Set-theoretic complete intersections

Abstract:

In this article we establish that: (1) Every monomial curve in ${\mathbf {P}}_k^n$ is a set-theoretic complete intersection, where $k$ is a field of characteristic $p$ (and thus generalize a result of R. Hartshorne [3]). (2) Let $k$ be an algebraically closed field of characteristic $p$ and $C$ a curve of ${\mathbf {P}}_k^n$. If there is a linear projection $\tau :{\mathbf {P}}_k^n \to {\mathbf {P}}_k^2$ with center of $\tau$ disjoint of $C$, $\tau (C)$ is birational to $C$ and $\tau (C)$ has only cusps as singularities, then $C$ is a set-theoretic complete intersection (and thus generalize a result of D. Ferrand [2]).