$\DeclareMathOperator\gon{gon}$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 for a reference.
I am interested in the converse of the above property. To be precise,
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?)
Any comments are welcome!
Best Answer
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, 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 $g \leq 2d-2$.