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

By Maria R. Gonzalez-Dorrego

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 varied from 2). This Kummer floor is a quartic floor with 16 nodes as its simply singularities. those nodes supply upward push to a configuration of 16 issues and 16 planes in ${\mathbb P}^3$ such that every aircraft includes precisely six issues and every element belongs to precisely six planes (this is termed 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 on finite subgroups of $PGL_4(k)$. She makes use of this data to offer a whole class of Kummer surfaces with specific equations and specific descriptions in their singularities. additionally, the gorgeous connections to the idea of K3 surfaces and abelian forms are studied.

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58. All the planes Wj, 1 < j < 16 are distinct. We first prove the Corollary assuming the Lemma. Proof. Suppose that for some j , / , 1 < j < I < 16, we have Wj = w\. Let P,P' denote the entries in the 4 x 4 diagram (*) corresponding to Wj, wi respectively. We have # { ^ | 1 < p < 10, v p G Inc(P)} > 3 and similarly for P'. On the other hand, #{p \ 1 < p < 10, vp G Inc(P) D Inc(P')} < 2. Hence, one of the two planes (say, Wj) must contain a point vp, 1 < p < 10, such that vp £ Inc(P). Take p i , p2 and p 3 , 1 < pi,P2,P3 < 10, such that vPt G Inc(P) for 1 < t < 3.

This projective space has 15 planes and 35 lines. Each line contains exactly 3 points. To give a non-degenerate symplectic form on F | is equivalent to fixing an isomorphism Fj. —> (Pjp )* such that the image of each point in Pp is a plane which contains this point. This is also equivalent to specifying all the isotropic planes in ¥% or, equivalently, their projectifications in P | , which we call isotropic lines. To identify the set of 15 pairs (a, 6), a, b G K, with the 15 points of the projective space P | equipped with a symplectic structure in the above sense, we need to specify the sets of isotropic and non-isotropic lines in JPjp .

64. (1) N is the subgroup of PGL^k) ' 1 0 0 ,0 0 0 1 0 0 i 0 0 0^ 0 0 generated by the matrices (16,6) CONFIGURATIONS AND GEOMETRY OF KUMMER SURFACES IN P 3 . 47 (2) #N = 2 8 • 3 2 • 5 = 11520. N is a non-split extension of S6 by F | . 1) 1 -» F0 -> N -* 5 6 - • 1, In particular, N is not a semi-direct product of F4, with ^6. 1) is given by identifying Se = 5^4(2), acting on the 4-dimensional vector space F 4 . 65. It is well-known that Sp4(F2) = S6 [4, p. [4], lines 5, 22]. To visualize this identification, consider the set K = { 1 , 2 , 3 , 4 , 5 , 6 } of 6 elements.

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