§1. T H E CLASSIFICATION O F (16,6) CONFIGURATIONS. Let k be an algebraically closed field of characteristic ^ 2. This section is de- voted to the study of (16,6) configurations. We define (16,6) configurations of type (*) (this definition is purely combinatorial: it amounts to specifying a particular incidence matrix) and the notion of a non-degenerate (16,6) configuration. The motivation for studying such configurations is the fact (which will be proved below) that the singular locus of a Kummer surface always forms such a configuration and, conversely, any non-degenerate (16,6) configuration of points and planes in P 3 is of type (*) and is the singular locus of a uniquely determined Kummer surface. Thus classifying Kummer surfaces is equivalent to classifying non-degenerate (16,6) configurations. The main point of this section is to prove that any non-degenerate (16,6) configuration of points and planes in P 3 is of type (*) and, up to a linear au- tomorphisms of P 3 , looks like the orbit of a sufficiently general point (a, 6, c, d) G IP3 (where we give a precise definition (1.2.1) of "sufficiently general") under a certain finite subgroup Fo of PGL^{k) of order 16. Hudson [12, p. 7] makes this claim with- out proof and without mentioning the non-degeneracy hypothesis, without which the statement is false, (see Example 1.11). We introduce the notion of an abstract (16,6) configuration, in order to separate the combinatorial aspects of the problem from the geometric ones. This section contains the following subsections: I. The classification of abstract non-degenerate (16,6) configurations. II. The (8,4) configuration. III. A non-degenerate (16,6) configuration of points and planes in P 3 is of type (*)• IV. A nondegenerate (16,6) configuration in P 3 is of the form (a, 6, c, d) of (1.4.1). V. Moduli of non-degenerate (16,6) configurations. Definition 1.1. A (16,6) configuration is a set of 16 planes and 16 points in Pi such that every plane contains exactly 6 of the 16 points and every point lies in exactly 6 of the 16 planes. When talking about a given (16,6) configuration, we shall sometimes refer to the 16 planes as special planes (this is the classical terminology). The purpose of this section is to classify the (16,6) configurations satisfying an additional assumption of non-degeneracy: Definition 1.2. A (16,6) configuration is non-degenerat e if every two special planes share exactly two points of the configuration and every pair of points is contained in exactly two special planes (note that the second part of the deBnition follows automatically from the first by a counting argument). We classify these configurations both as subsets of P 3 and as subsets of P 3 mod- ulo projective linear transformations. The results of this section will allow us to completely classify Kummer surfaces, both as subvarieties of P 3 and as abstract surfaces. Since there is a bijection between the set of Kummer surfaces and the set of non-singular curves of genus 2 (Propositions 4.22 and 4.23), we also get a new way of expressing M2 as a quotient of an open subset of P 3 by a finite group. 3
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