Biliaison of sheaves
- PDF / 404,222 Bytes
- 21 Pages / 439.37 x 666.142 pts Page_size
- 17 Downloads / 200 Views
Mathematische Zeitschrift
Biliaison of sheaves Mengyuan Zhang1 Received: 3 August 2020 / Accepted: 1 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We define an equivalence relation among coherent sheaves on a projective variety called biliaison. We prove the existence of sheaves that are minimal in a biliaison class in a suitable sense, and show that all sheaves in the same class can be obtained from a minimal one using certain deformations and other basic moves. Our results generalize the main theorems of liaison theory of subvarieties to sheaves, and provide a framework to study sheaves and subvarieties simultaneously.
Introduction Liaison theory originated from the work of M. Noether in 1882 that classified algebraic curves 3 of degree ≤ 20. It has since become instrumental in the study of the Hilbert scheme of in PC projective spaces [4,19]. Briefly, a (geometric) link is a pair of subschemes that are residual to each other in a complete intersection. An even linkage class consists of those subschemes that can be obtained from one another using even numbers of links. Tremendous progress has been made in the past decade in understanding the structure of an even linkage class in codimension two. The situation is less clear in higher codimensions and an alternative notion of a Gorenstein linkage has been proposed and studied, see for example [7,17,21,25]. The following is a summary of the main results of liaison theory in codimension two. 1. The degrees of subschemes in an even linkage class are bounded below, and those with the minimal degree differ by a deformation preserving cohomologies [5,22]. 2. Any subscheme can be obtained from a minimal one in its even linkage class using certain elementary moves [2,13,18,26,32]. 3. Even linkage classes are in bijection with stable equivalence classes of primitive bundles [28,30]. In particular, the numerical invariants of subschemes in an even linkage class can be systematically deduced from those of a minimal one. We refer to the book [23] for an introduction to liaison theory, and to [13, Introduction] for a survey of the above results.
B 1
Mengyuan Zhang [email protected] Department of Mathematics, University of California, Berkeley, CA 94720, USA
123
M. Zhang
In this article we extend the equivalence relation of even linkage between codimension two subschemes to an equivalence among sheaves on a projective variety, which we call biliaison of sheaves. The following main results are established in this article. (a) There are minimal sheaves in each biliaison class under a suitable preorder (Theorem C). Those that are minimal can be obtained from each other using a rational deformation preserving cohomologies (Proposition 4.7). (b) Any sheaf can be obtained from a minimal one in its biliaison class using rigid deformations and certain basic moves (Sect. 4). (c) Biliaison classes of sheaves are in bijection with stable equivalence classes of primitive sheaves. The results (a)–(c) give us satisfactory extensions of (1)–(3)
Data Loading...
