Containments of symbolic powers of ideals of generic points in $\PP^3$

@article{Dumnicki2012ContainmentsOS,
  title={Containments of symbolic powers of ideals of generic points in \$\PP^3\$},
  author={Marcin Dumnicki},
  journal={arXiv: Algebraic Geometry},
  year={2012}
}
  • M. Dumnicki
  • Published 2012
  • Mathematics
  • arXiv: Algebraic Geometry
We show that the Conjecture of Harbourne and Huneke, $I^{(Nr-(N-1))} \subset M^{(r-1)(N-1)}I^{r}$ holds for ideals of generic (simple) points in $\PP^3$. As a result, for such ideals we prove the following bounds, which can be recognized as generalizations of Chudnovsky bounds: $\alpha(I^{(3m-k)}) \geq m\alpha(I)+2m-k$, for any $m \geq 1$ and $k=0,1,2$. Moreover, we obtain lower bounds for the Waldshmidt constant for such ideals. 
On the containment problem for fat points ideals and Harbourne’s conjecture
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.Expand
A containment result in $\mathbb{P}^n$ and the Chudnovsky conjecture
In the paper we prove the containment $I^{(nm)}\subset M^{(n-1)m}I^m$, for a radical ideal $I$ of $s$ general points in $\mathbb{P}^n$, where $s\geq 2^n$. As a corollary we get that the ChudnovskyExpand
Symbolic powers of ideals defining F-pure and strongly F-regular rings
Given a radical ideal $I$ in a regular ring $R$, the Containment Problem of symbolic and ordinary powers of $I$ consists of determining when the containment $I^{(a)} \subseteq I^b$ holds. By work ofExpand
The Waldschmidt constant for squarefree monomial ideals
Given a squarefree monomial ideal $$I \subseteq R =k[x_1,\ldots ,x_n]$$I⊆R=k[x1,…,xn], we show that $$\widehat{\alpha }(I)$$α^(I), the Waldschmidt constant of I, can be expressed as the optimalExpand
Lower Bounds for Waldschmidt Constants of Generic Lines in $${\mathbb {P}}^3$$P3 and a Chudnovsky-Type Theorem
The Waldschmidt constant $${{\,\mathrm{{\widehat{\alpha }}}\,}}(I)$$α^(I) of a radical ideal I in the coordinate ring of $${\mathbb {P}}^N$$PN measures (asymptotically) the degree of a hypersurfaceExpand
A stable version of Harbourne's Conjecture and the containment problem for space monomial curves
Abstract 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 withExpand
Resurgences for ideals of special point configurations in ${\bf P}^N$ coming from hyperplane arrangements
Abstract Symbolic powers of ideals have attracted interest in commutative algebra and algebraic geometry for many years, with a notable recent focus on containment relations between symbolic powersExpand
The Least Generating Degree of Symbolic Powers of Ideals of Fermat Configuration of Points
Let n ≥ 2 be an integer and consider the defining ideal of the Fermat configuration of points in P: In = (x(y n − z), y(z − x), z(x − y)) ⊂ R = C[x, y, z]. In this paper, we compute explicitly theExpand
The Least Generating Degree of Symbolic Powers of Fermat-like Ideals of Planes and Lines Arrangements
We compute explicitly the least degree of generators of almost all symbolic powers of the defining ideal of Fermat-like configuration of lines in P C and provide efficient bounds for remaining cases.Expand
LOWER BOUNDS FOR WALDSCHMIDT CONSTANTS OF GENERIC LINES IN P AND A CHUDNOVSKY-TYPE THEOREM
The Waldschmidt constant α̂(I) of a radical ideal I in the coordinate ring of P measures (asymptotically) the degree of a hypersurface passing through the set defined by I in P . Nagata’s approach toExpand
...
1
2
3
...

References

SHOWING 1-10 OF 16 REFERENCES
Symbolic powers of ideals of generic points in P^3
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 homogeneousExpand
Comparing powers and symbolic powers of ideals
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 anExpand
ARE SYMBOLIC POWERS HIGHLY EVOLVED
Searching for structural reasons behind old results and conjectures of Chudnovksy regarding the least degree of a nonzero form in an ideal of fat points in projective N-space, we make conjecturesExpand
Regularity and non-emptyness of linear systems in $\mathbb P^n$
The main goal of this paper is to present a new algorithm bounding the regularity and ``alpha'' (the lowest degree of existing hypersurface) of a linear system of hypersurfaces (in $\mathbb P^n$)Expand
Linear subspaces, symbolic powers and Nagata type conjectures
Abstract Inspired by results of Guardo, Van Tuyl and the second author for lines in P 3 , we develop asymptotic upper bounds for the least degree of a homogeneous form vanishing to order at least mExpand
The resurgence of ideals of points and the containment problem
We relate properties of linear systems on X to the question of when I r contains I (m) in the case that I is the homogeneous ideal of a finite set of distinct points p 1 ,...,p n ∈ P 2 , where X isExpand
On the postulation of s^d fat points in P^d
In connection with his counter-example to the fourteenth problem of Hilbert, Nagata formulated a conjecture concerning the postulation of r fat points of the same multiplicity in P2 and proved itExpand
An algorithm to bound the regularity and nonemptiness of linear systems in Pn
  • M. Dumnicki
  • Computer Science, Mathematics
  • J. Symb. Comput.
  • 2009
TLDR
A new algorithm is presented bounding the regularity and ''alpha'' (the lowest degree of existing hypersurface) of a linear system of hypersurfaces in P^n passing through multiple points in a general position and a new theorem is formulated and proved, which allows to show the non-speciality of alinear system by splitting it into non- special and simpler systems. Expand
Uniform bounds and symbolic powers on smooth varieties
The purpose of this note is to show how one can use multiplier ideals toestablish effective uniform bounds on the multiplicative behavior of certainfamilies of ideal sheaves on a smooth algebraicExpand
A primer on Seshadri constants
Seshadri constants express the so called local positivity of a line bundle on a projective variety. They were introduced by Demailly. The original idea of using them towards a proof of the FujitaExpand
...
1
2
...