Using the Grothendieck-Lefschetz theory (see cite{[SGA2]}) and a generalization (due to Cutkosky cite{C}) of a result from cite{[SGA2]} concerning the simple connectedness, we prove that many closed subvarieties of $mathbb P^n$ of dimension $geq 2$ need at least $n-1$ equations to be defined in $mathbb P^n$ set-theoretically, i.e. their arithmetic rank is $geq n-1$ (Theorem 1 of the Introduction). As applications we give a number of relevant examples. In the second part of the paper we prove that the arithmetic rank of a rational normal scroll of dimension $dgeq 2$ 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).
Grothendieck-Lefschetz theory, set-theoretic complete intersections and rational normal scrolls
BADESCU, LUCIAN SILVESTRU;VALLA, GIUSEPPE
2010-01-01
Abstract
Using the Grothendieck-Lefschetz theory (see cite{[SGA2]}) and a generalization (due to Cutkosky cite{C}) of a result from cite{[SGA2]} concerning the simple connectedness, we prove that many closed subvarieties of $mathbb P^n$ of dimension $geq 2$ need at least $n-1$ equations to be defined in $mathbb P^n$ set-theoretically, i.e. their arithmetic rank is $geq n-1$ (Theorem 1 of the Introduction). As applications we give a number of relevant examples. In the second part of the paper we prove that the arithmetic rank of a rational normal scroll of dimension $dgeq 2$ 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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