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$\DeclareMathOperator\gon{gon}$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!

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    $\begingroup$ 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. $\endgroup$
    – Will Sawin
    Commented May 4, 2022 at 19:35

1 Answer 1

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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 $g \leq 2d-2$.

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    $\begingroup$ 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} $. $\endgroup$
    – abx
    Commented May 6, 2022 at 9:32
  • $\begingroup$ Re the comment by @abx - for the genus 6 case see mathoverflow.net/questions/295624/… $\endgroup$ Commented May 14, 2022 at 17:07

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