The Lazarsfeld-Rao property on an arithmetically Gorenstein variety
Liaison techniques have been intensively used in the classification of curves in \(\mathbb{P}^ 3\) and, more generally, of two-codimensional subschemes of \(\mathbb{P}^ n\). But already since the fundamental paper of \textit{P. Rao} [Math. Ann. 258, 169-173 (1981; Zbl 0493.14009)] liaison is considered in a more general setting, in particular when the variety where we work is a smooth arithmetically Gorenstein subscheme \(X\) of \(\mathbb{P}^ n\). The main purpose of this paper is to furnish in this relevant situation (e.g., hypersurfaces of \(\mathbb{P}^ n\), the Grassmann variety Gr(1,3), etc.) all the machinery of liaison in codimension two as developed in the projective situation. So, many of the results here are just an adaptation of results known in the projective case, and hence our discussion will sometimes simply describe the connections with the known cases and sketch the proofs. We define the notion of basic double link, the notion of shift and of minimal shift of an even liaison class, and we prove a structure theorem for even liaison classes, which generalizes the Lazarsfeld-Rao property (LR-property) known in \(\mathbb{P}^ n\). -- We devote \S2 to the licci (linkage class of a complete intersection) liaison class, characterizing the possible locally free resolution of licci (in \(X)\) two-codimensional subschemes of \(X\) and, when \(\dim X \leq 4\), of the smooth ones. We hope that this will help in the study of licci subscheme of \(\mathbb{P}^ n\) in codimension \(>2\). -- The LR-property gives a description of all possible locally free resolutions of elements of a fixed liaison class, and also geometric consequences, such as deformation to reducible subschemes with nice irreducible components. -- In the last section we sketch some applications of these results to the Grassmann variety Gr(1,3).
linkage class of a complete intersection
scientific article; zbMATH DE number 492736
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