# Special divisors on smooth plane curves

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

Such a divisor cannot exist. Let $$H$$ be the divisor of a line. By the base-point free pencil trick, we have an exact sequence $$0\rightarrow H^0(\Delta -H)\rightarrow H^0(\Delta)^2\rightarrow H^0(\Delta +H)\,;$$since $$\deg (\Delta +H)=16$$, we have $$h^0(\Delta +H)\leq 8$$, hence $$h^0(\Delta -H)\geq 2$$. Then $$D:=\Delta -H$$ is a $$g^1_4$$, thus base-point free since $$X$$ is not trigonal. Now applying again the base-point free pencil trick, we get an exact sequence $$0\rightarrow H^0(H-D)\rightarrow H^0(H)^2\rightarrow H^0(\Delta )$$ which tells us $$h^0(H-D)>0$$. Thus $$D\equiv H-p-q$$, for some points $$p,q$$ in $$X$$; but this implies $$h^0(D)=1$$, a contradiction.

• From $\text{deg}(\Delta+H) =16$ aren't we getting by Clifford's theorem that $h^0(\Delta+H) -1 \leq \frac{16}{2}=8$, i.e $h^0(\Delta+H) \leq 9$ and hence $h^0(\Delta-H) \geq 1$ ? Please correct me if this is wrong.
– User
Commented Mar 8, 2021 at 13:34
• This is correct, but what do you do with that? For my argument I need $h^0(\Delta -H)\geq 2$.
– abx
Commented Mar 8, 2021 at 14:03
• can we say something if we know that $\Delta^2=8$?
– User
Commented Mar 8, 2021 at 15:28
• What do you call $\Delta^2$?
– abx
Commented Mar 8, 2021 at 15:56
• by $\Delta^2$ I mean its self intersection number.
– User
Commented Mar 8, 2021 at 15:58