13
$\begingroup$

We know from Hartshorne's "Algebraic geometry" that if a curve $C$ is smooth then there exists a closed immersion of $C$ into $\mathbb{P}^3$. Does this still hold if $C$ is not generically reduced or singular at finitely many points?

$\endgroup$
2
  • 5
    $\begingroup$ No. You always have a lower bound based on the dimension of the tangent space at any point. $\endgroup$ Jan 21 '14 at 10:31
  • 1
    $\begingroup$ See also mathoverflow.net/a/14405 for smooth varieties over finite fields. $\endgroup$
    – Cantlog
    Jan 21 '14 at 14:13
25
$\begingroup$

No. If a curve embeds into $\mathbb{P}^3$, its tangent space at every point has dimension $\leq 3$. This is a strong restriction on the possible singularities. For instance, a curve locally isomorphic to the union of the lines $x=y=z=0$, $x=y=t=0$, $x=z=t=0$, $y=z=t=0$ in $\mathbb{A}^4$ cannot be embedded in $\mathbb{P}^3$.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.