Download e-book for iPad: 16, 6 Configurations and Geometry of Kummer Surfaces in P3 by Maria R. Gonzalez-Dorrego

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 is a K3 floor (here $k$ is an algebraically closed box of attribute assorted from 2). This Kummer floor is a quartic floor with 16 nodes as its simply singularities. those nodes supply 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 decided by way of its set of nodes. Gonzalez-Dorrego classifies (16,6) configurations and experiences their manifold symmetries and the underlying questions on finite subgroups of $PGL_4(k)$. She makes use of this data to provide an entire category of Kummer surfaces with particular equations and specific descriptions in their singularities. additionally, the attractive connections to the idea of K3 surfaces and abelian types are studied.

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

Sample text

Then, since 3, 4 and P are not collinear by (c), they span 4', so 4' = P , a contradiction. We have now constructed: V

Similarly, V fl 2 fl 2' = 0. 40. Hence we may assume that V fl 11' = V fl 22' = 0. 35(b)) to find a line in P 3 which meets all the lines of (a) but not V. We take the line 12. The reasoning in case (6) is analogous. The line 3 fl 3 ' meets all the four lines in (a) but not V. • Take a line L satisfying (a)-(c). Let P := L n 3 D 3 ' , R := L fl 4 H 4'. 41. Let us consider a standard (8,4) configuration Let L be a line satisfying (a)-(c). Let P, R be as above. 2). 1). a unique 4', P, R diagram Proof.

Let L be the isotropic line corresponding to (ei,e2), as above, and / the stabilizer of L. By (3) we now know that y- = S4 x F 2 [4, p. [4], line 44]. 2) Fo S4 x F 2 (16,6) CONFIGURATIONS AND GEOMETRY OF KUMMER SURFACES IN P 3 . 2) does not split. 3) 1 -» Fo -> W - ^ 5 3 x F 2 -> 1. 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.

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

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Categories: Algebraic Geometry