A variety $V$ of codimension $r$ in a projective space $\mathbb{P}^n$ is called a set-theoretic complete intersection if $V$, as a set, is the intersection of exactly $r$ hypersurfaces in $\mathbb{P}^n$. I will discuss the history of the general problem, which varieties $V$ are s.t.c.i., with special attention to the still open problem, is every irreducible nonsingular curve in $\mathbb{P}^3$ a set-theoretic complete intersection? In particular I will mention several algebraic criteria, including local cohomology that can in principle be used to show that certain varieties are not s.t.c.i.
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