# On emptiness of certain $G^r_d(X)$ on a smooth plane curve

Let $$X$$ be a smooth plane projective curve of degree $$6$$ and genus $$10$$ (over complex numbers). Then my question is the following :

Question : Is it possible that there exists a special divisor $$D$$ of degree $$9$$ on $$X$$ admitting exactly $$4$$ independent sections?

Observations : $$(i)$$ From Clifford's theorem: we have, $$h^0(\mathcal O_X(D)) -1 =3\leq \frac{\text{deg}(D)}{2} =4.5$$. Therefore, this theorem says that such divisor may exists on $$X$$.

$$(ii)$$ Note that, if such a divisor exists on $$X$$, then it belongs to $$G^3_9(X)$$. Since, $$\rho(10,3,9)=-6<0$$, we can't guarantee the non-emptiness of $$G^3_9(X)$$.

Any insight or remark from anyone is welcome.

No, this is not possible. Let $$H$$ be the divisor of a line. Use the base point free pencil trick to get an exact sequence $$0\rightarrow H^0(D-H)\rightarrow H^0(D)^2\rightarrow H^0(D+H)$$ Since $$\deg(D+H)=15$$, we have by Riemann-Roch $$h^0(D+H)\leq 7$$, hence $$h^0(D-H)\geq 1$$. Thus $$D\equiv H+E$$ with $$E\geq 0$$ of degree 3. But by Serre duality, $$h^0(H+E)=h^0(2H-E)$$. Now $$E$$ imposes independent conditions on conics, so $$h^0(2H-E)=h^0(2H)-3=3$$, contradicting $$h^0(D)=4$$.

• The system $\lvert 2H\rvert$ is cut down by conics in $\mathbb{P}^2$, so $\lvert 2H-E\rvert$ is cut down by conics passing through $E$. Now any length 3 subscheme of $\mathbb{P}^2$ imposes 3 conditions on conics — see Lemma 0.1.1 in Beltrametti-Sommese Zero cycles and $k$-th order embeddings of smooth projective surfaces, INDAM Symposia Mathematica, 32,(1992), 33-48.
– abx
Commented Mar 16, 2021 at 20:27
• What is your question if $\deg(D)=8$? You can have $h^0=3$, and certainly not 4 — the exact sequence would give you a $g^1_2$.
– abx
Commented Mar 17, 2021 at 16:53
• $E$ cannot be a $g^1_2$, $X$ would be hyperelliptic.
– abx
Commented Mar 17, 2021 at 17:37
• If it is 4, $D+p$ for any $p\in X$ contradicts my previous answer.
– abx
Commented Mar 17, 2021 at 18:26
• Yes, this is what I mean.
– abx
Commented Mar 17, 2021 at 19:27