The number of arcs in ${\mathbb{F}}_q^2$ of a given cardinality

A subset of ${\mathbb{F}}_q^2$ is called an arc if it does not contain three collinear points. We show that there are at most $\left(\genfrac{}{}{0px}{}{(1+o(1))q}{m}\right)$ arcs of size $m\gg q^{1\mathrm{/}2}(\log q{)}^{3\mathrm{/}2}$, nearly matching a trivial lower bound $\left(\genfrac{}{}{0px}{}{q}{m}\right)$. This was previously known to hold for $m\gg q^{2\mathrm{/}3}(\log q{)}^3$, due to Bhowmick and Roche-Newton. The lower bound on $m$ is best possible up to a logarithmic factor.