A hyperelliptic curve can be understood as the set of points satisfying an equation of the form
$$\displaystyle z^2 = f(x,y),$$
where $f(x,y)$ is a binary form of degree $d = 2g+2$. In this case, $g$ can be shown to be the genus of the curve.


It turns out that when $g=1$, the curve is an elliptic curve and in fact all curves of genus 1 curves over $\mathbb{C}$. However, this is not the case for higher genus; in particular, most curves of genus $g \geq 2$ are not hyperelliptic,


My questions are:


1) Is there a description, for a fixed genus $g$, an infinite family of non-hyperelliptic curves of genus $g$? By that I mean an equation similar to the one above that characterizes all curves in the family.


2) Bhargava recently proved that a proportion tending to 100 percent as $g \rightarrow \infty$ of hyperelliptic curves have no rational points and morever, the failure is accounted for by the Brauer-Manin obstruction. I believe that it is conjectured that all curves either satisfy the Hasse principle or if it fails the Hasse principle, the failure is accounted for by the Brauer-Manin obstruction. Since 'most' genus $g \geq 2$ curves are not hyperelliptic, Bhargava's theorem does not account for the situation for all genus $g$ curves. What is the best known result on the density of general curves of fixed genus $g$ with no rational points?


Thanks for any insight.



Non-hyperelliptic curves of genus at least two
Perhaps a couple of things should be pointed out.


1) All curves of genus $g=2$ are hyperelliptic. 


2) Your equation $z^2=f(x, y)$ is not the equation of an hyperelliptic curve. If you take $f(x, y)$ to be a binary form of degree $2g+2$ then the (projective) equation of the curve is
$$ z^2 y^{2g} = f(x, y),$$
where the discriminant $\Delta (f) \neq 0$. 


3) A natural generalization of hyperelliptic curves are superelliptic curves, i.e, curves which have affine equation $y^n=f(x)$. 


Perhaps, one can attempt to investigate what Bhargava type statements mean for superelliptic curves first or at least fix the "type" of equation. For example, take non-hyperelliptic curves of genus 3, which are ternary quartics. 


