By Maria R. Gonzalez-Dorrego

ISBN-10: 0821825747

ISBN-13: 9780821825747

This monograph stories the geometry of a Kummer floor in ${\mathbb P}^3_k$ and of its minimum desingularization, that is a K3 floor (here $k$ is an algebraically closed box of attribute diverse from 2). This Kummer floor is a quartic floor with 16 nodes as its basically singularities. those nodes provide upward thrust to a configuration of 16 issues and 16 planes in ${\mathbb P}^3$ such that every airplane includes precisely six issues and every aspect belongs to precisely six planes (this is termed a '(16,6) configuration').A Kummer floor is uniquely made up our minds by way of its set of nodes. Gonzalez-Dorrego classifies (16,6) configurations and stories their manifold symmetries and the underlying questions about finite subgroups of $PGL_4(k)$. She makes use of this data to offer an entire class of Kummer surfaces with specific equations and specific descriptions in their singularities. moreover, the attractive connections to the speculation of K3 surfaces and abelian kinds are studied.

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**Example text**

3). 3), respectively:

3) together with det ei = 1 imply that both e$ and e 4 commute with e2. 2), e 4 must commute with both e 2 and e3, hence e 4 must be a diagonal matrix with eigenvalues 1,-1,-1. But then e 4 belongs to the subgroup of SL^k) generated by e 2 and e3, which is a contradiction. The Sublemma is proved. • From now on we assume that e2j = 1, 1 < j < 4. Then each of the e7- is a diagonaHzable matrix with eigenvalues 1,1,-1,-1. 2) imply that for 2 < j < 4, e^ej = ±eje\. Next, we may assume that at least two of the ej, 2 < j < 4, actually commute with e\.

We have a natural action of N on Fo by conjugation. View Fo as a 4-dimensional vector space over F2 on which N acts by linear automorphisms. 1) ^ — GL4(¥2) (note that GL4(¥2) = PGL4(¥2) = SL4(¥2))The point of the next theorem is to show that -p- can be identified with Sp4{¥2) for a suitable symplectic structure on F 0 = F*. Next we describe the symplectic structure on Fo which is preserved by N. Namely, consider any four element subgroup H of Fo. We may think of such a subgroup as a 2-plane in our 4-dimensional vector space F | .

### 16, 6 Configurations and Geometry of Kummer Surfaces in P3 by Maria R. Gonzalez-Dorrego

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