Tangent cone to a theta divisor
= Tangent cone to a theta divisor
{title2=$\det(\xi\smile-)=0$}
The <tangent cone to a theta divisor> at a degree-$g-1$ <line bundle> $L$ is cut out in $H^1(\mathcal O_X)$ by the <determinant> of $\xi\smile-:H^0(L)\to H^1(L)$. Under <Serre duality> its <matrix> entries are $\langle\xi,s_i t_j\rangle$ for bases of $H^0(L)$ and $H^0(K_X\otimes L^{-1})$. For $h^0(L)=1$, its tangent hyperplane is annihilated by the nonzero <holomorphic differential form> $s_1t_1$.