69 research outputs found
On sets defining few ordinary lines
Let P be a set of n points in the plane, not all on a line. We show that if n
is large then there are at least n/2 ordinary lines, that is to say lines
passing through exactly two points of P. This confirms, for large n, a
conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for
this problem, as well as all sets having fewer than n - C ordinary lines for
some absolute constant C. We also solve, for large n, the "orchard-planting
problem", which asks for the maximum number of lines through exactly 3 points
of P. Underlying these results is a structure theorem which states that if P
has at most Kn ordinary lines then all but O(K) points of P lie on a cubic
curve, if n is sufficiently large depending on K.Comment: 72 pages, 16 figures. Third version prepared to take account of
suggestions made in a detailed referee repor
A finite version of the Kakeya problem
Let be a set of lines of an affine space over a field and let be a
set of points with the property that every line of is incident with at
least points of . Let be the set of directions of the lines of
considered as points of the projective space at infinity. We give a geometric
construction of a set of lines , where contains an grid and
where has size , given a starting configuration in the plane.
We provide examples of such starting configurations for the reals and for
finite fields. Following Dvir's proof of the finite field Kakeya conjecture and
the idea of using multiplicities of Dvir, Kopparty, Saraf and Sudan, we prove a
lower bound on the size of dependent on the ideal generated by the
homogeneous polynomials vanishing on . This bound is maximised as
plus smaller order terms, for , when contains
the points of a grid.Comment: A few minor changes to previous versio
From Pappus Theorem to parameter spaces of some extremal line point configurations and applications
In the present work we study parameter spaces of two line point
configurations introduced by B\"or\"oczky. These configurations are extremal
from the point of view of Dirac-Motzkin Conjecture settled recently by Green
and Tao. They have appeared also recently in commutative algebra in connection
with the containment problem for symbolic and ordinary powers of homogeneous
ideals and in algebraic geometry in considerations revolving around the Bounded
Negativity Conjecture. Our main results are Theorem A and Theorem B. We show
that the parameter space of what we call configurations is a three
dimensional rational variety. As a consequence we derive the existence of a
three dimensional family of rational configurations. On the other hand
the moduli space of configurations is shown to be an elliptic curve with
only finitely many rational points, all corresponding to degenerate
configurations. Thus, somewhat surprisingly, we conclude that there are no
rational configurations.Comment: 17 pages, v.2. title modified, material reorganized, introduction new
rewritten, discussion more streamline
On the Sylvester-Gallai and the orchard problem for pseudoline arrangements
We study a non-trivial extreme case of the orchard problem for
pseudolines and we provide a complete classification of pseudoline arrangements
having triple points and double points. We have also classified those
that can be realized with straight lines. They include new examples different
from the known example of B\"or\"oczky. Since Melchior's inequality also holds
for arrangements of pseudolines, we are able to deduce that some combinatorial
point-line configurations cannot be realized using pseudolines. In particular,
this gives a negative answer to one of Gr\"unbaum's problems. We formulate some
open problems which involve our new examples of line arrangements.Comment: 5 figures, 11 pages, to appear in Periodica Mathematica Hungaric
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