Subadjoint ideals and hyperplane sections (Q2736610)

The paper under review deals with the study of the behaviour of the notion of ``subadjoint ideal to a projective variety'' with respect to general hyperplane sections. The author works over an algebraically closed field of characteristic zero. The paper is related to a modern revisitation of the classical theory of adjoint hypersurfaces due to \textit{F. Enriques} [Introduzione alla geometria sopra le superficie algebriche, Memorie della Societa Italiana dell Scienze (detta dei XL) 10, 1-81 (1896; JFM 27.0518.02)] and others, which is far from being well understood [see also, in this context, the paper by \textit{P. Blass} and \textit{J. Lipman} in Am. J. Math. 101, 331-336 (1979; Zbl 0436.14011)].NEWLINENEWLINENEWLINEThe main result of the present paper is a ``going up'' and ``going down'' theorem for subadjointness with respect to hyperplane sections (theorem 1.5). A corollary of the previous theorem is that the classical definitions of subadjoint hypersurface given by Enriques [op. cit.] and \textit{O. Zariski} [``An introduction to the theory of algebraic surfaces'' (Lect. Notes Math. 83, Berlin-Heidelberg-New York 1969; Zbl 0177.49001)] are equivalent. NEWLINENEWLINENEWLINEThe techniques used by the author belong mainly to commutative algebra: exact sequences, Nakayama's lemma, integral elements, the long exact sequence of Ext etc., applied further to projective geometry using some Bertini-type theorems [see \textit{J. P. Jouanolou}, in: Théorèmes de Bertini et applications, Prog. Math. 42 (Boston-Basel-Stuttgart 1983; Zbl 0519.14002)].NEWLINENEWLINENEWLINE[Editor's comment: This paper is a word by word copy of the paper of \textit{N.NEWLINEChiarli} and \textit{S. Greco} with the same title in Proc. Am. Math. Soc. 124,NEWLINENo.4, 1035-1041 (1996; Zbl 0874.14001)]