Are symbolic powers highly evolved?

Abstract

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 conjectures which explain them, and we prove the conjectures in certain cases, including the case of general points in the projective plane. Our conjectures were also partly motivated by the Eisenbud-Mazur Conjecture on evolutions, which concerns symbolic squares of prime ideals in local rings, but in contrast we consider higher symbolic powers of homogeneous ideals in polynomial rings.

13 pages; for version 3 a minor change was made to the acknowledgments but no change was made to mathematical content; for version 2 a reference to a paper of Esnault and Viehweg has been added; related to this a new section, 4.2, has been included with additional questions. Otherwise, version 2 is the same as version 1

References in corpus (2)

• Reduction method for linear systems of plane curves with base fat points
• Limit linear systems and applications

Cited by in corpus (34)

• Counterexamples to the $I^{(3)}\subset I^2$ containment
• Resurgences for ideals of special point configurations in ${\mathbf{P}}^N$ coming from hyperplane arrangements
• Linear subspaces, symbolic powers and Nagata type conjectures
• On the containment problem
• Points fattening on P^1 x P^1 and symbolic powers of bi-homogeneous ideals
• Containment results for ideals of various configurations of points in P^N
• Symbolic powers of edge ideals of graphs
• A vanishing theorem and symbolic powers of planar point ideals
• Containment problem for points on a reducible conic in ${\mathbb{P}}^2$
• Uniform Harbourne-Huneke Bounds via Flat Extensions
• Containment Counterexamples for ideals of various configurations of points in ${\mathbf{P}}^N$
• On resurgence via asymptotic resurgence
• Singular loci of reflection arrangements and the containment problem
• Symbolic powers versus regular powers of ideals of general points in P^1 x P^1
• Configurations of Linear Spaces of Codimension Two and the Containment Problem
• The Alpha Problem & Line Count Configurations
• The effect of points fattening on Hirzebruch surfaces
• Symbolic powers of ideals of generic points in P^3
• Local effectivity in projective spaces
• Inclics, galaxies, star configurations and Waldschmidt constants
• A containment result in ${\mathbb{P}}^n$ and the Chudnovsky conjecture
• Containments of symbolic powers of ideals of generic points in $\PP^3$
• Fat lines in P^3: powers versus symbolic powers
• Symbolic Rees algebras
• Ideals generated by $a$-fold products of linear forms have linear graded free resolution
• On the containment problem for fat points ideals
• An Elementary Approach to Containment Relations Between Symbolic and Ordinary Powers of Certain Monomial Ideals
• Symbolic powers of generalized star configurations of hypersurfaces
• Chudnovsky's Conjecture for very general points in ${\mathbb{P}}_k^N$
• Symbolic powers, set-theoretic complete intersection and certain invariants
• Demailly's conjecture on Waldschmidt constants for sufficiently many very general points in ${\mathbb{P}}^n$
• Asymptotic Hilbert Polynomial and limiting shapes
• Waldschmidt constants for Stanley-Reisner ideals of a class of Simplicial Complexes
• Counterexamples to containment problems for fat points schemes in the projective plane via multiplier ideals