50 research outputs found
Rank Function Equations and their solution sets
We examine so-called rank function equations and their solutions consisting
of non-nilpotent matrices. Secondly, we present some geometrical properties of
the set of solutions to certain rank function equations in the nilpotent case.Comment: 8 pages, all comments welcome
Seshadri constants and special configurations of points in the projective plane
In the present note, we focus on certain properties of special curves that
might be used in the theory of multi-point Seshadri constants for ample line
bundles on the complex projective plane. In particular, we provide three
Ein-Lazarsfeld-Xu-type lemmas for plane curves and a lower bound on the
multi-point Seshadri constant of under the
assumption that the chosen points are not very general. In the second part, we
focus on certain arrangements of points in the plane which are given by line
arrangements. We show that in some cases the multi-point Seshadri constants of
centered at singular loci of line
arrangements are computed by lines from the arrangement having some extremal
properties.Comment: 13 pages, 1 figure. This is the final version which incorporates the
referee remarks. To appear in Rocky Mountain Journal of Mathematic
On line and pseudoline configurations and ball-quotients
In this note we show that there are no real configurations of lines
in the projective plane such that the associated Kummer covers of order
are ball-quotients and there are no configurations of lines
such that the Kummer covers of order are ball-quotients. Moreover, we
show that there exists only one configuration of real lines such that the
associated Kummer cover of order is a ball-quotient. In the second
part we consider the so-called topological -configurations and we
show, using Shnurnikov's inequality, that for there do not exist
-configurations and and for there do not exist
-configurations.Comment: 7 pages, one figure. This is the final version, incorporating the
suggestions of the referee, to appear in ARS Mathematica Contemporane
Curve configurations in the projective plane and their characteristic numbers
In this paper we study the concept of characteristic numbers and Chern slopes
in the context of curve configurations in the real and complex projective
plane. We show that some extremal line configurations inherit the same
asymptotic invariants, namely asymptotic Chern slopes and asymptotic Harbourne
constants which sheds some light on relations between the bounded negativity
conjecture and the geography problem for surfaces of general type. We discuss
some properties of Kummer extensions, especially in the context of
ball-quotients. Moreover, we prove that for a certain class of smooth curve
configurations in the projective plane their characterstic numbers are bounded
by .Comment: 12 page
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
Bounded negativity, Harbourne constants and transversal arrangements of curves
The Bounded Negativity Conjecture predicts that for every complex projective
surface there exists a number such that holds for
all reduced curves . For birational surfaces there have
been introduced certain invariants (Harbourne constants) relating to the effect
the numbers , and the complexity of the map . These invariants
have been studied previously when is the blowup of all singular points of
an arrangement of lines in , of conics and of cubics. In the
present note we extend these considerations to blowups of at
singular points of arrangements of curves of arbitrary degree . We also
considerably generalize and modify the approach witnessed so far and study
transversal arrangements of sufficiently positive curves on arbitrary surfaces
with the non-negative Kodaira dimension.Comment: This is the final version, incorporating the suggestions of the
referee, to appear in Annales de l'Institut Fourier Grenobl