New PDF release: 16, 6 Configurations and Geometry of Kummer Surfaces in P3

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 merely 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 decided via its set of nodes. Gonzalez-Dorrego classifies (16,6) configurations and reports their manifold symmetries and the underlying questions on finite subgroups of $PGL_4(k)$. She makes use of this data to offer a whole type of Kummer surfaces with particular equations and specific descriptions in their singularities. additionally, the attractive connections to the idea of K3 surfaces and abelian kinds are studied.

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Extra resources for 16, 6 Configurations and Geometry of Kummer Surfaces in P3

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1), these four points will be coplanar if and only if (a2 — d2)(b2 — c 2 ) — 0. 1). 1) (l f ) abed / 0 , and 9 7 9 7 (2') all of the ± a 2 , ± 6 2 , ± c 2 , ±d2 |N are distinct. 1) have the rank p(S) of Neron-Severi group equal to 2 (for a generic Kummer surface, p(S) = 1) [7]. 1) corresponds to the locus of bielliptic curves in M2 [7]. 57. Consider four points {vjt}i

3) implies that Ta = aT for any a £ Fo. In other words, T belongs to the centralizer of F0 in PGL±(k). 50. Fo is its own centralizer in PGL^{k). Proof. Pick and fix once and for all an element i of k such that i2 + 1 = 0. We have the 4-1 covering 7T : SL4(k) - • PGL4(k), whose kernel is { 1 , - 1 , i , — i } (here we identify constant multiples of the identity matrix with elements of fc). 1) e3 0 0 1 0 0 -1 0 0 <0 1 0 0 0 ( 0 0 0 \^0 0 1 0 0 -1 0 0 0 0 0 0 e2 -1 0 f°0 00 01 and e4 = 0 1 0 \1 0 0 1—1 ei = 1 0 0 0 0 o -1/ 0 j 1 0/ 0\ 0 0 \) 1 0 0 0 (16,6) CONFIGURATIONS AND GEOMETRY OF KUMMER SURFACES IN P 3 .

3). 3), respectively:

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


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