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 $\pi_*(I_p\otimes O_X(B))$?
Best Answer
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 $X$.