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Let the canonical curve $C$ $\subset$ $\mathbb{P}^5$ lie on the Veronese surface. How to see that $C$ is a smooth plane quintic?

Leo
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3 Answers3

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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?

Tony
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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)

Ehsan M. Kermani
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  • The problem is that the proof of Enriques-Petri theorem in Griffiths-Harris uses this fact. – Leo Feb 26 '13 at 09:50
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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.

Brenin
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  • The same problem. We can't refer to this theorem, because its proof uses the fact about plane quintic. – Leo Feb 26 '13 at 09:53
  • Can you please explain to me what you mean exactly? It is fun because usually a proof does not use the claim to prove the claim :) – Brenin Feb 27 '13 at 21:27