The structure of an even liaison class

Abstract

We describe a structure called the Lazarsfeld-Rao property for even liaison classes in projective space. This property holds for many even liaison classes of curves in P 3 {{\mathbf {P}}^3} . We give a procedure for showing that an even liaison class in codimension 2 2 possesses this property, and we prove it for a family of even liaison classes in codimension 2 2 in any P n , n ⩾ 3 {{\mathbf {P}}^n},\;n \geqslant 3 . However, we conjecture that it in fact holds for every even liaison class in codimension 2 2 , so we want to give consequences for an even liaison class that possesses this property. The main element in describing this structure is the notion of a basic double link. The Lazarsfeld-Rao property says that there exist minimal elements of the even liaison class and that any element of the even liaison class can be deformed to a curve obtained by a sequence of basic double links beginning with any minimal element. We show that there is a unique standard type of sequence for any given element of the even liaison class. As a result, we can express the even liaison class as a disjoint union of irreducible nonempty families parameterized by certain finite sequences of integers. The standard numerical invariants of the elements of any family can be computed from the associated sequence of integers. We apply this to surfaces in P 4 {{\mathbf {P}}^4} . Our main tool for these results is a deformation technique related to liaison in codimension 2 2 . We also study Schwartau’s procedure of Liaison Addition in codimension 2 2 from the point of view of vector bundles. Using this, we give a different sort of structure for an even liaison class with the Lazarsfeld-Rao property.