributed to Kieran O'Grady.

4 recordsLinked to original sources
arXiv2020

Hilbert squares of K3 surfaces and Debarre-Voisin varieties

The Debarre-Voisin hyperkähler fourfolds are built from alternating $3$-forms on a $10$-dimensional complex vector space, which we call trivectors. They are analogous to the Beauville-Donagi fourfolds associated with cubic fourfolds. In this article, we study several trivectors whose associated Debarre-Voisin variety is degenerate, in the sense that it is either reducible or has excessive dimension. We show that the Debarre-Voisin varieties specialize, along general $1$-parameter degenerations to these trivectors, to varieties isomorphic or birationally isomorphic to the Hilbert square of a K3 surface.

math.AG↗
arXiv2018

GIT versus Baily-Borel compactification for $K3$'s which are double covers of $\mathbb P^1\times\mathbb P^1$

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.

math.AG↗
arXiv2017

GIT versus Baily-Borel compactification for quartic K3 surfaces

Looijenga has introduced new compactifications of locally symmetric varieties that give a complete understanding of the period map from the GIT moduli space of plane sextics to the Baily-Borel compactification of the moduli space polarized K3's of degree 2, and also of the period map of cubic fourfolds. On the other hand, the period map of the GIT moduli space of quartic surfaces is significantly more subtle. In our previous paper, we introduced a Hassett-Keel-Looijenga program for certain locally symmetric varieties of Type IV. As a consequence, we gave a complete conjectural decomposition into a product of elementary birational modifications of the period map for the GIT moduli spaces of quartic surfaces. The purpose of this note is to provide compelling evidence in favor of our program. Specifically, we propose a matching between the arithmetic strata in the period space and suitable strata of the GIT moduli spaces of quartic surfaces. We then partially verify that the proposed matching actually holds.

math.AG↗