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Let $X$ be a normal complex projective variety over $\mathbb C$. In order to define the intersection product of the Chow ring, one usually requires $X$ to be smooth. How to weak the smoooth assumption is an interesting question.

Intersection on Singular Varieties and Fulton's book (e.g Chapter $18$) discuss the question before.

I am interested to the case beyond divisors and curves. For instance, one can consider intersection of surfaces in a fourfold. Normality is not enough to define an intersection product in higher dimensions.

If $X$ can be embedded into a smooth variety $Y$, we can at least intersect any cycle on $X$ with the pullback of a cycle on $Y$, by transfering the intersection to $Y$.

Assume now $X$ is a Cartier divisor on a smooth projective variety $Y$. Under mild assumption on $X$ (which seems to be necessary), how to define intersection theory of cycles on $X$ using $Y$?

If the pullback is surjective on the level of Chow groups, one can apply the pullback trick.

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    $\begingroup$ One can always define the intersection product if one of the arguments is a linear combination of subschemes whose structure sheaves are perfect complexes. Note that this property is automatically satisfied for pullbacks of cycles from an ambient smooth variety (because on a smooth variety every coherent sheaf is a perfect complex). $\endgroup$ – Sasha Feb 9 at 7:51
  • $\begingroup$ I'm no algebraic geometer, but doesn't one more often approximate singular varieties with smooth ones in the dual way? That is, resolution of singularities in $X$ involves a map $Z \to X$ where $Z$ is smooth, not an embedding map $X \to Y$. I guess it's notable that intersection theory doesn't really seem to need resolution of singularities to run, so maybe it's desirable to keep avoiding it? $\endgroup$ – Tim Campion Feb 9 at 14:36
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Welcome to Mathoverflow!

One can not define intersection product on the Chow groups for a singular variety, even when it is embedded as a divisor in a smooth one: see the quadric cone example in Hartshorne A.1.1.2. In that example one can define rational intersection multiplicities, but this should not be true in general. The basic issue is that cycles on singular varieties tend to intersect in a wrong dimension, and this can not be solved by moving them: in the example of the cone above, every divisor equivalent to a ruling of the cone passes through the vertex. A related problem is that Chow groups do not in general admit pull-backs.

In a broader context, Chow groups are homology (Borel-Moore homology to be precise, that is made of cycles which don't need to have compact support), and they are not supposed to form a ring. However, if $X$ is a divisor in a smooth $Y$, then $CH_*(X)$ is a module over the ring $CH_*(Y)$.

One can replace Chow groups by an appropriate cohomology theory, one is operational Chow groups, another is K-theory. Cohomology theories have an obvious intersection product coming from external product and the pull-back (see Hartshorne A.1.A5).

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  • $\begingroup$ Thanks, I only want a rational intersection theory. For Cartier divisor $D \rightarrow X$, it's possible to pullback things (as the immersion is regular). The notion of operational Chow groups is interesting. If one can define intersection products here for our X, what would it be for some intersection with unbounded Tor? $\endgroup$ – p-adic worker Feb 10 at 7:55

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