On varieties whose degree is small with respect to codimension
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 degree. In particular, it shows that the classification of varieties of degree \(\leq 8\), which was obtained by the author [for \(d\leq 7\) see Algebraic geometry, Proc. internat. Conf., Bucharest/Rom. 1982, Lect. Notes Math. 1056, 142-186 (1984)], was possible, while the classification of varieties of degree \(\geq 9\) is difficult. An important step in the proof of the biregular classification of all (smooth, projective) 3-folds whose hyperplane section is a birationally ruled surface. This in turn depends on the study of the adjunction mapping of Sommese, Van de Ven and the author, on some results of S. Mori for 3-folds whose canonical bundles are not numerically effective, and on a result of L. Bădescu describing all 3-folds supporting a geometrically ruled surface as an ample divisor.
scientific article; zbMATH DE number 3859303