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

By Maria R. Gonzalez-Dorrego

This monograph reviews 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 merely singularities. those nodes supply 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 made up our minds through 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 provide an entire type of Kummer surfaces with particular equations and particular descriptions in their singularities. furthermore, the gorgeous connections to the idea of K3 surfaces and abelian forms are studied.

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**Sample text**

Then VjA = Wfcj fl Wk2 H wjfe8, where 1 < ^1,^2,^3 < 6 and p £ {ki, &2, ^3}. Since the planes wj, 1 < j < 6 are in general position, VjA £ wp, as desired. Case 2. No three of the Vjt, 1 < / < 4, are contained in a plane wp with 1 < p < 6. Associated with each Vjt we have a three element set Mi = {ku, &2i, k$i} C { 1 , . . 2) p | MI = 0. 2), U J = 1 M j = { 1 , . . , 6 } . 4) #(M1nM3) = #(M2nM4) = l #(Mj n Mi) = # ( M 2 n M») = o. 4. (16,6) CONFIGURATIONS AND GEOMETRY OF KUMMER SURFACES IN P 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. 45. 6, up to an automorphism of P 3 . in P 3 is of the form de- Proof. 1). As usual, let us denote a plane in P 3 by a 4-tuple of elements of k. We may view this 4-tuple as a point in P 3 . Let V denote the open subset of (P 3 ) 6 consisting of all the ordered 6-tuples of planes in general linear position.

59. For a subset M of the entries of the 4 x 4 diagram, let the s p an of M be the set of all rows and all columns of the matrix which contain at least one element of M. It is easy to see from definitions that a four element subset M of {vj}i