Related papers: Pairwise incident planes and hyperkaehler four-fol…
We show that the maximum cardinality of an anti-chain composed of intersections of a given set of n points in the plane with half-planes is close to quadratic in n. We approach this problem by establishing the equivalence with the problem…
We study the geometry of the moduli space of planes in a general cubic 5-fold and its deformation. We show that this moduli space is a smooth projective surface whose canonical bundle is ample. We also show that the variation of degree 1…
We show that the maximal number of planes in a complex smooth cubic fourfold in ${\mathbb P}^5$ is $405$, realized by the Fermat cubic only; the maximal number of real planes in a real smooth cubic fourfold is $357$, realized by the…
Existing results of Fu show that, if two finite sets of roots of unity are projectively equivalent by a projective automorphism that does not act bijectively on the set of all roots of unity, then these sets consist of at most 14 points.…
It is shown that the maximum size of a binary subspace code of packet length $v=6$, minimum subspace distance $d=4$, and constant dimension $k=3$ is $M=77$; in Finite Geometry terms, the maximum number of planes in $\operatorname{PG}(5,2)$…
A classical result asserts that the complex projective plane modulo complex conjugation is the 4-dimensional sphere. We generalize this result in two directions by considering the projective planes over the normed real division algebras and…
The maximum cut problem for a quintic del Pezzo surface ${\rm Bl}_{4}(\mathbb{P}^2)$ asks: Among all partitions of the 10 exceptional curves into two disjoint sets, what is the largest possible number of pairwise intersections? In this…
We show that the maximal $1$-system of loops in a punctured projective plane is unique up to the mapping class group action if and only if the number of punctures is at most five.
We give some new explicit examples of putatively optimal projective spherical designs. i.e., ones for which there is numerical evidence that they are of minimal size. These form continuous families, and so have little apparent symmetry in…
We consider an incidence problem in $\mathbb{R}^4$ which asks, for a set of $L$ lines and a set of $S$ planes in general position, what the maximum number of line-plane incidences is. A line-plane incidence is defined as a point where a…
We show that the order of the cardinality of maximal complete $1$-systems of loops on non-orientable surfaces is $\sim |\chi|^{2}$. In particular, we determine the exact cardinality of maximal complete $1$-systems of loops on punctured…
We present various new constructions and bounds for arcs in projective Hjelmslev planes over finite chain rings of nilpotency index 2. For the chain rings of cardinality at most 25 we give updated tables with the best known upper and lower…
The largest possible average diameter of a bounded cell of a simple hyperplane arrangement is conjectured to be not greater than the dimension. We prove that this conjecture holds in dimension 2, and is asymptotically tight in fixed…
We describe a maximal exceptional collection on the Cayley plane, the minimal homogeneous projective variety of $E_6$. This collection consists in a sequence of 27 irreducible homogeneous bundles.
A convex geometric hypergraph or cgh consists of a family of subsets of a strictly convex set of points in the plane. There are eight pairwise nonisomorphic cgh's consisting of two disjoint triples. These were studied at length by Bra{\ss}…
The purpose of this note is to study configurations of lines in projective planes over arbitrary fields having the maximal number of intersection points where three lines meet. We give precise conditions on ground fields F over which such…
Let p be a puncture of a punctured sphere, and let Q be the set of all other punctures. We prove that the maximal cardinality of a set of arcs pairwise intersecting at most once, which start at p and end in Q, is |X|(|X| + 1). We deduce…
In this paper we prove birational superrigidity of finite covers of degree $d$ of the $M$-dimensional projective space of index 1, where $d\geqslant 5$ and $M\geqslant 10$, with at most quadratic singularities of rank $\geqslant 7$,…
We present constructions and bounds for additive codes over a finite field in terms of their geometric counterpart, i.e., projective systems. It is known that the maximum number of $(h-1)$-spaces in PG$(2,q)$, such that no hyperplane…
We consider the problem of finding the maximum number $e_d(n)$ of pairs of touching circles in a packing of $n$ congruent circles of diameter $d$ in the hyperbolic plane of curvature $-1$. In the Euclidean plane, the maximum comes from a…