I'm trying to prove that given 3 disjoint lines in $\mathbb{P}^{3}$ there exists a non-singular quadric containing them. The exercise is from the following link:


I think you have to use page 19 of the source and make an argument based on dimension, but I'm really not sure. Any hint/help appreciated.

  • 2
    $\begingroup$ Hint: a quadric in P^3 has 10 coefficients. To contain a given line imposes 3 conditions. $\endgroup$ – user64687 Mar 11 '14 at 9:34
  • $\begingroup$ To flesh out the hint: $10-3 \cdot 3 = 1>0$. $\endgroup$ – user64687 Mar 11 '14 at 20:31
  • $\begingroup$ @AsalBeagDubh: Just to be sure, your $1$ is an "affine" $1$, right? I mean, the wanted quadric should be the point $\mathbb P(V)$, where $V$ is a $1$-dimensional vector space. $\endgroup$ – Brenin Mar 12 '14 at 11:59
  • $\begingroup$ Dear @Brenin: yes, my 1 is the dimension of the space of forms that vanish on the 3 lines. $\endgroup$ – user64687 Mar 12 '14 at 13:05

Your Answer

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

Browse other questions tagged or ask your own question.