Kummer surfaces and quadratic line complexes in characteristic two

We construct the coarse moduli space $${\mathcal{M}}_{qc}(\sigma)$$ of quadratic line complexes with a fixed Segre symbol σ as well as the moduli space $${\mathcal{M}}_{ss}(\sigma)$$ of the corresponding singular surfaces. We show that the map associating to a quadratic line complex its singular surface induces a morphism $$\pi: {\mathcal{M}}_{qc}(\sigma) \rightarrow {\mathcal{M}}_{ss}(\sigma)$$ . Finally we deduce that the varieties of cosingular quadratic line complexes are almost always curves.

We study the relation between Cremona transformations in space and quadratic line complexes. We show that it is possible to associate a space Cremona transformation to each smooth quadratic line complex once we choose two distinct lines contained in the complex. Such Cremona transformations are cubo-cubic and we classify them in terms of the relative position of the lines chosen. It turns out that the base locus of such a transformation contains a smooth genus two quintic curve. Conversely, we show that given a smooth quintic curve C of genus 2 in ℙ3 every Cremona transformation containing C in its base locus factorizes through a smooth quadratic line complex as before. We consider also some cases where the curve C is singular, and we give examples both when the quadratic line complex is smooth and singular.

Projective-geometric aspects of homogeneous third-order Hamiltonian operators

Linearly degenerate partial differential equations and quadratic line complexes

Equal Sums of Sixth Powers and Quadratic Line Complexes

Two of C. Segre’s earliest papers, (Segre 1883a) and (Segre 1884), dealt with the classification of quadratic line complexes, a central topic in line geometry. These papers, the first written together with Gino Loria, were submitted to Felix Klein in 1883 for publication in Mathematische Annalen. Together with the two lengthier works that comprise Segre’s dissertation, (Segre 1883b) and (Segre 1883c), they took up and completed a topic that Klein had worked on a decade earlier (when he was known primarily as an expert on line geometry). Using similar ideas, but a new and freer approach to higher-dimensional geometry, Segre not only refined and widened this earlier work but also gave it a new direction. Line geometry, as well described by Alessandro Terracini in his obituary for his mentor, proved to be an excellent starting point for both Segre and Italian algebraic geometry. The present account begins by looking back at the early work of Klein and Adolf Weiler on quadratic complexes in order to show how Segre’s two papers for Klein’s journal represented a new start that reawakened interest in a topic that had been dormant for nearly a decade.

The intersection of two quadrics is called a biquadric. If we mark a non-singular quadric in the pencil of quadrics through a given biquadric, then the given biquadric is called a marked biquadric. In the classical papers of Plücker and Klein, a Kummer surface was canonically associated with every three-dimensional marked biquadric (that is, with a quadratic line complex provided that the Plücker–Klein quadric is marked). In Reid’s thesis, this correspondence was generalized to odd-dimensional marked biquadrics of arbitrary dimension . In this case, a Kummer variety of dimension corresponds to every biquadric of dimension . Reid only constructed the generalized Plücker–Klein correspondence. This map was not studied later. The present paper is devoted to a partial solution of the problem of creating the corresponding theory.

Introduction. If a set of four directions in a Riemannian four-dimensional space, V4, is orthogonal, then the ds2 can be expressed in terms of their sixteen parameters, hka(XO, X1l x2, X3), as in Einstein's recent papers. The first purpose of this paper is to set up sixteen invariant linear firstorder partial differential equations in these parameters (?2). The solutions of these equations include all solutions for empty space of the Einstein field equations of 1917. There is a restriction which excludes some special cases. In addition to the ka, these sixteen equations contain four linear combinations of the components of the curvature tensor. These four combinations are to be taken as independent variables, Xk. Since only alternating tensors appear it is convenient to use Cartan's notationt for symbolic differential forms and for their derivatives and products. Covariant differentiation in the sense of the absolute differential calculus is not used, except in ?4. The components of the curvature tensor may be taken as coefficients in the equation of a quadratic line complex in a three-dimensional projective space,P34 ?. The directions hka correspond to the vertices of a tetrahedron in P3. Bivector and simple bivector correspond to linear complex and special complex. Where there is no danger of misunderstanding, the language of V4 will be used interchangeably with that of P3. The second purpose of this paper is the application of some of the theory of quadratic complexes to the study of the curvatures in V4.? The lines which lie in a plane in P3 and which belong to the quadratic complex are tangent to a conic. The envelope of planes for which this conic

