By Brill-Noether theory, the generic genus $6$ curve is birational to a sextic plane curve in $\mathbb{P}^2$. I was wondering if there is a direct/natural construction of this birational map. In other words,

do we know a divisor $D$ on the curve inducing this map? Is it given by a complete system, or by some natural subspace thereof?

This question came up while talking with other people, we were trying to recollect the beautiful picture relating genus $6$ curves, del Pezzo surface, and plane sextics.