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 $\pi\_\*(I\_p\otimes O\_X(B))$?



Chern classes of the direct image of an ideal sheaf resp. skyscraper sheaf
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 $X$.


