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Suppose I have a singular projective variety $X\subset \mathbb{P}^n$ that is reducible with $X=\bigcup_i X_i$ smooth irreducible components. That is, the irreducible components are smooth but $X$ is singular along the intersections of the $X_i$.

I would like to compute the Segre classes $s_t(X,\mathbb{P}^n)$. In particular I (optimistically) wonder whether one can obtain these classes in terms of the Segre classes $s_j(X_i,\mathbb{P}^n)$ of the irreducible componentes. If not, what would I need more to compute $s_t(X,\mathbb{P}^n)$?

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  • $\begingroup$ Segre classes of what vector bundle do you want to compute? $\endgroup$ – Sasha Sep 28 '16 at 9:45
  • $\begingroup$ It is the Segre classes of $X$ inside $\mathbb{P}^n$, so I think they are the classes of the normal cone of $X$. $\endgroup$ – IMeasy Oct 1 '16 at 17:34
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Unfortunately Segre class doesn't share very good inclusion-exclusion property. For example, Let $Y$ ve irreducible smooth, and let $X=\cup_t C_t\subset Y$ is a union of curves. Assume that the intersection of curves are only nodal type point (no three points collapse together, and no tangent points). Then we have the following $$ s(X,Y)=\sum_t s(C_t,Y)+\sum [C_a\cap C_b] . $$ For general case, it will get only more complicated. For a detailed discussion, you can read this paper by Aluffi (https://www.math.fsu.edu/~aluffi/archive/paper165.pdf)

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