Rigid special presentation conjecture for higher Brill–Noether loci

Let $i\geqslant 4$ and $g>i^2-i+1$ be integers. Let $C$ be a curve of genus $g$ with general moduli, and let $d$ satisfy

$$2g-2\leqslant d<2g-2-i+\frac{g-\epsilon}{i-1},$$

where $\epsilon\in\{0,1\}$ is chosen so that $d+g-(i-1)k_i\equiv\epsilon\pmod{2}$. Write $k_i$ for the integer used in the definition of the Brill–Noether locus $\widetilde{B_C^{k_i}}(d)$, $D_{i-1}$ for the divisor associated with a general divisor $D\in C^{(i-1)}$, and let $\mathcal V_d^{2g-1-i,1,i-1}$, $\Lambda_{i-1}$, $i(\mathcal F)$ and $s(\mathcal F)$ denote the loci and invariants defined in the paper.

Rigid special presentation conjecture. There exists an irreducible, regular component $\mathcal B\subseteq\widetilde{B_C^{k_i}}(d)$ such that

$$\mathcal B=\overline{\mathcal V_d^{2g-1-i,1,i-1}}.$$

Moreover, a general point $[\mathcal F]\in\mathcal B$ is stable, satisfies $i(\mathcal F)=i$ and $s(\mathcal F)\geqslant g-(i-1)k_i-\epsilon>0$, and has a rigid special presentation

$$0\to N\to\mathcal F\to\omega_C(-D_{i-1})\to 0,$$

where $D\in C^{(i-1)}$ is general, $N\in\operatorname{Pic}^{d-2g+1+i}(C)$ is general (special and non-effective), and $\mathcal F=\mathcal F_v$ for general $v\in\Lambda_{i-1}\subseteq\operatorname{Ext}^1(N,\omega_C(-D))$ in a good component.

The conjecture proposes existence and regularity of a specified component of the higher-rank Brill–Noether locus, together with a rigid special presentation of its general stable bundle. It is presented as a less ambitious extension of the paper’s existence results for $i\geqslant4$; the supplied source gives no resolution.