Quartics and moduli

Even restricted to surfaces (of Kod. dim 0), moduli has various viewpoints.  I will discuss them centering three famous quartic hypersurfaces.

1. Igusa 3-fold and Enriques surfaces: This quartic was first found as the projective dual of Segre's 10-nodal cubic, the moduli of 6 points on the projective line.  It was re-discovered as moduli of p.p.a.s's by Igusa(1964).  I explain its new interpretation (Contemp.  Math., 2012) as the moduli of Enriques surfaces of certain root type.

2. Coble 6-fold and K3 surfaces: The moduli space of polarized K3 surfaces is of general type except for finitely many genus.  As an example of exception I explain the uni-rationality for genus 13 using the Coble quartic in  P7, which is the moduli space  N+  of 2-bundles on a plane quartic curve.  The generic K3 is described explicitly in the moduli space  N-, the Hecke partner of N+ (joint work with A. Kanemitsu).

3. Inose surfaces and type II degenerations: The 2-parameter family of quartic surfaces found by Inose (1978) contains infinitely many 1-parameter subfamilies which are mirror of polarized K3 surfaces in the sense of Dolgachev(1996).  Their Picard lattices and nef cones control the type II degenerations.  I discuss the Hilbert square of Inose quartics, first their birational automorphisms and next a connection with type II degenerations of holomorphic symplectic 4-folds of deformation type K3[2].