= Projective dual variety
{title2=$Y^*\subset\mathbb P(V^*)$}
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 $(Y^*)^*$: the family of tangent <projective hyperplanes> determines the original <projective variety>.
For a nondegenerate smooth <projective curve> $C\subset\mathbb P^n$ with $n\geq2$, the incidence pairs $(p,H)$ with $T_pC\subset H$ 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 $C^*$ 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.<Projectively Dual Varieties by Evgueni Tevelev>[section1.2, Theorems1.7,1.10,1.12 and Example1.13]{ref}
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