# Natural model for genus $6$ curves

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.

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.