By Maria R. Gonzalez-Dorrego

ISBN-10: 0821825747

ISBN-13: 9780821825747

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 various from 2). This Kummer floor is a quartic floor with 16 nodes as its in simple terms singularities. those nodes supply upward push to a configuration of 16 issues and 16 planes in ${\mathbb P}^3$ such that every airplane includes precisely six issues and every element belongs to precisely six planes (this is named a '(16,6) configuration').A Kummer floor is uniquely made up our minds 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 knowledge to provide a whole type of Kummer surfaces with particular equations and particular descriptions in their singularities. additionally, the gorgeous connections to the speculation of K3 surfaces and abelian types are studied.

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Example text

Hence, if four of our planes passed through the same point in P 3 , this point would belong to the (16,6) configuration. 1), which proves that they are in general linear position. • N o t e o n proofs by i n s p e c t i o n of t h e diagram . 46 is an example of what we consider an acceptable proof by inspection of the diagram. There are two statements in the proof, each of which involves checking (4 ) = 15 cases, and each case takes about one second to check. Writing down a proof of these statements would be tedious and pointless.

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.

F> is a finite-to-one morphism between two irreducible quasi-projective varieties of the same dimension, so Im((j)) is Zariski-dense in V. (Below we shall prove that Im((f)) = V, so is, in fact, a 16-1 covering of V). Note that S is irreducible, since 0 is injective. 45 is the statement that TJ is surjective. We consider an arbitrary (16,6) configuration of type (*) (denote it by ft). 48. 1). Let us denote these 6 planes by {^;}i Wi € Fz . 46 the wi, 1 < i < 6, are in general linear position, so they determine a point in V.

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