Rank conjecture for the restricted ramification divisor map

Let $X$ be a compact Riemann surface, let $L$ and $J$ be line bundles, and let $d$ denote the parameter appearing in the restricted ramification divisor map $\rho$. Choose nonzero sections

$$0\neq i_1\in H^0\big(\operatorname{Hom}(L,J)\big),\qquad 0\neq i_2\in H^0\big(\operatorname{Hom}(L,J^{-1})\big).$$

Rank conjecture. The rank of the complex derivative of $\rho$ at $(i_1,i_2)$ satisfies:

• If $J^2\neq\mathcal O_X$, then

$$\operatorname{rank}\rho_{*(i_1,i_2)}=2d-1,$$

and

$$\operatorname{Ker}\rho_{*(i_1,i_2)}=\mathbb C(i_1,-i_2).$$

• If $J^2=\mathcal O_X$ and $i_1\neq\lambda i_2$ for every $\lambda\in\mathbb C$, then

$$\operatorname{rank}\rho_{*(i_1,i_2)}=2d-3,$$

and

$$\operatorname{Ker}\rho_{*(i_1,i_2)}=\mathbb C(i_1,-i_2)\oplus\mathbb C(i_2,0)\oplus\mathbb C(0,i_1).$$

The conjectured rank would yield a subset of the complete linear system with larger Hausdorff dimension whose divisors are represented by reducible metrics; the source does not report a resolution of this conjecture.