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.