Über das Normalenbündel von projektiven Varietäten. (On the normal bundle of projective varieties).

*(German)*Zbl 0704.14010
Bayreuth: Universität Bayreuth, Diss. 182 S. (1989).

Chapter 1 of this thesis is devoted to the question: when is the (-1)- twist N(-1) of the normal bundle N of a smooth variety X in \(P_ N\) ample? The main result in this direction has the following form:

Let S be a smooth rational non-degenerate non-special surface in \(P_ 4\); then:

(1) If S is rigid then \(N_{S/P_ 4}(-1)\) is not ample;

(2) If S is not rigid and \(N_{S/P_ 4}(-1)\) is not ample then S can be deformed to a surface \(S'\) of the same type on which \(N_{S'/P_ 4}(- 1)\) is ample.

“Of the same type” means that the degree of S and the genus of its hyperplane section coincide. “Rigid” means that all surfaces of the same type are projectively equivalent. Cubic scrolls and the Veronese surfaces in \(P_ 4\) are precisely the rigid nonspecial surfaces in \(P_ 4\). (S is non-special if \(h^ 1(S,{\mathcal O}_ S(1))=0.)\) There is a very useful table connecting the degree of S and the ampleness properties of N(-1).

The second chapter develops the methods from G. Ellingsrud, L. Gruson, C. Peskine and S. A. Strømme [Invent. Math. 80, 181-184 (1985; Zbl 0629.14026)] to prove a similar statement when X is an effective Cartier divisor on an n-dimensional (maybe singular) variety Y in \(P_ N\). The precise statement is:

Let Y be an n-dimensional variety in \(P_ N (2\leq n\leq N)\) and X an effective Cartier divisor on Y having a neighborhood U in Y which is a locally complete intersection in \(P_ N\). Then

(1) If the exact sequence of the normal bundles \(0\to N_{X/Y}\to N_{X/P_ N}\to N_{X/P_ N}|_ X\to 0\) splits then some positive multiplicity of X is numerically equivalent to a hyperplane section of Y;

(2) If in addition Y is regular in codimension 1 then there is an integer \(d_ 0\) such that the converse of (1) is true if the degree of X is \(\geq d_ 0\) \((d_ 0\) depends only on the embedding of Y in \(P_ N)\).

Let S be a smooth rational non-degenerate non-special surface in \(P_ 4\); then:

(1) If S is rigid then \(N_{S/P_ 4}(-1)\) is not ample;

(2) If S is not rigid and \(N_{S/P_ 4}(-1)\) is not ample then S can be deformed to a surface \(S'\) of the same type on which \(N_{S'/P_ 4}(- 1)\) is ample.

“Of the same type” means that the degree of S and the genus of its hyperplane section coincide. “Rigid” means that all surfaces of the same type are projectively equivalent. Cubic scrolls and the Veronese surfaces in \(P_ 4\) are precisely the rigid nonspecial surfaces in \(P_ 4\). (S is non-special if \(h^ 1(S,{\mathcal O}_ S(1))=0.)\) There is a very useful table connecting the degree of S and the ampleness properties of N(-1).

The second chapter develops the methods from G. Ellingsrud, L. Gruson, C. Peskine and S. A. Strømme [Invent. Math. 80, 181-184 (1985; Zbl 0629.14026)] to prove a similar statement when X is an effective Cartier divisor on an n-dimensional (maybe singular) variety Y in \(P_ N\). The precise statement is:

Let Y be an n-dimensional variety in \(P_ N (2\leq n\leq N)\) and X an effective Cartier divisor on Y having a neighborhood U in Y which is a locally complete intersection in \(P_ N\). Then

(1) If the exact sequence of the normal bundles \(0\to N_{X/Y}\to N_{X/P_ N}\to N_{X/P_ N}|_ X\to 0\) splits then some positive multiplicity of X is numerically equivalent to a hyperplane section of Y;

(2) If in addition Y is regular in codimension 1 then there is an integer \(d_ 0\) such that the converse of (1) is true if the degree of X is \(\geq d_ 0\) \((d_ 0\) depends only on the embedding of Y in \(P_ N)\).

Reviewer: V.K.Vedernikov

##### MSC:

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

14J10 | Families, moduli, classification: algebraic theory |

14N05 | Projective techniques in algebraic geometry |

14C20 | Divisors, linear systems, invertible sheaves |