Klein quadric

Abstract: The theory of slice regular functions of a quaternion variable is applied to the study of orthogonal complex structures on domains \Omega\ of R^4. When \Omega\ is a symmetric slice domain, the twistor transform of such a function is a holomorphic curve in the Klein quadric. The case in which \Omega\ is the complement of a parabola is studied in detail and described by a rational quartic surface in the twistor space CP^3.

Abstract: The problem is considered of constructing a maximal set of lines, with no three in a pencil, in the finite projective geometry PG(3, q) of three dimensions over GF(q) (A pencil is the set of q+1 lines in a plane and passing through a point) It is found that an orbit of lines of a Singer cycle of PG(3, q) gives a set of size q3 + q2 + q + 1 which is definitely maximal in the case of q odd A (q3 + q2 + q + 1)-cap contained in the hyperbolic (or ‘Klein’) quadric of PG(5, q) also comes from the construction (A k-cap is a set of k points with no three in a line) This is generalized to give direct constructions of caps in quadrics in PG(5, q) For q odd and greater than 3 these appear to be the largest caps known in PG(5, q) In particular it is shown how to construct directly a large cap contained in the Klein quadric, given an ovoid skew to an elliptic quadric of PG(3, q) Sometimes the cap is also contained in an elliptic quadric of PG(5, q) and this leads to a set of q3 + q2 + q + 1 lines of PG(3,q2) contained in the non-singular Hermitian surface such that no three lines pass through a point These constructions can often be applied to real and complex spaces

Abstract: We discuss a unified approach to a class of geometric combinatorics incidence problems in two dimensions, of the Erdos distance type. The goal is obtaining the second moment estimate. That is, given a finite point set $S$ in two dimensions, and a function $f$ on $S\times S$, find the upper bound for the number of solutions of the equation $f(p,p') = f(q,q') eq 0, (p,p',q,q')\in S\times S\times S\times S$; e.g., $f$ is the Euclidean distance in the plane, sphere, or a sheet of the two-sheeted hyperboloid. Our ultimate tool is the Guth--Katz incidence theorem for lines in $\mathbb{RP}^3$, but we focus on how the original problem in two dimensions gets reduced to its application. The corresponding procedure was initiated by Elekes and Sharir, based on symmetry considerations. The point we make here is that symmetry considerations, not necessarily straightforward and potentially requiring group representation machinery can be bypassed or made implicit. The classical Plucker--Klein formalism for line geometry...

Abstract: In this note we give a shortened proof of a theorem of Rudnev, which bounds the number of incidences between points and planes over an arbitrary field. Rudnev's proof uses a map that goes via the four-dimensional Klein quadric to a three-dimensional space, where it applies a bound of Guth and Katz on intersection points of lines. We describe a simple geometric map that directly sends point-plane incidences to line-line intersections in space, allowing us to reprove Rudnev's theorem with fewer technicalities.