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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 $$P_ N)$$.
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
##### Keywords:
ampleness of twist of normal bundle