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

Available Formats:
Electronic ISBN: 978-1-4704-0089-7
Product Code: MEMO/107/512.E
List Price: $36.00 MAA Member Price:$32.40
AMS Member Price: $21.60 Click above image for expanded view$(16,6)$Configurations and Geometry of Kummer Surfaces in${\mathbb P}^3$Available Formats:  Electronic ISBN: 978-1-4704-0089-7 Product Code: MEMO/107/512.E  List Price:$36.00 MAA Member Price: $32.40 AMS Member Price:$21.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 1071994; 101 pp
MSC: Primary 51; 14;

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.

Research mathematicians.

• Chapters
• 0. Introduction
• 1. The classification of (16,6) configurations
• 2. The classification of Kummer surfaces in $\mathbb {P}^3$
• 3. Divisors on a Kummer surface and its minimal desingularization
• 4. Geometry of a Kummer surface in $\mathbb {P}^3$ and the associated abelian variety
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Volume: 1071994; 101 pp
MSC: Primary 51; 14;

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.

• 2. The classification of Kummer surfaces in $\mathbb {P}^3$
• 4. Geometry of a Kummer surface in $\mathbb {P}^3$ and the associated abelian variety