Question about plane quintic Let the canonical curve $C$ $\subset$ $\mathbb{P}^5$ lie on the Veronese surface. How to see that $C$ is a smooth plane quintic?
 A: If it's on a Veronese surface, the Veronese embedding exhibits it as isomorphic to a smooth plane curve. Since it has degree 10, can't we simply conclude that it is the image of a smooth plane quintic? 
A: This is indeed included in the content of Enriques-Petri theorem, which you may find it in "Principles of Algebraic geometry" by Griffiths-Harris, page 535. Its proof is here (in case you have access to SpringerLink)
A: This has to do with a theorem of Enriques and Petri. For a nonhyperelliptic canonical curve $C$ of genus $g\geq 3$, there are two possibilities: its ideal sheaf may or may not be generated by quadrics, and the theorem says that the second possibility occurs if and only if $C$ is contained in a surface of (minimal) degree $g-2$, if and only if $C$ is either trigonal or (for $g=6$) isomorphic to a smooth plane quintic. 
Now, our $C$ is embedded in the Veronese surface (of degree $4=g-2$), and the gonality of $C$ is $[(6+3)/2]=4\neq 3$, so $C$ is isomorphic to a smooth plane quintic.
You can look up at ACGH's Geometry of Algebraic Curves (Volume I, around p. 244, but also p. 209) for the details.
