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For an irreducible projective variety $Y\subset \mathbb{P}(V)$ over the complex numbers, its projective dual variety is the closure in the Zariski topology of the projective hyperplanes containing the embedded tangent space at some smooth point of $Y$. The projective biduality theorem recovers $Y$ as $(Y^{\ast }{)}^{\ast }$: the family of tangent projective hyperplanes determines the original projective variety.

For a nondegenerate smooth projective curve $C\subset {\mathbb{P}}^n$ with $n\ge 2$, the incidence pairs $(p,H)$ with $T_{p}C\subset H$ form a projective bundle over $C$ with fiber ${\mathbb{P}}^{n-2}$, hence dimension $n-1$. The general contact locus is a projective subspace of $C$; since $C$ is not a projective line, it is a single point. Thus $C^{\ast }$ is an irreducible projective hypersurface, and a general member represents one ordinary tangency. For a smooth plane conic, the tangent-line equation gives another smooth plane conic in the dual projective plane.

The ambient-dimension hypothesis excludes $C={\mathbb{P}}^1\subset {\mathbb{P}}^1$: no projective hyperplane contains its whole tangent space, so its dual is empty. The stated reflexivity and general tangency assertions also use characteristic zero; inseparable behavior in positive characteristic requires additional hypotheses.^[ref]