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Symbolic Powers Versus Regular Powers of Ideals of General Points in $\mathbb{P}^1 \times \mathbb{P}^1$

  Published:2012-11-13
 Printed: Aug 2013
  • Elena Guardo,
    Dipartimento di Matematica e Informatica, Viale A. Doria, 6-95100-Catania, Italy
  • Brian Harbourne,
    Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA
  • Adam Van Tuyl,
    Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON P7B 5E1
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Abstract

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 asymptotic numerical characters of symbolic powers of the ideals. Most of the work done up to now has been done for ideals defining 0-dimensional subschemes of projective space. Here we focus on certain subschemes given by a union of lines in $\mathbb{P}^3$ which can also be viewed as points in $\mathbb{P}^1 \times \mathbb{P}^1$. We also obtain results on the closely related problem, studied by Hochster and by Li-Swanson, of determining situations for which each symbolic power of an ideal is an ordinary power.
Keywords: symbolic powers, multigraded, points symbolic powers, multigraded, points
MSC Classifications: 13F20, 13A15, 14C20 show english descriptions Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25]
Ideals; multiplicative ideal theory
Divisors, linear systems, invertible sheaves
13F20 - Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25]
13A15 - Ideals; multiplicative ideal theory
14C20 - Divisors, linear systems, invertible sheaves
 

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