Let $X$ be a smooth, plane projective curve of degree $6$ and genus $10$ (over complex numbers).

**Question :** Is it possible that there exists a special divisor $\Delta$ of degree $10$ on $X$ such that it has exactly $5$ independent sections?

**Observations :** $(1)$ From Clifford's Theorem, we have $h^0(\mathcal O_X(\Delta)) -1 =4 \leq \frac {\text{deg}(\Delta)}{2}=5$ and therefore it says this can happen.

$(2)$ If the curve $X$ is general, then from Theorem-$A$ of this paper, we have this is not possible.(Please correct me if I am wrong). This is also true from Brill-Noether theorem for general curves.

Can we say anything more in this situation when the curve $X$ is not necessarily general and if we know what $\Delta^2, K_X$ are?

Any insight from anyone is welcome