Let $X$ be a variety and $i:Y\hookrightarrow X$ be a regularly embedded closed subvariety, and write $\tilde X$ for $Bl_Y X$ the blow-up of $X$ along $Y$. Let $A^*(X)$ refer to the Fulton-Macpherson operational Chow groups of $X$.

If one assumes $i^*:A^*(X)\rightarrow A^*(Y)$ is surjective, there is a result of Keel (1992, Trans. Amer. Math. Soc., the appendix) that describes $A^*(\tilde X)$ in terms of $A^*(X)$ and the Chern classes $c_i(N_Y X)$ of the normal bundle to $Y$ in $X$. In more detail, $A^*(\tilde X)$ is isomorphic to $A^*(X)[T]/(P(T),(T\cdot ker(i^*)))$, where $P(T)\in A^*(X)[T]$ is any polynomial with constant term $[Y]$ and whose restriction to $A^*(Y)$ is the Chern polynomial of the normal bundle $N_Y X$.

If $Y$ is not regularly embedded, then instead of a normal bundle there is only the normal cone $C_Y X$. Cones do not have Chern classes, but they have Segre classes, and so I am wondering:

Has anyone developed a description of $A^*(\tilde X)$ in terms of $A^*(X)$ and the Segre classes $s_i(C_Y X)$? Even special cases (e.g. toric varieties) are welcome.

  • $\begingroup$ A bit confused about which variety is embedded in which: if $Y\hookrightarrow X$ the normal bundle of $X$ in $Y$ looks weird. $\endgroup$ – Fan Zheng May 28 '18 at 1:58
  • $\begingroup$ I am following the conventions of Fulton's Intersection Theory. Y is embedded in X, $N_Y X$ is the normal bundle to $Y$ in $X$. $\endgroup$ – A. S. May 28 '18 at 2:08
  • $\begingroup$ What confuses me is "the Chern classes $c_i(N_YX)$ of the normal bundle of $X$ in $Y$," but on a second reading I guess that "in $Y$" part refers to the Chern class, nor the normal bundle, right? $\endgroup$ – Fan Zheng May 28 '18 at 2:13
  • $\begingroup$ Corrected, thanks. Actually I will try to add a bit more since I think some clarification is necessary. $\endgroup$ – A. S. May 28 '18 at 2:14
  • $\begingroup$ Non regular blow ups are difficult to control. If $Y \subset X$ is not a regular embedding then the blow up morphism may not even induce a pull-back on the Chow groups. The simplest example to consider is the blow up of the vertex of the cone $X$: $xy = z^2$, which is its resolution. Furthermore $A^*(X)$ will not embed into $A^*(\widetilde{X})$: in the quadric cone case the Chow groups of the cone have torsion but the resolution has torsion-free Chow groups. $\endgroup$ – Evgeny Shinder May 30 '18 at 8:28

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