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, 



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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.



Natural model for genus $6$ curves
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.


