# Grothendieck-Lefschetz Theory, Set-Theoretic Complete Intersections and Rational Normal Scrolls - Mathematics > Algebraic Geometry

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Abstract: Using the Grothendieck-Lefschetz theory see \cite{SGA2} we prove acriterion to deduce that certain subvarieties of $\mathbb P^n$ of dimension$\geq 2$ are not set-theoretic complete intersections see Theorem 1 of theIntroduction. As applications we give a number of relevant examples. In thelast 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 $N-2$, byproducing an explicit set of $N-2$ homogeneous equations which define thesescrolls set-theoretically see Theorem 2 of the Introduction.

Author: ** Lucian Badescu, Giuseppe Valla**

Source: https://arxiv.org/