Related papers: Geproci sets on skew lines in $\mathbb P^3$ with t…
In this short note we develop new methods toward the ultimate goal of classifying geproci sets in $\mathbb P^3$. We apply these methods to show that among sets of $16$ points distributed evenly on $4$ skew lines, up to projective…
We call a set of points $Z\subset{\mathbb P}^{3}_{\mathbb C}$ an $(a,b)$-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point $P$ to a plane is a complete intersection of curves of degrees…
This article introduces a previously unrecognized combinatorial structure underlying configurations of skew lines in $\mathbb{P}^3$, and reveals its deep and surprising connection to the algebro-geometric concept of geproci sets. Given any…
In the present note we give an example of a finite set of points in $\mathbb{P}^{3}$ which has the so-called geproci property, but it is neither a grid nor a half-grid. This answers a question on the existence of such sets rised by Pokora,…
The geproci property is a recent development in the world of geometry. We call a set of points $Z\subseteq\mathbb{P}_k^3$ an $(a,b)$-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point $P$…
In this note we introduce the notion of $(b,d)$-geprofi sets and study their basic properties. These are sets of $bd$ points in $\mathbb{P}^4$ whose projection from a general point to a hyperplane is a full intersection, i.e., the…
The motivating problem addressed by this paper is to describe those non-degenerate sets of points $Z$ in $\mathbb P^3$ whose general projection to a general plane is a complete intersection of curves in that plane. One large class of such…
We investigate the enumerative geometry of point configurations in projective space. We define "projective configuration counts": these enumerate configurations of points in projective space such that certain specified subsets are in fixed…
Given a smooth complex variety $X$, an algebraically skew embedding of $X$ is an embedding of $X$ into a complex projective space $\mathbb{P}^N$ such that for any two points $x,y\in X$, their embedded tangent spaces in $\mathbb{P}^N$ do not…
Let $S\subset \mathbb{R}^d$ $(d\geq 2)$. A set $S$ is said to be $m$-point convex, if for every $m$ distinct points in $S$, at least one of the line-segments determined by them lies in $S$. We also say that $S$ has property $P_m$. Let…
All sets of lines providing a partition of the set of internal points to a conic C in PG(2,q), q odd, are determined. There exist only three such linesets up to projectivities, namely the set of all nontangent lines to C through an external…
The skewer of a pair of skew lines in space is their common perpendicular. To configuration theorems of plane projective geometry involving points and lines (such as Pappus or Desargues) there correspond configuration theorems in space:…
We show that $m$ points and $n$ two-dimensional algebraic surfaces in $\mathbb{R}^4$ can have at most $O(m^{\frac{k}{2k-1}}n^{\frac{2k-2}{2k-1}}+m+n)$ incidences, provided that the algebraic surfaces behave like pseudoflats with $k$ degrees…
Kelly's theorem states that a set of $n$ points affinely spanning $\mathbb{C}^3$ must determine at least one ordinary complex line (a line passing through exactly two of the points). Our main theorem shows that such sets determine at least…
In this paper, we study maximal sets of skew lines on Hermitian surfaces. We give a new algorithm to compute these sets and give some computational results for Hermitian surfaces of degrees 3,4, and 5. In more generality, this algorithm…
We introduce a notion of genus range as a set of values of genera over all surfaces into which a graph is embedded cellularly, and we study the genus ranges of a special family of four-regular graphs with rigid vertices that has been used…
In this paper, we study the restrictions on the number $m$ of conic-line curves in special pencils. The most general result we obtain is the relation between upper bounds on $m$ and the number $p$ of concurrent lines in these pencils. We…
We completely describe the structure of the connected components of transversals to a collection of n line segments in R^3. We show that n>2 arbitrary line segments in R^3 admit 0, 1, ..., n or infinitely many line transversals. In the…
We show that plane bipolar posets (i.e., plane bipolar orientations with no transitive edge) and transversal structures can be set in correspondence to certain (weighted) models of quadrant walks, via suitable specializations of a bijection…
A dessin is a 2-cell embedding of a connected $2$-coloured bipartite graph into an orientable closed surface. A dessin is regular if its group of orientation- and colour-preserving automorphisms acts regularly on the edges. In this paper we…