Let $X$ be a smooth projective variety over $\mathbb{C}$. Is there an ample line bundle $L$ such that $L^{\otimes m}$ is very ample, but $L^{\otimes(m+1)}$ is not very ample?

I expect such an $L$ to exist, though I have not been able to construct one. Of course, there is a threshold $M > 0$ such that $L^{\otimes m}$ is very ample for all $m \geq M$ (this is Matsusaka's theorem), but my interest is in the interval of $m$'s before this threshold is reached.

If this is true or false under different hypotheses (say, over a field of positive characteristic or by removing/weakening smoothness), I'd be interested to hear it!

up vote 16 down vote accepted

Take a smooth quartic curve $C$ in the projective plane with a point $P\in C$ such that the tangent line at $P$ meets it only in $P$ (four times). For example, you may take $C$ to be defined by $x^4+y^4+yz^3=0$ and $P=(0,0,1)$. Then $\mathcal{O}_C(4P)=\mathcal{O}_C(1)=K_C$ and thus very ample. But $\mathcal{O}_C(5P)=K_C+P$ always has $P$ as a base point and thus not very ample.

  • Very nice example! – msteve Oct 30 '15 at 18:38

Your Answer

 

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

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