# Curves having only one linear system realizing its gonality

$$\DeclareMathOperator\gon{gon}$$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!

• I don't think so - having a single bundle realizing the gonality should be the generic situation, and should place almost no restrictions on the degree and genus. Commented May 4, 2022 at 19:35

## 1 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 $$g \leq 2d-2$$.

• Just for fun, let me add that the case $g=2d-2$ is indeed an exception: Castelnuovo showed that a generic curve of genus $2d-2$ admits $C_{d-1}\ g^1_{d}$, where $C_{d-1}$ is the Catalan number $\frac{1}{d}\binom{2d-2}{d-1}$.
– abx
Commented May 6, 2022 at 9:32
• Re the comment by @abx - for the genus 6 case see mathoverflow.net/questions/295624/… Commented May 14, 2022 at 17:07