Symbolic powers of codimension two Cohen-Macaulay ideals. Issue 11 (1st November 2020)

• Record Type: Journal Article
• Authors: Cooper, Susan; Fatabbi, Giuliana; Guardo, Elena; Lorenzini, Anna; Migliore, Juan; Nagel, Uwe; Seceleanu, Alexandra; Szpond, Justyna; Tuyl, Adam Van
• Is Part Of: Communications in algebra. Volume 48:Issue 11(2020)
• Page Start: 4663
• Page End: 4680
• Publication Date: 2020-11-01
• Subjects: Symbolic powers -- codimension two -- arithmetically Cohen-Macaulay -- locally complete intersection -- points in P1×P1; 13C40 -- 13F20 -- 13A15 -- 14C20 -- 14M05; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2020.1769120
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: Let IX be the saturated homogeneous ideal defining a codimension two arithmetically Cohen-Macaulay scheme X ⊆ 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 ≥ 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 P 1 × P 1 ; (2) verify a conjecture of Guardo, Harbourne, and Van Tuyl, and (3) provide additional evidence to a conjecture of Römer.