Natural model for genus $6$ curves

Question

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.

Answer 1

On the canonical model $C_{10} \subset \mathbb{P}^5$ of a smooth curve of genus six there exist five special $g_4^1$, obtained as follows: we take an arbitrary point on the curve, and the remaining three are determined so that the four points lie on a plane. We can also see these five $g^1_4$ as cut on the $C_{10}$ by the five pencils of conics of the Del Pezzo surface containing it.

Projecting the canonical model $C_{10}$ from one set of four coplanar points belonging to one of the $g_4^1$, we obtain a birational plane model $X_6 \subset \mathbb{P}^2$, namely a sextic with four nodes. The five $g_4^1$ are cut on $X_{6}$ by the lines through a node and by the conics through the four nodes.

For more details you can look at the classical book

H. F. Baker, Principles of Geometry, Vol. 5, Note II page 97.