Let k be an algebraically closed field of characteristic different from 2. A Kummer
surface in P 3 is a quartic surface with 16 nodes. A quartic surface in P 3 with only
nodes as singularities can have at most 16 nodes. In this sense Kummer surfaces
are extremal, so we may expect certain special geometric characteristics. These 16
nodes together with the 16 planes which correspond to the 16 nodes of the dual
Kummer surface (called special planes or tropes by the classical authors) form a
(16,6) configuration in P 3 which determines the Kummer surface uniquely (Lemma
2.16). This configuration has many symmetries and is a beautiful combinatorial
object to study. The purpose of this work is to understand in detail the geometry
of a Kummer surface in P 3 .
Kummer surfaces appear in many different contexts: they are related to abelian
surfaces and to the quadric line complex. Different authors define a Kummer surface
in different ways; for example, sometimes it is defined to be the quotient of an
abelian surface by the involution. The minimal desingularization of a Kummer
surface, which is always a K3 surface, is often called a non-singular Kummer surface.
The fact that the quotient of the Jacobian of a non-singular curve of genus 2 by the
involution is a Kummer surface in our sense is well-known and easy to prove using
theta divisors. The converse, namely, the statement that any Kummer surface can
be obtained from some principally polarized abelian variety by taking a quotient
by the involution, is also known. We give a new point of view of this in §4. Our
approach is based on an a priori classification of (16,6) configurations which we
call non-degenerate (this means that every two special planes have exactly two
points of the configuration in common). We prove that all the non-degenerate
(16,6) configurations of points and planes in P 3 are combinatorially the same. In
other words, for a (16,6) configuration of points and planes in P 3 imposing the non-
degeneracy hypothesis determines the incidence matrix uniquely. Thus, in contrast
to the traditional approach to Kummer surfaces, we do not need the theory of
theta divisors to conclude that the (16,6) configuration associated with a Kummer
surface has a certain specified incidence matrix (which we call a (16,6) configuration
of type (*)). The classical authors [12] assumed this fact, and the classification of
non-degenerate (16,6) configurations in P3 without proofs.
§1 is devoted to a systematic study of (16,6) configurations. We show that a
(16,6) configuration associated with a Kummer surface is always non-degenerate.
We define (16,6) configurations of type (*) (this definition is purely combinatorial:
it amounts to specifying a particular incidence matrix) and prove that any non-
degenerate (16,6) configuration of points and planes in P 3 is of type (*). To do
that, we first classify abstract non-degenerate (16,6) configurations (which are, by
definition, 16x16 matrices (a»j) whose entries are zeroes and ones, with exactly 6
ones in each row and each column, such that for every two rows i and j there are
exactly two columns / and I' such that an = dji 1). We prove that there are
Received by the editor January 16, 1991, and in revised form May 18, 1992
Previous Page Next Page