Nilpotent lci structures on global complete intersections
Let \(X\) be a smooth subvariety of \(\mathbb{P}^n\). A closed subscheme \(Y\) of \(\mathbb{P}^n\) is said to be a nilpotent structure on \(X\) if \(Y \neq X\) and \(Y_{\text{red}} = X\). The study of schemes with some ``mild'' nilpotent structure (e.g. Cohen-Macaulay or locally complete intersection) appears in connection with the study of set-theoretic complete intersections [see for instance \textit{L. Szpiro}, ``Lectures on equations defining space curves'', Tata Inst. Lect. Notes No. 62 (1979; Zbl 0442.14010)] and also in the study of vector bundles via the zero sets of their sections [\textit{R. Hartshorne}, Math. Ann. 238, 229-280 (1978; Zbl 0411.14002)]. A Cohen-Macaulay nilpotent structure \(Y\) on \(X\) has the property that the multiplicities of all local rings of \(Y\) are equal, and so one may define the multiplicity \(\mu (Y)\) of \(Y\) as the multiplicity of any of its local rings. In the present paper, using previous results of himself [Math. Nachr. 167, 157-202 (1994)] and \textit{G. Faltings} [Invent. Math. 62, 393- 401 (1981; Zbl 0456.14027)], the author gives an explicit description of the nilpotent locally complete intersection structures, up to multiplicity 6, on smooth complete intersection subvarieties \(X\) of \(\mathbb{P}^n\) with \(\text{codim} (X) < (n + 1)/3\). As a consequence, one sees that all these locally complete intersection structures are again globally complete intersection in \(\mathbb{P}^n\) (in the paper \(\mathbb{P}^n\) denotes the projective space over an algebraically closed field of characteristic \(\neq 2,3)\).
scientific article; zbMATH DE number 795029