Generalized Castelnuovo varieties
In this paper we will show that if \(V \subset \mathbb{P}^ r_ \mathbb{C}\) is an irreducible nondegenerate projective variety of dimension \(n\) and degree \(d\), not contained in any irreducible subvariety of \(\mathbb{P}^ r_ \mathbb{C}\) of dimension \(n + 1\) and degree \(< s\), and if \(d\) is large with respect to \(s\), then the genus of \(V\) (arithmetic if \(n = 1\); geometric if \(n \geq 2)\) is bounded above by a number \(G = d^{n+1}/(n + 1)! s^ n + O(d^ n)\) which depends only on \(n,r,d\) and \(s\). We will see that this upper bound \(G\) is sharp (at least in some cases): generalized Castelnuovo varieties are varieties whose genus is \(G\). In the case of nondegenerate curves (i.e. \(n = 1\), \(s = r - 1)\) this result is well known since the end of the last century and at the beginning of the 1980's it has been proved for \(n = 1\), \(r = 3\), \(s \geq 2\), \(d \gg s\) and for \(n = 1\), \(r \geq 3\), \(r - 1 \leq s \leq 2r - 3\), \(d \gg s\); the general result has been only recently proved for \(n = 1\), \(r \geq 3\), \(s \geq r - 1\) and \(d \gg s\) [cf. \textit{L. Chiantini}, \textit{C. Ciliberto} and the author, Duke Math. J. 70, No. 2, 229-245 (1993; Zbl 0799.14011)]. The case \(n \geq 1\) has been analyzed by \textit{J. Harris} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 8, 35-68 (1981; Zbl 0467.14005)] for \(s = r - n\) (in this case the varieties which achieve the bound \(G\) are called ``Castelnuovo varieties'') and in 1990 by \textit{U. Nagel} and \textit{W. Vogel} [in: Topics in Algebra, Part 2: Commutative rings and algebraic groups, Pap. 31st Semester Class. Algebraic Struct., Warsaw 1988, Banach Cent. Publ. 26, Part 2, 163-183 (1990; Zbl 0735.14032)] for codimension two arithmetically Cohen-Macaulay subvarieties \(V\) of \(\mathbb{P}^ r\) [in this case, when \(d \gg s\), our bound is more precise than the bound obtained by U. Nagel and W. Vogel (loc. cit.)].
scientific article; zbMATH DE number 554387
scientific article