= Determinant description of the theta divisor
Choose an effective <Cartier divisor> $E$ of degree at least $g$ on a smooth projective genus-$g$ <algebraic curve>. For a family $M$ of degree-$g-1$ <line bundles>, the evaluation map from $\pi_*M(E)$ to $\pi_*(M(E)|_E)$ is between <vector bundles> of the same rank. Its <determinant> vanishes exactly when $H^0(C,M)$ is nonzero. Changing $E$ changes the presentation by cancelling an invertible determinant factor; the zero <Cartier divisor> is the <theta divisor>. At a point with $h^0(M)=h^1(M)=1$, the derivative is the cup-product map $H^1(\mathcal O_C)\to\operatorname{Hom}(H^0(M),H^1(M))$. By <Serre duality> it is nonzero, since the product of nonzero sections of $M$ and $K_C\otimes M^{-1}$ is nonzero. Hence this <Cartier divisor> is generically reduced.
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