Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I've been having problem actually restricting a Line bundle $L$ defined on some projective space $\mathbb C \mathbb P^{N-1}$ to a subvariety $X$.

I know how to do this on an abstract level, but actually computing what's going on, seems quite mysterious.

From Fulton's "Intersection Theory" I have

$c_1(L) \cap [X] = [C]$ where $C$ is the divisor corresponding to $\mathcal O_X(C) \simeq L\vert_X$

Now, I have $c_1(L)$ given by $-N[H]$ where $[H]$ denotes the hyperplane class in $\mathbb C \mathbb P^{N-1}$. I also have some polynomial $P$ whose zero locus defines $X$. I even know $c_1(TX) = 0$ and have computed that if $X$ is taken to be a divisor in $\mathbb C \mathbb P^{N-1}$, the corresponding line bundle would satisfy $c_1(\mathcal{O}(X)) = N[H]$ (Not quite sure yet if this helps).

But, what I really would like to know is $c_1(L\vert_X)$?

My attempts so far have been to find an actual section $s$ of $L$, use the equation for the zero locus and actually intersect that with $X$. However, finding such a section has proven difficult.

How would one normally go about this? Am I on the right track? Any help is highly appreciated!

share|cite|improve this question
up vote 4 down vote accepted

In general (i.e. for restricting line bundles from a variety to a subvariety) one has $c_1(L_{\vert X}) = c_1(L) \cdot X$ (the intersection, thought of as a divisor on $X$).

In your case one can be more explicit, since $L = \mathcal O(d) = \mathcal O(1)^{\otimes d}$ for some $d$ (all line bundles on projective space are of this form). Thus, if we let $H_X$ denote a generic hyperplane section of $X$, then $c_1(L_{\vert X}) = d H_X$.

share|cite|improve this answer
Thanks very much! I was kind of hoping that the first chern class vanishes. I guess the only case where this happens is if $d=0$, or are there other possibilities? – Mike Jun 29 '12 at 5:20
@Mike: Dear Mike, No, there are no possibilities besides the trivial line bundle ($d = 0$). A basic fact is that $H_X$ gives a non-trivial class in $H^2$ of $X$. (The so-called Kahler class.) Regards, – Matt E Jun 29 '12 at 12:35
Thanks again :) – Mike Jun 29 '12 at 16:52

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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