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

TL;DR

This paper applies Grothendieck-Lefschetz theory to identify when subvarieties of projective space are not set-theoretic complete intersections and determines the explicit equations defining rational normal scrolls.

Contribution

It introduces a criterion using Grothendieck-Lefschetz theory for non-complete intersections and explicitly computes the arithmetic rank of rational normal scrolls.

Findings

Subvarieties of dimension ≥ 2 are not set-theoretic complete intersections under certain conditions.

The arithmetic rank of rational normal scrolls is exactly N-2.

Explicit defining equations for rational normal scrolls are provided.

Abstract

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).