Set-theoretic complete intersections

Abstract

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