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On the canonical model $C_{10} \subset \mathbb{P}^5$ of your curve 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.

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.

On the canonical model $C_{10} \subset \mathbb{P}^5$ of your curve 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 $g_1^4$ 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.

On the canonical model $C_{10} \subset \mathbb{P}^5$ of your curve 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.

On the canonical model $C_{10} \subset \mathbb{P}^5$ of your curve 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 over which the $C_{10}$ lie.

Projecting the curve $C_{10}$ from one set of four coplanar points belonging to one of the $g_4^1$, we obtain a plane sextic $X_6 \subset \mathbb{P}^2$ with $4$ nodes birational to it. The five $g_1^4$ 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.

On the canonical model $C_{10} \subset \mathbb{P}^5$ of your curve 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 $g_1^4$ 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.