Exact rank of the derivative of the restricted ramification divisor map

Determine the exact rank of the complex derivative of the restricted ramification divisor map ρ at a pair (i1, i2), where i1 ∈ H^0(X, Hom(L, J)) and i2 ∈ H^0(X, Hom(L, J^{-1})). Specifically, prove that rank(ρ_{*(i1,i2)}) = 2d − 1 when J^2 ≠ O_X, and rank(ρ_{*(i1,i2)}) = 2d − 3 when J^2 = O_X and i1 is not a scalar multiple of i2, where d = dim_C H^0(X, Hom(L, J)) = dim_C H^0(X, Hom(L, J^{-1})). Equivalently, characterize Ker(ρ_{*(i1,i2)}) as C·(i1, −i2) in the first case, and as C·(i1, −i2) ⊕ C·(i2, 0) ⊕ C·(0, i1) in the second case.

Background

The restricted ramification divisor map ρ, introduced in Proposition 5.1, sends pairs of sections (i1, i2) defining an embedding L → J ⊕ J{-1} to the second fundamental form β* ∈ H0(X, Hom(L, L{-1} ⊗ K_X)). Its derivative controls how divisors represented by reducible metrics vary with deformations of the embedding.

Lemma 5.2 provides lower bounds on rank(ρ_{*(i1,i2)}), but does not establish the exact rank. The conjecture gives precise values tied to whether J is 2-torsion and whether i1 and i2 are co-linear, which would sharpen dimension estimates for subsets of linear systems consisting of divisors representable by reducible metrics.

Proving this conjecture would improve the Hausdorff-dimension bounds for the arcwise connected Borel subsets of |D| consisting of divisors represented by reducible metrics, strengthening Theorems 1.2 and 1.3.

References

Hereafter, we present a conjecture regarding the rank of the complex derivative of the restricted ramification divisor map. Assume that $0\neq i_{1}\in H0\big({\rm Hom}(L,\,J)\big)$ and $0\neq i_{2}\in H0\big({\rm Hom}(L,\,J{-1})\big)$, then the rank of the complex derivative of the restricted ramification divisor map $\rho$ at $(i_{1},i_{2})$ is as follows: \begin{itemize} \item If $J{2}\neq{\mathcal O}X$, then $\operatorname{rank} \rho{(i_{1},i_{2})}= 2d-1$. In particular, $\operatorname{Ker}\rho_{(i_{1},i_{2})}= \mathbb{C}(i_{1},-i_{2})$. \n\item If $J{2}={\mathcal O}X$ and $i{1}\neq \lambda i_{2}$ for any $\lambda\in\mathbb{C}$, then $\operatorname{rank} \rho_{(i_{1},i_{2})}= 2d-3$. In particular, $\operatorname{Ker}\rho_{(i_{1},i_{2})}= \mathbb{C}(i_{1},-i_{2})\oplus \mathbb{C}(i_{2},0)\oplus \mathbb{C}(0,i_{1})$. \end{itemize}

— Existence and non-uniqueness of cone spherical metrics with prescribed singularities on a compact Riemann surface with positive genus  (2405.12673 - Feng et al., 2024) in Conjecture 6.2, Section 6.2