2008/04/03 by Juan Migliore, Migliore, Juan, Uwe Nagel +1
Mathematics · #13C40 #13D02 #14F05 #14M06 #14M07 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Geometric and Algebraic Topology #math.AC #math.AG #msc:13C40 #msc:13D02 #msc:14F05 #msc:14M06 #msc:14M07
paper · pdf · doi:10.48550/arxiv.0804.0590
25 pages
arxiv created 2008/04/03 · openalex publication_date 2008/04/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If V is an equidimensional codimension c subscheme of an n-dimensional projective space, and V is linked to V' by a complete intersection X, then we say that V is \em minimally linked to V' if X is a codimension c complete intersection of smallest degree containing V. Gaeta showed that if V is any arithmetically Cohen-Macaulay (ACM) subscheme of codimension two then there is a finite sequence of minimal links beginning with V and arriving at a complete intersection. We extend this work in the following ways: 1) In the codimension 2 non-ACM case, we show that for any n ≥ 3 there are examples of subschemes that are not minimal in their even liaison class, and cannot be minimally linked in any number of steps to a minimal subscheme. 2) Nevertheless, there are examples of non-ACM liaison classes of curves in projective 3-space where all elements are minimally linked in a finite number of steps to a minimal curve. 3) Extending previous work of the authors with Huneke and Ulrich (about the licci case), we show that also in the non-ACM case in any higher codimension there are non-minimal subschemes that are not minimally linked to a minimal subscheme in the even liaison class. 4) J. Watanabe had shown many years ago that codimension 3 graded Gorenstein ideals of any dimension are licci. Here we show that any such ideal is minimally linked in a finite number of steps to a complete intersection, and that it admits a sequence of strictly decreasing CI-biliaisons down to a complete intersection, extending work of Hartshorne, Sabadini and Schlesinger.