To a general quadratic line complex and any pair of different lines in it one can associate a cubo-cubic Cremona transformation of projective 3-space in a natural way. This gives a map $$\gamma $$ from the space of pairs of lines in general quadratic complexes to the space of equivalence classes of these cubo-cubic transformations. We compute the dimension of a general fibre of $$\gamma $$ as well as the codimension of subspace of the transformations arizing in this way in the corresponding space of Cremona transformations.

In this work, several classical ideas concerning the geometry of the inertia of a rigid body are revisited. This is done using a modern approach to screw theory. A screw, or more precisely a twist, is viewed as an element of the Lie algebra to the group of proper rigid-body displacements. Various moments of inertia, about lines, planes and points are considered as geometrical objects resulting from least-squares problems. This allows relations between the various inertias to be found quite simply. A brief review of classical line geometry is given; this includes an outline of the theory of the linear line complex and a brief introduction to quadratic line complexes. These are related to the geometry of the inertia of an arbitrary rigid body. Several classical problems concerning the mechanics of rigid bodies subject to impulsive wrenches are reviewed. We are able to correct a small error in Ball’s seminal treatise. The notion of spatial percussion axes is introduced, and these are used to solve a problem concerning the diagonalisation of the mass matrix of a two-joint robot.

We discuss the geometry of transverse linear sections of the spinor tenfold , the connected component of the orthogonal Grassmannian of 5-dimensional isotropic subspaces in a 10-dimensional vector space endowed with a non-degenerate quadratic form. In particular, we show that if the dimension of a linear section of is at least 5, then its integral Chow motive is of Lefschetz type. We discuss the classification of smooth linear sections of of small codimension. In particular, we check that there is a unique isomorphism class of smooth hyperplane sections and exactly two isomorphism classes of smooth sections of codimension 2. Using this, we define a natural quadratic line complex associated with a linear section of . We also discuss the Hilbert schemes of linear spaces and quadrics on and its linear sections.

Groups acting properly and discontinuously on the Cartesian product $$\mathbb {H}^{2}\times \mathbb {H}^{2}$$ of two hyperbolic planes are termed hyperabelian by Picard. The automorphism group $$\mathrm {Aut}f$$ of a quaternary integral quadratic form f of index 2 is an example of a hyperabelian group. Hence the quotient orbifold $$Q_{f}$$ of the action of $$\mathrm {Aut}f$$ on $$\mathbb {H}^{2}\times \mathbb {H}^{2}$$ is a 4-dimensional arithmetic orbifold, endowed with a natural $$\mathbb {H}^{2}\times \mathbb {H}^{2}$$ -geometry. Plucker coordinates are used to understand $$Q_{f}$$ . A real automorphism U of $$\mathbb {R}^{4}$$ induces a real automorphism $$\mathbf {K(}U)$$ of $$(\mathbb {R}^{6},k)$$ in such a way that if $$U\in SL(4,\mathbb {Z})$$ then $$\mathbf {K(}U)\in SL(6,\mathbb {Z})$$ is an automorph of the Klein quadratic form k. It is proved that the converse is true. That is, given an automorph $$M\in SL(6,\mathbb {Z})$$ of k there is $$U\in SL(4,\mathbb {Z})$$ such that $$\mathbf {K(}U)=\pm M$$ , so that the proper automorphism group of the Klein quadric is isomorphic to $$SL(4,\mathbb {Z})$$ via $$\mathbf {K}$$ . This is used to obtain the automorphism group of the quadratic line complex of line tangents to a quadric in projective space $$P^{3}$$ . With this, a description is given of the automorphism group of a quaternary integral quadratic form of index 2.

On the Largest Caps Contained in the Klein Quadric of PG(5, q), q Odd