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A few shades of interpolation
The topic of this snapshot is interpolation. In the ordinary sense, interpolation means to insert something of a different nature into something else. In mathematics, interpolation means constructing new data points from given data points. The new points usually lie in between the already-known points. The purpose of this snapshot is to introduce a particular type of interpolation, namely, polynomial interpolation. This will be explained starting from basic ideas that go back to the ancient Babylonians and Greeks, and will arrive at subjects of current research activity
On the containment problem
The purpose of this note is to provide an overview of the containment problem
for symbolic and ordinary powers of homogeneous ideals, related conjectures and
examples. We focus here on ideals with zero dimensional support. This is an
area of ongoing active research. We conclude the note with a list of potential
promising paths of further research.Comment: 13 pages, 1 figur
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
A few introductory remarks on line arrangements
Points and lines can be regarded as the simplest geometrical objects. Incidence relations between them have been studied since ancient times. Strangely enough our knowledge of this area of mathematics is still far from being complete. In fact a number of interesting and apparently difficult conjectures has been raised just recently. Additionally a number of interesting connections to other branches of mathematics have been established. This is an attempt to record some of these recent developments