= Schur complement presentation of a theta singularity
{c}
{title2=$\det A=\text{unit}\cdot\det M,\quad M(0)=0$}
For a square evaluation matrix representing a degree-$g-1$ <line bundle> $L$, remove an invertible minor of size $N-h^0(L)$. <Holomorphic> row and column operations leave a square matrix $M$ of size $r=h^0(L)$ with zero constant term. Its linear part in a tangent direction $\xi\in H^1(\mathcal O_X)$ is $\xi\smile-:H^0(L)\to H^1(L)$. The <invertible cup-product direction for a line bundle on a curve> makes the leading degree-$r$ <determinant> nonzero, establishing order exactly $r$ rather than just a lower bound.
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