By Maria R. Gonzalez-Dorrego

ISBN-10: 0821825747

ISBN-13: 9780821825747

This monograph reports the geometry of a Kummer floor in ${\mathbb P}^3_k$ and of its minimum desingularization, that's 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 merely singularities. those nodes provide upward thrust to a configuration of 16 issues and 16 planes in ${\mathbb P}^3$ such that every aircraft includes precisely six issues and every aspect belongs to precisely six planes (this is named a '(16,6) configuration').A Kummer floor is uniquely decided by means of its set of nodes. Gonzalez-Dorrego classifies (16,6) configurations and experiences their manifold symmetries and the underlying questions about finite subgroups of $PGL_4(k)$. She makes use of this data to offer an entire type of Kummer surfaces with specific equations and particular descriptions in their singularities. furthermore, the gorgeous connections to the idea of K3 surfaces and abelian kinds are studied.

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**Additional info for 16, 6 Configurations and Geometry of Kummer Surfaces in P3**

**Example text**

MARIA R. 1). By assumption, we start out with < ? < ? - . 13 n PR ^ 0, a contradiction. The two points H and / in the upper half of the third column

Let V be one of the lines (l)-(6) above. 35(b), it is sufficient to prove independence of V from any three lines from among the four lines mentioned in (a). In the cases (1), (3) and (4) V meets exactly two of the lines in (a); in case (2) exactly one of the lines of (a). 40. Consider case (5). By definition of the standard (8,4) configuration, V is disjoint from 1 fl 1' (otherwise, V — 33' would have to be contained in either 1 or 1', which is a contradiction). Similarly, V fl 2 fl 2' = 0. 40.

Does split (for instance, we may send the generator of 1 x F2 to ( - 1 0 0 0 ^ 0 1 0 0 0 0 1 0 0 0 0 1, and the elements of S$ X 1 to permutations on the last three variables). Here and below the identity element in any group will be denoted by 1, even when the group is F2. Let t\ := 7r((7i). Suppose there was a section s : S4 x F2 —>• I . Then s(tj) would be an element of W which commutes (in PGL±(k)) with both e\ and 62. The only elements of W with this property are S4 x 1 conjugates of (j\. Conjugating the whole situation by an element of £4 X 1, if necessary, we may assume that s(i\) = G\.

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

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