3
$\begingroup$

Given a double cover $\pi: X\rightarrow \mathbb{P}^2$ of the projective plane by choosing a square root $S$ of $O_{\mathbb{P}^2}(Q)$, where $Q$ is a quartic in the plane.

Choose a closed point $p\in X$, then we have the exact sequence:

$$0\rightarrow I_p\otimes O_X(B) \rightarrow O_X(B) \rightarrow k(p) \rightarrow 0,$$

where $I_p$ is the ideal sheaf of $p$. Now since $\pi$ is affine $\pi_*$ is exact and we get the exact sequence:

$$0\rightarrow \pi_*(I_p\otimes O_X(B)) \rightarrow \pi_*O_X(B) \rightarrow \pi_*k(p) \rightarrow 0.$$

What are the Chern classes of these bundles?

I already found $c_1(\pi_*O_X(B))=\pi_*B-S$. Here on MO I also found a quite nontrivial formula for $c_2(\pi_*O_X(B))$.

So it is enough to compute the classes for one of the two sheaves, probably for $\pi_*k(p)$.

But here I'm stuck, i know that as sheaves on $X$ we have $c_1(k(p))=0$ and $c_2(k(p))=-p$. How do i get to $c_i(\pi_*k(p))$? Or is it easier to compute the Chern classes of $\pi_*(I_p\otimes O_X(B))$?

$\endgroup$
1
  • 1
    $\begingroup$ Isn't $\pi_*k(p)=k(\pi(p))$? $\endgroup$ Feb 23, 2011 at 18:54

1 Answer 1

1
$\begingroup$

If you want to compute the Chern character of a pushforard you can use Grothendieck-Riemann-Roch. But if you are just interested in $c_i(\pi_* k(p))$ then it is very easy. Just note that $k(p) = i_* k$, where $i: p \to X$ is the embedding of the point. Hence $\pi_* k(p) = \pi_*i_* k = (\pi\circ i)_*k = k_{\pi(p)}$. So, the pushforward of the skyscraper sheaf is the skyscraper sheaf of the image point. Consequently, its Cher classes can be computed in the same way as on $X$.

$\endgroup$
3
  • 2
    $\begingroup$ I think that to see that $\pi_*k(p)=k(\pi(p))$ one can simply use the definition of the push-forward. $\endgroup$ Feb 23, 2011 at 18:58
  • 1
    $\begingroup$ Also, for these sheaves GRR may be a little too big a weapon. One can figure out what the push-forward of these sheaves are directly. $\endgroup$ Feb 23, 2011 at 19:00
  • 1
    $\begingroup$ Ahh, again, so obvious and i didn't see it. Thanks for your help. I don't like GRR that much, because there one also has to compute the Todd classes. I just found a quite elementary derivation for $c_i(\pi_{\*}O_X(B))$ in R.Friedman's book about surfaces and vector bundles, pp. 47-49. $\endgroup$
    – TonyS
    Feb 23, 2011 at 19:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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