Degrees of Segre and Chern classes of a $\mathbb{P}^2$-bundle over a smooth curve - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T06:59:57Z http://mathoverflow.net/feeds/question/95000 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95000/degrees-of-segre-and-chern-classes-of-a-mathbbp2-bundle-over-a-smooth-curve Degrees of Segre and Chern classes of a $\mathbb{P}^2$-bundle over a smooth curve gio 2012-04-24T06:23:24Z 2012-04-24T06:23:24Z <p>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).</p> <p>I am interested to know the <strong>degrees</strong> 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 $\mathcal{T}_X$)).</p> <p>Note 1: By multiplicativity property of the topological Euler characteristic we have $c_3(X)=c_2(\mathbb{P}^2)c_1(C)=3c_1(C)=6(1-g(C))$.</p> <p>Note 2: On Besana and Biancofiore's article, avaidable <a href="http://condor.depaul.edu/gbesana/papers/BebiForumI.pdf" rel="nofollow" title="NUMERICAL CONSTRAINTS FOR EMBEDDED PROJECTIVE MANIFOLDS">here</a> , 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.</p> <p>Thanks in advance.</p>