Degrees of Segre and Chern classes of a $\mathbb{P}^2$-bundle over a smooth curve

2012-04-24T06:23:24Z

Let $X\subset\mathbb{P}^N$ be a smooth irreducible complex projective $3$-dimensional variety and let $H_X$ be its hyperplane section. Assume that there exists a smooth curve $C$ and a surjective morphism $\pi:X\to C$ such that all its fibers are isomorphic to $\mathbb{P}^2$ and $\mathcal{O}_{X}(H_X)$ induces $\mathcal{O}_{\mathbb{P}^2}(d)$ on each fiber ($d\geq1$ fixed).

I am interested to know the degrees of the Segre (resp. Chern) classes of $X$ (i.e. of the normal bundle $\mathcal{N}_{X,\mathbb{P}^N}$ (resp. of the tangent bundle $\mathcal{T}_X$)).

Note 1: By multiplicativity property of the topological Euler characteristic we have $c_3(X)=c_2(\mathbb{P}^2)c_1(C)=3c_1(C)=6(1-g(C))$.

Note 2: On Besana and Biancofiore's article, avaidable here , at page 12, there are formulas expressing $s_1,s_2,s_3$ as functions of $q:=h^1(X,\mathcal{O}_X)$ and $d=\deg(X)$. This would be just what I want, but I do not understand how these formulas are derived.

Thanks in advance.

Degrees of Segre and Chern classes of a $\mathbb{P}^2$-bundle over a smooth curve - MathOverflow

Degrees of Segre and Chern classes of a $\mathbb{P}^2$-bundle over a smooth curve