Related papers: On the containment problem for fat points
Given an ideal $I$ in a Noetherian ring, one can ask the containment question: for which $m$ and $r$ is the symbolic power $I^{(m)}$ contained in the ordinary power $I^r$? C. Bocci and B. Harbourne study the containment question in a…
In this note we show that Harbourne's conjecture is true for symbolic powers of ideals of points, we check that the stable version of this conjecture is valid for ideals of very general points (resp. generic points) in $\mathbb P_{\mathbb…
In this note we address the relation between symbolic and ordinary powers of the ideal of a reduced set or points in projective space: the so-called containment problem. In particular, we obtain sharp lower bounds on the Waldschmidt…
In this note we consider two configurations of twelve lines with nineteen triple points (i.e., points where three lines meet). Both of them have the same combinatorial features. In both configurations nine of twelve lines have five triple…
Guided by evidence coming from a few key examples and attempting to unify previous work of Chudnovsky, Esnault-Viehweg, Eisenbud-Mazur, Ein-Lazarsfeld-Smith, Hochster-Huneke and Bocci-Harbourne, Harbourne and Huneke recently formulated a…
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…
By defining a fat point subscheme of $P^2$ to be a 0-dimensional subscheme defined by a sheaf of integrally closed ideals one extends the notion of fat point subschemes to allow infinitely near points. With this notion of fat points, this…
B. Harbourne and C. Huneke conjectured that for any ideal $I$ of fat points in $P^N$ its $r$-th symbolic power $I^{(r)}$ should be contained in $M^{(N-1)r}I^r$, where $M$ denotes the homogeneous maximal ideal in the ring of coordinates of…
Motivated by the work of Chudnovsky and the Eisenbud-Mazur Conjecture on evolutions, Harbourne and Huneke give a series of conjectures that relate symbolic and regular powers of ideals of fat points in $\mathbb P^n$. The conjectures involve…
We investigate the minimal graded free resolutions of ideals of at most n+1 fat points in general position in P^n. Our main theorem is that these ideals are componentwise linear. This result yields a number of corollaries, including the…
The purpose of this note is to show a new series of examples of homogeneous ideals $I$ in ${\mathbb K}[x,y,z,w]$ for which the containment $I^{(3)}\subset I^2$ fails. These ideals are supported on certain arrangements of lines in ${\mathbb…
In this note we consider two simplicial arrangements of lines and ideals $I$ of intersection points of these lines. There are $127$ intersection points in both cases and the numbers $t_i$ of points lying on exactly $i$ configuration lines…
We develop tools to study the problem of containment of symbolic powers $I^{(m)}$ in powers $I^r$ for a homogeneous ideal $I$ in a polynomial ring $k[{\bf P}^N]$ in $N+1$ variables over an algebraically closed field $k$. We obtain results…
The main result provides an algorithm for determining the minimal free resolution of ideals of fat point subschemes of ${\bf P}^2$ involving up to 8 general points with arbitrary multiplicities; the results hold over algebraically closed…
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…
Given a symbolic power of a homogeneous ideal in a polynomial ring, we study the problem of determining which powers of the ideal contain it. For ideals defining 0-dimensional subschemes of projective space, as an immediate corollary of our…
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,…
The symbolic powers $I^{(n)}$ of a radical ideal $I$ in a polynomial ring consist of the functions that vanish up to order $n$ in the variety defined by $I$. These do not necessarily coincide with the ordinary algebraic powers $I^n$, but it…
Let I be the ideal corresponding to a set of general points $p_1,...,p_n \in P^2$. There recently has been progress in showing that a naive lower bound for the Hilbert functions of symbolic powers $I^{(m)}$ is in fact attained when n>9.…
This paper is concerned with determining the number of generators in each degree for minimal sets of homogeneous generators for saturated ideals defining fat point subschemes $Z=m_1p_1+ ... +m_rp_r$ for general sets of points $p_i$ of…