Related papers: On the containment problem
We say that an ideal I is homogeneous, if its restriction to any I-positive subset is isomorphic to I. The paper investigates basic properties of this notion -- we give examples of homogeneous ideals and present some applications to…
This paper provides insights into the role of symmetry in studying polynomial functions vanishing to high order on an algebraic variety. The varieties we study are singular loci of hyperplane arrangements in projective space, with emphasis…
Some basic features of confinement are reviewed, in particular the symmetry patterns of the dual dynamics. Open problems and possible directions of progress are discussed.
In this survey article we give a brief history of symbolic powers and its connection with the interesting problem of set-theoretic complete intersection. We also state a few problems and conjectures. Recently, in connection to symbolic…
Recent work of Ein-Lazarsfeld-Smith and Hochster-Huneke raised the problem of which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci-Harbourne developed methods to address this problem, which involve…
The purpose of this note is twofold. We present first a vanishing theorem for families of linear series with base ideal being a fat points ideal. We apply then this result in order to give a partial proof of a conjecture raised by Bocci,…
We survey classical and recent results on symbolic powers of ideals. We focus on properties and problems of symbolic powers over regular rings, on the comparison of symbolic and regular powers, and on the combinatorics of the symbolic…
In this short note, we deal with a problem posted by Huneke on a numerical invariant attached to symbolic powers of homogenous ideals.
We investigate the higher divisorial ideal $D(I):= Ann(Ext^g_R(R/I,R))$ associated to an ideal I of grade g. Our main focus is the containment problem $D(I) \subseteq \overline{I}$. We show that this inclusion holds for broad classes of…
In this paper, we investigate containment statements between symbolic and ordinary powers and bounds on the Waldschmidt constant of defining ideals of points in projective spaces. We establish the stable Harbourne conjecture for the…
We establish a criterion for the (failure of) the containment $I^{(m)}\subset I^r$ for 3-generated ideals $I$ defining reduced sets of points in $\mathbb{P}^2$. Our criterion arises from studying the minimal free resolutions of the powers…
The study deals with a minimal energy problem in the presence of an external field over noncompact classes of vector measures of infinite dimension in a locally compact space. The components are positive measures (charges) satisfying…
The idea of monotonicity (or positive-definiteness in the linear case) is shown to be the central theme of the solution theories associated with problems of mathematical physics. A "grand unified" setting is surveyed covering a…
The main theme of the article is the study of discrete systems of material points subjected to constraints not only of a geometric type (holonomic constraints) but also of a kinematic type (nonholonomic constraints). The setting up of the…
In this note, we find a monomization of a certain power ideal associated to a directed graph. This power ideal has been studied in several settings. The combinatorial method described here extends earlier work of other, and will work on…
This article investigates under which conditions the symbolic powers of the extension of an ideal is the same as the extension of the symbolic powers. Our result generalizes the known scenarios. As an application, we prove formulas for the…
We introduce and study rational symbolic powers of ideals in Noetherian rings. We give membership criteria for rational symbolic powers and discuss settings where they agree with integer symbolic powers. We investigate the binomial…
The purpose of this note is to give counterexamples to the containment $I^{(3)}\subset I^2$ over the real numbers.
In the present note we study certain arrangements of codimension $2$ flats in projective spaces, we call them "Fermat arrangements". We describe algebraic properties of their defining ideals. In particular, we show that they provide…
In this article the cosmological constant problems, as well as the astronomical evidence for a cosmologically significant homogeneous exotic energy density with negative pressure (quintessence), are reviewed for a broad audience of…