48 research outputs found
Symbolic powers of planar point configurations II
We study initial sequences of various configurations of planar points. We
answer several questions asked in our previous paper (Symbolic powers of planar
point configurations), and we extend our considerations to the asymptotic
setting of Waldschmidt constants. We introduce the concept of Bezout
Decomposition which might be of independent interest.Comment: This article is a sequel to arXiv:1205.6002. 15 page
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
Nagata type statements
Nagata solved Hilbert's 14-th problem in 1958 in the negative. The solution
naturally lead him to a tantalizing conjecture that remains widely open after
more than half a century of intense efforts. Using Nagata's theorem as starting
point, and the conjecture, with its multiple variations, as motivation, we
explore the important questions of finite generation for invariant rings, for
support semigroups of multigraded algebras, and for Mori cones of divisors on
blown up surfaces, and the rationality of Waldschimdt constants. Finally we
suggest a connection between the Mori cone of the Zariski-Riemann space and the
continuity of the Waldschmidt constant as a function on the space of
valuations.Comment: 45 pages. These notes correspond to the course of the same title
given by the first author in the workshop "Asymptotic invariants attached to
linear series" held in the Pedagogical University of Cracow from May 16 to
20, 201
Mini-Workshop: Seshadri Constants
Seshadri constants were defined by Demailly around 30 years ago using the ampleness criterion of Seshadri. Demailly was interested in studying problems related to separation of jets of line bundles on projective varieties, specifically in the context of the well-known Fujita Conjecture. However, Seshadri constants turned out to be objects of fundamental importance in the study of positivity of linear series and many other areas. Consequently, in the past three decades, they have become a central object of study in numerous directions in algebraic geometry and commutative algebra. In this mini-workshop, we studied some of the most interesting current research problems concerning Seshadri constants. We expect that this exploration will help focus research on some of the most important questions in this area in the years to come
Mini-Workshop: Linear Series on Algebraic Varieties
Linear series have long played a central role in algebraic geometry. In recent years, starting with seminal papers by Demailly and Ein-Lazarsfeld, local properties of linear series â in particular local positivity, as
measured by Seshadri constants â have come into focus. Interestingly, in their multi-point version they are closely related to the famous Nagata conjecture on plane curves. While a number of important basic results are available by now, there are still a large number of open questions and even completely open lines of research
A Contribution to Metric Diophantine Approximation : the Lebesgue and Hausdorff Theories
This thesis is concerned with the theory of Diophantine approximation from the point of
view of measure theory. After the prolegomena which conclude with a number of conjectures set
to understand better the distribution of rational points on algebraic planar curves, Chapter 1
provides an extension of the celebrated Theorem of Duffin and Schaeffer. This enables one to
set a generalized version of the DuffinâSchaeffer conjecture. Chapter 2 deals with the topic of
simultaneous approximation on manifolds, more precisely on polynomial curves. The aim is
to develop a theory of approximation in the so far unstudied case when such curves are not
defined by integer polynomials. A new concept of soâcalled âliminf setsâ is then introduced in
Chapters 3 and 4 in the framework of simultaneous approximation of independent quantities.
In short, in this type of problem, one prescribes the set of integers which the denominators of
all the possible rational approximants of a given vector have to belong to. Finally, a reasonably
complete theory of the approximation of an irrational by rational fractions whose numerators
and denominators lie in prescribed arithmetic progressions is developed in chapter 5. This
provides the first example of a Khintchine type result in the context of soâcalled uniform
approximation