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I am a bit confused about a statement in Exercise IV.6.1 of Hartshorne. The exercise is:

6.1 A rational curve of degree $4$ in $\mathbb P^3$ is contained in a unique quadric surface $Q$, and $Q$ is necessarily nonsingular.

I haven't tried to prove this but I also know that in classical geometry the Viviani curve is the intersection of two quadrics (cylinder and sphere) and is a rational space curve of degree 4. So this confuses me a bit on why $Q$ is unique or is there something I am missing for this quartic curve?

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  • $\begingroup$ Hartshorne assumes that all curves are nonsingular (check the introduction to chapter IV), while the Viviani curve is singular at its self-intersection point. $\endgroup$
    – KReiser
    Commented Oct 29, 2021 at 5:48
  • $\begingroup$ I see. Thanks, that clarifies it. I guess it's not a good idea to skip sections (esp. the first section of the chapter) when reading Hartshorne. $\endgroup$
    – quantum
    Commented Oct 29, 2021 at 5:54
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    $\begingroup$ If you'd like, I can record that as below so that the question can be marked as answered. $\endgroup$
    – KReiser
    Commented Oct 29, 2021 at 5:55
  • $\begingroup$ Yes please. Thank you again. $\endgroup$
    – quantum
    Commented Oct 29, 2021 at 5:56

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Hartshorne assumes that all curves are nonsingular (check the introduction to chapter IV), while the Viviani curve is singular at its self-intersection point.

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  • $\begingroup$ just a minor correction. He puts it in the first section of the chapter and not in the introduction of the chapter (which is the one page before the first section). He starts the section with "In this chapter we will use the word curve to mean..." $\endgroup$
    – quantum
    Commented Oct 29, 2021 at 6:08

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