$\DeclareMathOperator\gon{gon}$Let $C$ be a smooth irreducible projective curve defined over complex numbers. Recall that the gonality of $C$, $\gon(C)$, is defined to be the minimal possible degree of a dominant morphism $C\to\mathbb P^1$.


I am interested in curves $C$ such that there is only one linear system $g\_d^1$ satisfying $d=\gon(C)$. Examples of such curves are hyperelliptic curves of genus $\geq 2$ or trigonal curves of genus $\geq 5$, or more generally, $p$-gonal curves of genus $\geq (p-1)^2+1$ for $p$ a prime number. See, for instance, [Mathew - Hyperelliptic and trigonal curves](https://amathew.wordpress.com/2013/05/22/hyperelliptic-and-trigonal-curves) for a reference.


I am interested in the converse of the above property. To be precise,



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> Let $C$ be a smooth irreducible projective smooth curve defined over $\mathbb C$ such that there is only one linear system $g\_d^1$ satisfying $d=\gon(C)$. What can we say aboout $d$, $g=\operatorname{genus}(C)$? Do we have any restrictions for relations of $d$ and $g$ (e.g., some nontrivial inequalities not predicted by Brill–Noether–Petri theory?)
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> 
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Any comments are welcome!



Curves having only one linear system realizing its gonality
A generic $d$-gonal curve of genus $g$ satisfies this property unless $g \leq 2d-2$. So the only possible restriction for curves with this property is $g \geq 2d-1$, which I believe follows from Brill-Noether-Petri theory for $d>2$.


Indeed, let $C$ be a generic such curve and $\pi : C \to \mathbb P^1$ the projection map. Let $L$ be a line bundle realizing its gonality, i.e. a line bundle of degree $d$ with a two-dimensional space of global sections, and consider $\pi\_\* L$ as a vector bundle on $\mathbb P^1$.


Since $\pi\_\* L$ has a two-dimensional space of global sections, we must have $\pi\_\* L \cong \mathcal O\_{\mathbb P^1}(1) \oplus W$ for $W$ a vector bundle of rank $d-1$ or $\mathcal O\_{\mathbb P^1} \oplus \mathcal O\_{\mathbb P^1} \oplus V$ for $V$ a vector bundle of rank $d-2$.


In the first case, $L(-1)$ has a nontrivial global section and since $\mathcal O\_C(1)$ and $L$ both have degree $d$, so $L(-1)$ has degree $0$ and thus $L \cong \mathcal O\_C(1)$.


In the second case, we have


$$ \dim H^1(\mathbb P^1, \operatorname{Hom} ( \pi\_\* L, \pi\_\* L ) ) \leq \dim H^1(\mathbb P^1, \operatorname{Hom} (\mathcal O\_{\mathbb P^1} \oplus \mathcal O\_{\mathbb P^1}, V ) ) = 2 \dim H^1(\mathbb P^1, V) = 2 \dim H^1(\mathbb P^1, \pi\_\* L) = 2 \dim H^1(C, L) = 2 (g+1 - d) $$


By Theorem 1.2 of [A Refined Brill-Noether Theory over Hurwitz Spaces by Hannah Larsen](https://arxiv.org/pdf/1907.08597.pdf), the dimension of the space of $L$ with $\pi\_\* L$ of this form is exactly $g - 2 (g+1-d)$ and there are no such $L$ unless this dimension is nonnegative, i.e. unless $g\leq 2d-2$.


So there are no such $L$ except $\mathcal O\_C(1)$ unless $g \leq 2d-2$.


