Let $g\geq 1$. I would like to write down (for all $g$) a smooth projective geometrically connected curve $X$ over $\mathbf{Q}$ of genus $g$ with precisely one rational point.
Is this possible?
For which $g$ is this possible?
I think for $g=1$ this is possible. I just don't know an explicit equation, but I should be able to find it. (We just write down an elliptic curve without torsion of rank zero over $\mathbf{Q}$.)
For $g\geq 2$ things get more complicated for me.
I would really like the curve to be of gonality at least $4$, but I'll think about that later.