# rational quartic in $\mathbb P^3$, confusion

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?

• Hartshorne assumes that all curves are nonsingular (check the introduction to chapter IV), while the Viviani curve is singular at its self-intersection point. Commented Oct 29, 2021 at 5:48
• 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. Commented Oct 29, 2021 at 5:54
• If you'd like, I can record that as below so that the question can be marked as answered. Commented Oct 29, 2021 at 5:55
• Yes please. Thank you again. Commented Oct 29, 2021 at 5:56