Grothendieck-Lefschetz Theory, Set-Theoretic Complete Intersections and Rational Normal Scrolls

Details Grothendieck-Lefschetz Theory, Set-Theoretic Complete Intersections and Rational Normal Scrolls Grothendieck-Lefschetz Theory, Set-Theoretic Complete Intersections and Rational Normal Scrolls

2009-10-20

arxiv.org/abs/0910.3847v1

Mathematics

Algebraic Geometry

24 pages

Scientific paper

Using the Grothendieck-Lefschetz theory (see \cite{[SGA2]}) we prove a criterion to deduce that certain subvarieties of $\mathbb P^n$ of dimension $\geq 2$ are not set-theoretic complete intersections (see Theorem 1 of the Introduction). As applications we give a number of relevant examples. In the last part of the paper we prove that the arithmetic rank of a rational normal $d$-dimensional scroll $S_{n_1,...,n_d}$ in $\mathbb P^N$ is $N-2$, by producing an explicit set of $N-2$ homogeneous equations which define these scrolls set-theoretically (see Theorem 2 of the Introduction).

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