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Let $S$ be a ruled surface over a curve of genus $g$. Is it possible to compute the second Chern class of $S$ in terms of $g$?

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Sure why not. Riemann-Roch for surfaces gives $1+p_a=\frac{1}{12}(K^2+c_2)$, see Appendix A, Example 4.1.2 in Hartshorne. You also have $K^2=8(1-g)$ and $p_a=-g$, see V.2.11 and V.2.4 respectively in Hartshorne. Putting this together you get that $c_2=4(1-g)$.

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