On The Projective Normality of Smooth Surfaces of degree nine

Details On The Projective Normality of Smooth Surfaces of degree nine On The Projective Normality of Smooth Surfaces of degree nine

1997-10-07

arxiv.org/abs/alg-geom/9710009v1

Mathematics

Algebraic Geometry

22 pages, AmsLatex, see home pages http://www.emunix.emich.edu/~gbesana/ and http://www.math.kth.se/~sandra/Welcome

Scientific paper

We investigate the projective normality of smooth, linearly normal surfaces of degree 9. All non projectively normal surfaces which are not scrolls over a curve are classified. Results on the projective normality of surface scrolls are also given. One of the reasons that brought us to look at this question is our desire to find examples for a long standing problem in adjunction theory. Andreatta followed by a generalization by Ein and Lazarsfeld posed the problem of classifying smooth n-dimensional varieties (X,L) polarized with a very ample line bundle L, such that the adjoint linear system |H| = |K + (n-1)L| gives an embedding which is not projectively normal. After a detailed check of the non projectively normal surfaces found in this work no examples were found except possibly a blow up of an elliptic P^1-bundle whose existence is uncertain.

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