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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 $\pi_*(I_p\otimes O_X(B))$?

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Isn't $\pi_*k(p)=k(\pi(p))$? – Sándor Kovács Feb 23 '11 at 18:54
up vote 1 down vote accepted

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 $X$.

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I think that to see that $\pi_*k(p)=k(\pi(p))$ one can simply use the definition of the push-forward. – Sándor Kovács Feb 23 '11 at 18:58
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. – Sándor Kovács Feb 23 '11 at 19:00
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. – TonyS Feb 23 '11 at 19:33

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