Symbolic powers of codimension two Cohen-Macaulay ideals

Abstract

Let I_X be the saturated homogeneous ideal defining a codimension two arithmetically Cohen-Macaulay scheme $X\subseteq {\mathbb{P}}^n,$ and let $I_X^{(m)}$ denote its m-th symbolic power. We are interested in when $I_X^{(m)}=I_X^m.$ We survey what is known about this problem when X is locally a complete intersection, and in particular, we review the classification of when $I_X^{(m)}=I_X^m$ for all $m\ge 1.$ We then discuss how one might weaken these hypotheses, but still obtain equality between the symbolic and ordinary powers. Finally, we show that this classification allows one to: (1) simplify known results about symbolic powers of ideals of points in ${\mathbb{P}}^1\times {\mathbb{P}}^1;$ (2) verify a conjecture of Guardo, Harbourne, and Van Tuyl, and (3) provide additional evidence to a conjecture of Römer.

Keywords:

• Symbolic powers
• codimension two
• arithmetically Cohen-Macaulay
• locally complete intersection
• points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13C40
• 13F20
• 13A15
• 14C20
• 14M05

Acknowledgments

This project was started at the Mathematisches Forschungsinstitut Oberwolfach (MFO) as part of the mini-workshop “Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems” organized by C. Bocci, E. Carlini, E. Guardo, and B. Harbourne. All the authors thank the MFO for providing a stimulating environment. We also thank Brian Harbourne and Tomasz Szemberg for their feedback on early drafts of the paper.

Additional information

Funding

Cooper acknowledges support from the NDSU Advance FORWARD program sponsored by the National Science Foundation, HRD-0811239, as well as financial support provided by NSERC. Fatabbi and Lorenzini were partially supported by GNSAGA (INdAM). Guardo acknowledges the financial support provided by Prin 2011 and GNSAGA (INdAM) and Piano della ricerca – Linea intervento 2, Università di Catania. Migliore and Nagel were partially supported by Simons Foundation grants (#309556 for Migliore, #317096 and #636513 for Nagel). Seceleanu received support from MFO’s NSF grant DMS-1049268, “NSF Junior Oberwolfach Fellows” as well as from NSF grant DMS–1601024 and EPSCoR grant OIA–1557417. Szpond was partially supported by the National Science Center, Poland, grant 2014/15/B/ST1/02197. Van Tuyl acknowledges the financial support provided by NSERC RGPIN-2019-05412.