Timeline for Why is flatness needed for the Segre classes of a family of cones to be equal in the Chow ring of the base
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| Apr 26, 2017 at 13:20 | comment | added | Jason Starr | . . . Note: this already comes up in Example 4.1.6(c). In some sense, it also comes up in Proposition 5.2, the fundamental result that the specialization to the normal cone respects rational equivalence (and thus is well-defined on cycle classes). For those earlier results, certainly Fulton would not use the properties of specialization to the normal cone. | |
| Apr 26, 2017 at 13:17 | comment | added | Jason Starr | Your statement is correct. It follows by Example 10.1.7(a) for the Segre class $\alpha=s(\mathcal{C})$ on $X\times \mathbb{A}^1$. The "claim" is Example 10.1.10 in the chapter "Families of Algebraic Cycles" that reconciles the new approach via deformation to the normal cone with the earlier approach via moving families of cycles. So it is reasonable that Fulton would point out that, under a flatness hypothesis, the "naive" construction (without using deformation to the normal cone or refined Gysin pullbacks) already produces rationally equivalent cycles . . . | |
| Apr 25, 2017 at 23:15 | history | asked | Tomo | CC BY-SA 3.0 |