Kummer quartic surfaces, strict self-duality, and more

Abstract

In this paper we first show that each Kummer quartic surface (a quartic surface $X$ with 16 singular points) is, in canonical coordinates, equal to its dual surface, and that the Gauss map induces a fixpoint free involution $\gamma$ on the minimal resolution $S$ of $X$. Then we study the corresponding Enriques surfaces $S/ \gamma$. We also describe in detail the remarkable properties of the most symmetric Kummer quartic, which we call the Cefal\'u quartic. We also investigate the Kummer quartic surfaces whose associated Abelian surface is isogenous to a product of elliptic curves through an isogeny with kernel $(\mathbb{Z}/2)^2$, and show the existence of polarized nodal K3 surfaces $X$ of any degree $d=2k$ with the maximal number of nodes, such that $X$ and its nodes are defined over $\mathbb{R}$. We take then as parameter space for Kummer quartics an open set in $\mathbb{P}^3$, parametrizing nondegenerate $(16_6, 16_6)$-configurations, and compare with other parameter spaces. We also extend to positive characteristic some results which were previously known over $\mathbb{C}$. We end with a section devoted to remarks on normal cubic surfaces, and providing some other examples of strictly selfdual hypersurfaces.