Smooth Kummer Surfaces in Projective Three-space

In this note we prove the existence of smooth Kummer surfaces in projective three-space containing sixteen mutually disjoint smooth rational curves of any given degree. Introduction Let X be a smooth quartic surface in projective three-space P. As a consequence of Nikulin’s theorem [6] X is a Kummer surface if and only if it contains sixteen mutually disjoint smooth rational curves. The classical examples of smooth Kummer surfaces in P are due to Traynard (see [8] and [4]). They were rediscovered by Barth and Nieto [2] and independently by Naruki [5]. These quartic surfaces contain sixteen skew lines. In [1] it was shown by different methods that there also exist smooth quartic surfaces in P containing sixteen mutually disjoint smooth conics. Motivated by these results it is then natural to ask if, for any given integer d ≥ 1, there exist smooth quartic surfaces in P containing sixteen mutually disjoint smooth rational curves of degree d. The aim of this note is to show that the method of [1] can be generalized to answer this question in the affirmative. We show: Theorem. For any integer d ≥ 1 there is a three-dimensional family of smooth quartic surfaces in P containing sixteen mutually disjoint smooth rational curves of degree d. We work throughout over the field C of complex numbers. 1. Preliminaries Let (A,L) be a polarized abelian surface of type ( 1, 2d + 1 ) , d ≥ 1, and let L be symmetric. Denote by e1, . . . , e16 the halfperiods of A. We are going to consider the non-complete linear system∣∣∣∣OA (2L)⊗ 16 ⊗