= Torelli theorem
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A <smooth projective curve> over the <complex numbers> is determined up to isomorphism by its <Jacobian variety> together with its canonical principal <polarization of a complex torus>. For genus at least three, the <branch locus of the theta Gauss map> recovers the canonical curve by <projective biduality theorem>, or recovers the canonical <rational normal curve> and the branch points of a <hyperelliptic curve>. In genus two the <theta divisor> itself is the curve; in genus one the <Jacobian variety> is the curve after choosing an origin. The polarization is essential data in this statement.
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