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We present a direct and fairly simple proof of the following incidence bound: Let $P$ be a set of $m$ points and $L$ a set of $n$ lines in ${mathbb R}^d$, for $dge 3$, which lie in a common algebraic two-dimensional surface of degree $D$ that does not contain any 2-flat, so that no 2-flat contains more than $s le D$ lines of $L$. Then the number of incidences between $P$ and $L$ is $$ I(P,L)=Oleft(m^{1/2}n^{1/2}D^{1/2} + m^{2/3}min{n,D^{2}}^{1/3}s^{1/3} + m + nright). $$ When $d=3$, this improves the bound of Guth and Katz~cite{GK2} for this special case, when $D$ is not too large. A supplementary feature of this work is a review, with detailed proofs, of several basic (and folklore) properties of ruled surfaces in three dimensions.
We study incidences between points and algebraic curves in three dimensions, taken from a family $C$ of curves that have almost two degrees of freedom, meaning that every pair of curves intersect in $O(1)$ points, for any pair of points $p$, $q$, the
We give a fairly elementary and simple proof that shows that the number of incidences between $m$ points and $n$ lines in ${mathbb R}^3$, so that no plane contains more than $s$ lines, is $$ Oleft(m^{1/2}n^{3/4}+ m^{2/3}n^{1/3}s^{1/3} + m + nright) $
We introduce the notion of r-th Terracini locus of a variety and we compute it for at most three points on a Segre variety.
Let $L$ be a set of $n$ lines in $R^3$ that is contained, when represented as points in the four-dimensional Plucker space of lines in $R^3$, in an irreducible variety $T$ of constant degree which is emph{non-degenerate} with respect to $L$ (see belo
The aim of this paper is to make a connection between design theory and algebraic geometry/commutative algebra. In particular, given any Steiner System $S(t,n,v)$ we associate two ideals, in a suitable polynomial ring, defining a Steiner configuratio