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 diversified from 2). This Kummer floor is a quartic floor with 16 nodes as its purely singularities. those nodes provide upward thrust to a configuration of 16 issues and 16 planes in ${\mathbb P}^3$ such that every airplane comprises precisely six issues and every element belongs to precisely six planes (this is named a '(16,6) configuration').A Kummer floor is uniquely made up our minds through its set of nodes. Gonzalez-Dorrego classifies (16,6) configurations and reviews their manifold symmetries and the underlying questions on finite subgroups of $PGL_4(k)$. She makes use of this knowledge to provide a whole class of Kummer surfaces with particular equations and particular descriptions in their singularities. furthermore, the gorgeous connections to the speculation of K3 surfaces and abelian types are studied.

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**Sample text**

9 V

Thus to classify the configurations of points and conies of type (B) as above, it is enough to solve the following problem. Take three distinct points P 1 4 , P2s and MARIA R. GONZALEZ-DORREGO 30 P36 lying on a line L. 1) Pij = LinLj for {ij) = (14), (25) and (36). 1). Consider the sextuple of points P14, P25, P13, Pi3? P43, P26, Ps6- We want to classify all the situations where this sextuple and all of its six images under the stabilizer in S$ of the unordered triple (14)(25)(36) of unordered pairs, lie on a conic.

Zf,XiYi,XiZi,YiZi), 1 < i < 6, is singular. Thus we have to calculate the determinants of six 6 x 6 matrices. 2) gives three zeroes in the matrix of quadratic forms), this calculation can easily be done by hand. 3) a + b-ac-2ab. Thus we get the desired configuration of points and conies if and only if a + b — ac — 2ab = 0. 3. 43. I V . A n o n - d e g e n e r a t e (16,6) configuration in P 3 is of t h e form (a, b, c, d) of ( 1 . 4 . 1 ) . 44. We say that a set of planes is in general linear p o s i t i on if no four of them pass through the same point.