# \((16,6)\) Configurations and Geometry of Kummer Surfaces in \(\mathbb P^{3}\)

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*Maria R. Gonzalez-Dorrego*

This monograph studies the geometry of a Kummer surface in \({\mathbb P}^3_k\) and of its minimal desingularization, which is a K3 surface (here \(k\) is an algebraically closed field of characteristic different from 2). This Kummer surface is a quartic surface with sixteen nodes as its only singularities. These nodes give rise to a configuration of sixteen points and sixteen planes in \({\mathbb P}^3\) such that each plane contains exactly six points and each point belongs to exactly six planes (this is called a “(16,6) configuration”). A Kummer surface is uniquely determined by its set of nodes. Gonzalez-Dorrego classifies (16,6) configurations and studies their manifold symmetries and the underlying questions about finite subgroups of \(PGL_4(k)\). She uses this information to give a complete classification of Kummer surfaces with explicit equations and explicit descriptions of their singularities. In addition, the beautiful connections to the theory of K3 surfaces and abelian varieties are studied.

#### Table of Contents

# Table of Contents

## $(16,6)$ Configurations and Geometry of Kummer Surfaces in $\mathbb P^{3}$

- Table of Contents v6 free
- §0. Introduction 18 free
- §1. The classification of (16,6) configurations 310 free
- I. The classification of abstract non-degenerate (16,6) configurations 815
- II. The (8,4) configuration 1320
- III. A non-degenerate (16,6) configuration of points and planes in P[sup(3) is of type (*) 2633
- IV. A non-degenerate (16,6) configuration in P[sup(3)] is of the form (a, b, c, d) 3138
- V. Moduli of non-degenerate (16,6) configurations 3946

- §2. The classification of Kummer surfaces in P[sup(3)] 5966
- §3. Divisors on a Kummer surface and its minimal desingularization 7582
- §4. Geometry of a Kummer surface in P[sup(3)] and the associated abelian variety 8491
- References 101108