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 $s_t(X,\mathbb{P}^n)$?

  • $\begingroup$ 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

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)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.