ORBIT PARAMETRIZATIONS FOR K3 SURFACES

Manjul Bhargava,Wei Ho,Abhinav Kumar

doi:10.1017/fms.2016.12

Abstract

We study moduli spaces of lattice-polarized K3 surfaces in terms of orbits of representations of algebraic groups. In particular, over an algebraically closed field of characteristic 0, we show that in many cases, the nondegenerate orbits of a representation are in bijection with K3 surfaces (up to suitable equivalence) whose Néron–Severi lattice contains a given lattice. An immediate consequence is that the corresponding moduli spaces of these lattice-polarized K3 surfaces are all unirational. Our constructions also produce many fixed-point-free automorphisms of positive entropy on K3 surfaces in various families associated to these representations, giving a natural extension of recent work of Oguiso.

Highlights

Just as 2 × 2 × 2 cubical matrices played a key role in the understanding of many prehomogeneous representations [7], and just as 3 × 3 × 3 and 2 × 2 × 2 × 2 matrices played a key role in the understanding of coregular representations associated to genus one curves [16], we find that 4 × 4 × 4 and 2 × 2 × 2 × 2 × 2 matrices appear as fundamental cases for our study of K3 surfaces
We study orbits on symmetrized versions of these spaces, which turn out to correspond to moduli spaces of K3 surfaces of higher rank
We prove that the orbits of the space of doubly triply symmetric penteracts are related to certain K3 surfaces with

Summary

Triply symmetric Rubik’s revenge

We consider the triply symmetric Rubik’s revenge, in order to understand the orbits of Gm × GL(V ) on Sym3V for a 4-dimensional vector space V. We may use that proof to construct a 4 × 4 × 4 cube A in V1 ⊗ V2 ⊗ V3, for certain 4-dimensional vector spaces Vi , where there is an isomorphism φ32 : V3 → V2 so that A is symmetric, that is, maps to an element of V1 ⊗ Sym2V2 under Id ⊗ Id ⊗ φ32. Let G be the quotient of G by the kernel of the multiplication map G5m → Gm. we will study the G(F)-orbits on V (F), and in particular, describe the relationship between (an open subvariety of) the orbit space V (F)/G(F) and the moduli space of certain K3 surfaces having Neron–Severi rank at least 4: THEOREM 7.1.

A11 A12 A21 A22

11. Doubly triply symmetric penteracts

13. Quintuply symmetric penteracts