Before I start, let me make a note on terminology. Curves are always smooth projective connected curves over an algebraically closed field of characteristic zero.


Let $\mathcal C$ be a class of curves. We say that $\mathcal C$ is dominant if, for all curves $X$, there exists a curve $Y$ in $\mathcal C$ and a finite morphism $Y\to X$.


Bogomolov and Tschinkel proved that the class of hyperelliptic curves and their unramified covers is dominant. Manin proved that the class of modular curves $X(n)$ and their unramified covers is dominant. Both proofs rely on Abhyankar's Lemma.


Let $k\geq 2$ be an integer. Let $\mathcal C\_k$ be the class of $k$-gonal morphisms, i.e., the class of curves for which the gonality equals $k$.


Q1. Is $\mathcal{C}\_2$ dominant?


Q2. Is $\mathcal{C}\_k$ dominant?


Q3. Is $\cup\_{2 \leq j \leq k} \mathcal{C}\_j$ dominant if $k>>0$?


Let me repeat this in words. Let $X$ be a curve. Does there exist a $k$-gonal curve $Y$ and a finite morphism $Y\to X$?


I'm mainly interested in the case $k=2$. In this case, it suffices to answer the following question.


Q1b. Let $X$ be a curve. Does $\mathbf{P}^1$ admit a closed immersion into the symmetric product $X^{(2)}$?



Is the class of $k$-gonal curves dominant
The points of $X^{(2)}$ correspond to divisors of degree 2; two points of $X^{(2)}$ are in the image of a morphism $\mathbb P^1 \to X^{(2)}$ if and only if the two corresponding divisors are linearly equivalent. Hence there is a non-trivial morphism $\mathbb P^1 \to X^{(2)}$ if and only if $X$ is hyperelliptic.


More generally, if $Y \to X$ is non-constant morphism and $Y$ is $k$-gonal, then $X$ is also $k$-gonal.


