Fresh out of the arXiv press is the remarkable result of Manjul Bhargava saying that most hyperelliptic curves over $\mathbf{Q}$ have no rational points. Don Zagier suggests the paraphrase : *Most hyperelliptic curves are pointless*.

Crucial to the precise mathematical formulation of the statement is a kind of canonical equation for hyperelliptic curves (of a fixed genus) permitting one to define the density of those which have no rational points.

What is the corresponding statement for *all* curves over $\mathbf{Q}$ ?

**Addendum** (2013/09/28) A very nice introduction to the work of Bhargava can be found in How many rational points does a random curve have? by Wei Ho.

a fortiori, the statement "most hyperelliptic curves are pointless," taken in any reasonable sense? This refers to your comment following JSE's answer. – Vesselin Dimitrov Sep 28 '13 at 14:25