On varieties whose degree is small with respect to codimension
type: Item
instance of: scholarly article
title: On varieties whose degree is small with respect to codimension
zbMATH Open document ID: 0541.14032
DOI: 10.1007/BF01456072
author: Paltin Ionescu
published in: Mathematische Annalen
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/163972
review text: Let \(X^ r_ d\subset P_{{\mathbb{C}}}^{r+s}\) be a smooth, connected, non-degenerate, projective algebraic variety of dimension \(r\geq 2\) and degree d. Let H be a generic hyperplane section of X. A divisor \(E\subset X\) is said to be exceptional if \(E\cong P^{r-1},{\mathcal O}_ E(E)\cong {\mathcal O}(-1), {\mathcal O}_ X(H)\otimes {\mathcal O}_ E\cong {\mathcal O}(1).\) If \(r\geq 3\), denote by (X',H') a pair such that: \(\sigma:X\to X'\) is the contraction of all exceptional divisors \(E_ 1,...,E_ k\) contained in X, and H' is the (unique) divisor on X' such that \(H=\sigma^*(H')-E_ 1-...-E_ k.\) The main result of this paper shows that if X is as above and d is ''small'' with respect to s, both the abstract and the embedded structure of X can be determined. Moreover, our bound turns out to be sharp. Main theorem. Let \(X^ r_ d\subset P_{{\mathbb{C}}}^{r+s}\) be as above. If \(d\leq 2s+1,\) (X,H) is one of the following: (1) a birationally ruled surface; (2) a one or two-dimensional family of linear varieties; (3) a one-dimensional family of hyperquadrics; (4) a variety with \(\omega_ X^{-1}={\mathcal O}_ X((r-1)H)\) (the classification of these varieties is known); (5) \(P^ r\), H a hyperplane; (6) a hyperquadric and H a hyperplane section; (7) the pair (X',H') is one of the following: \(X'=P^ 3\) and H'\(\in | {\mathcal O}(3)|\), or X' is a hyperquadric in \(P^ 4_{{\mathbb{C}}}\) and H'\(\in | {\mathcal O}(2)|\), or \(X'=P^ 4_{{\mathbb{C}}}\) and H'\(\in | {\mathcal O}(2)|\), or X' is a \(P^ 2_{{\mathbb{C}}}\)-bundle over a (smooth) curve such that \({\mathcal O}_{X'}(H')\) induces \({\mathcal O}(2)\) on every fibre. On the other hand, if \(d=2s+2\) and X is not as in situations (1)-(7), then either (8) X is a K3 surface, or (9) \(r\geq 3\) and \(\omega_ X^{-1}={\mathcal O}_ X((r- 2)H).\) Since a surface of degree 5 in \(P^ 3\) is of general type, the bound \(d\leq 2s+2\) is sharp. The above result explains the possibility of listing varieties of small degre…
Mathematics Subject Classification ID: 14J10
zbMATH DE Number: 3859303
zbMATH Keywords: varieties of small degree
zbMATH Keywords: classification of varieties
MaRDI profile type: Publication
cites work: On the hyperplane section principle of Lefschetz
cites work: On the structure of polarized manifolds with total deficiency one. I
cites work: On the structure of polarized manifolds with total deficiency one. II
cites work: Q4195061
cites work: Varieties of small codimension in projective space
cites work: Q4143433
cites work: FANO 3-FOLDS. I
cites work: Threefolds whose canonical bundles are not numerically effective
cites work: The topology of normal singularities of an algebraic surface and a criterion for simplicity
cites work: On rational surfaces I. Irreducible curves of arithmetic genus $0$ or $1$
cites work: Q4402617
cites work: Hyperplane sections of projective surfaces. I: The adjunction mapping
cites work: On the minimality of hyperplane sections of projective threefolds.
cites work: On the 2-connectedness of very ample divisors on a surface
description: scientific article; zbMATH DE number 3859303