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.

**Read Online or Download 16, 6 Configurations and Geometry of Kummer Surfaces in P3 PDF**

**Similar algebraic geometry books**

**The Geometry of Moduli Spaces of Sheaves**

Now again in print, this very hot publication has been up to date to mirror fresh advances within the idea of semistable coherent sheaves and their moduli areas, which come with moduli areas in optimistic attribute, moduli areas of critical bundles and of complexes, Hilbert schemes of issues on surfaces, derived different types of coherent sheaves, and moduli areas of sheaves on Calabi-Yau threefolds.

**Spaces of Homotopy Self-Equivalences: A Survey**

This survey covers teams of homotopy self-equivalence sessions of topological areas, and the homotopy kind of areas of homotopy self-equivalences. For manifolds, the entire workforce of equivalences and the mapping type team are in comparison, as are the corresponding areas. integrated are tools of calculation, quite a few calculations, finite new release effects, Whitehead torsion and different components.

**Galois Theory of Difference Equations**

This ebook lays the algebraic foundations of a Galois conception of linear distinction equations and exhibits its courting to the analytic challenge of discovering meromorphic services asymptotic to formal strategies of distinction equations. Classically, this latter query used to be attacked by way of Birkhoff and Tritzinsky and the current paintings corrects and tremendously generalizes their contributions.

- Geometric invariant theory and decorated principal bundles
- Homotopy Type and Homology
- Curved Spaces: From Classical Geometries to Elementary Differential Geometry
- Algebraic geometry 1: Schemes
- Algebraic Groups
- Geometric Aspects of Dwork Theory

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

**Example text**

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

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).

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

by Edward

4.1