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

By Maria R. Gonzalez-Dorrego

ISBN-10: 0821825747

ISBN-13: 9780821825747

This monograph experiences the geometry of a Kummer floor in ${\mathbb P}^3_k$ and of its minimum desingularization, that's 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 supply upward thrust to a configuration of 16 issues and 16 planes in ${\mathbb P}^3$ such that every aircraft comprises precisely six issues and every aspect belongs to precisely six planes (this is termed a '(16,6) configuration').A Kummer floor is uniquely decided through its set of nodes. Gonzalez-Dorrego classifies (16,6) configurations and stories their manifold symmetries and the underlying questions on finite subgroups of $PGL_4(k)$. She makes use of this knowledge to provide an entire type of Kummer surfaces with particular equations and specific descriptions in their singularities. moreover, the attractive connections to the idea of K3 surfaces and abelian kinds are studied.

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Once L and P14, P25 and P36 are fixed, we are still free to choose I q , L2 (which amounts to choosing P12. After these are fixed, we still have a 1parameter group of linear automorphisms of the plane acting on the situation, so we are free to put P23 wherever we like in L 2 \ {Pi2,P2s}- After that there are no more choices and the three remaining lines give the three parameters on which the situation depends. One might think that the six conies described above give six conditions, but it turns out that all the six conditons are equivalent, so the desired configuration of fifteen points and six conies exists and has 2-dimensional moduli.

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. Consider the map :Ux PGL4(k) - • V, defined as follows (here PGL4(k) is the group of non-singular 4 x 4 matrices with entries in k, modulo multiplication by non-zero elements of k). Let (a, 6, c, d) G U and M G PGL±(k). Then M acts on planes in P 3 as a projective linear transformation.

Hence these three points are collinear, which is absurd. This completes the proof for (A). For (B), we consider the dual diagram to (B) of six lines and fifteen points in a plane and apply Pascal's theorem. Now, each of the four hexagons in (B) gives us the same collinearity condition. Namely, let us denote by Pjj , 1 < j < j < 6 the point dual to the line P{Pj in (B) and by 1^, 1 < i < 6, the line dual to P{. Then if we assume that each of the four sextuples of points corresponding to the four hexagons in (B) lie on a conic, we get from Pascal's theorem that the poinnts Pi4, P25 and P36 are collinear (of course, the line L which contains them must be distinct from all of the L^ 1 < i < 6).

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

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