Let $C$ be a smooth projective curve of genus $g$ over the complex field, embedded into $\mathbb{P}^r$ via a line bundle $L$ of degree $n>>0$. Let $D$ be a divisor of degree $d$ with $h^0(C,D)=s$. Let's write the basis of $H^0(C,D)$ as $\{x_i\}$ and that of $H^0(C,L-D)$ as $\{y_j\}$. Let $M=(x_iy_j)_{1\leq i\leq s,1\leq j\leq r+1-d}$ be the $s\times (r+1-d)$ matrix whose entries are defined by the multiplication $H^0(C,D)\otimes H^0(C,L-D)\to H^0(C,L)$. How can we calculate the dimension of the variety defined by the vanishing of the maximal minors $I_s(M)$?

The only thing I know is that $I_s(M)$ vanishes on $(s-2)$-th order secant variety of the curve. In other words, the variety contains the $(s-2)$-th order secant variety. But generally they are not equal.



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