Gherardelli linkage and complete intersections. (Q5954561)

The authors characterize complete intersections \(X\subset \mathbb{P}^n_K\) (\(n\geq 3\) and \(K\) an algebraically closed field of characteristic 0) as locally Cohen-Macaulay subvarieties \(X\) which are subcanonical and self-linked. This generalizes a similar result NEWLINEof \textit{V. Beorchia} and \textit{P. Ellia} [Rend. Sem. Mat. Univ. Politec. Torino 48, No. 4, 553--562 (1990; Zbl 0779.14007)] where \(X\) was assumed to be smooth.NEWLINENEWLINEIn the first part, the Gheradelli linkage theorem is proved in a more general setup; namely, the dualizing sheafNEWLINE\(\omega_X\) of a subscheme \(X\subset P\), contained in a Gorenstein scheme \(P\) of pure dimension at least 2, is described if \(X\) is linked to a quasi-complete intersection \(Y\) (i.e., \(Y\) is defined scheme-theoretically by three polynomials). Then, the proof follows further the ideas of [loc. cit.], the main ingredient being the splitting of the normal bundle of \(X\). It is explicitely described how the self-linkage of \(X\) occurs.NEWLINENEWLINENEWLINE Finally, an example is given that the result does not hold if \(\mathbb{P}^n\) is replaced by a smooth hypersurface, and there is a counterexample in the characteristic two case due to \textit{J. Migliore} [Trans. Am. Math. Soc. 294, 177--185 (1986; Zbl 0596.14019)].