# Question regarding intersection product in Chow group of $\mathbb{P}^n\times\mathbb{P}^m$

Assume we are working over complex numbers. Let $$\pi:\mathbb{P}^n\times\mathbb{P}^m\rightarrow \mathbb{P}^n (m,n>0)$$ be the first projection. Suppose we are given a vector bundle $$E$$ over $$\mathbb{P}^n$$ such that $$\mathbb{P}(E)=\mathbb{P}^n\times\mathbb{P}^m$$ as bundles over $$\mathbb{P}^n$$ (for example, $$E$$ might be $$\mathbb{P}^n\times\mathbb{C}^{m+1}$$).

Let $$\mathcal{Q}$$ be the universal quotient bundle on $$\mathbb{P}^n\times\mathbb{P}^m$$ of rank $$m$$, coming from the short exact sequence \begin{align} 0\to \mathcal{O}_{\mathbb{P}(E)}(-1)\to \pi^*(E)\to \mathcal{Q}\to 0. \end{align}

Let $$l$$ be a line in $$\mathbb{P}^n$$. Then, in Chow groups, $$\pi ^*([l]) = [l\times \mathbb{P}^m]\in CH_{m+1}(\mathbb{P}^n\times\mathbb{P}^m)$$.

Question: In $$CH_1(\mathbb{P}^n\times\mathbb{P}^m)$$, do we have $$c_m(\mathcal{Q})\cap[l\times \mathbb{P}^m] = [l\times\{*\}]$$?,

where $$c_m(\_)$$ denotes the $$m$$-th chern class, and $$*$$ denotes any point in $$\mathbb{P}^m$$?

I believe that they should not be equal (e.g. taking $$m=n=1$$, I feel they should not be equal) , but I can't prove it concretely. Any help/reference would be highly appreciated.

• What do you mean by $\mathbf P(E) = \mathbf P^n \times \mathbf P^m$? There exists an isomorphism of varieties? Compatible with the projection to $\mathbf P^n$? – R. van Dobben de Bruyn Jul 20 '19 at 11:53
• @R.vanDobbendeBruyn : I mean they are isomorphic as $\mathbb{P}^m$-bundles over $\mathbb{P}^n$, as you said. Sorry I forgot to add it. – yojusmath Jul 20 '19 at 11:57

The proposed equality is true, at least when $$E$$ is the trivial rank $$m+1$$ bundle on $$\mathbb{P}^n$$. Otherwise, your requirement that $$P(E) \cong \mathbb{P}^n\times \mathbb{P}^m$$ forces $$E$$ to be isomorphic to the $$(m+1)$$st power of a line bundle on $$\mathbb{P}^n$$. A reference for this is Corollary 9.5 of Eisenbud and Harris' book 3264.

The following shows how one can compute the class in question assuming $$E$$ is trivial. By the sum formula for Chern classes and your exact sequence, the total Chern class of $$Q$$ is $$c(Q) = \frac{1}{c(O_{P(E)}(-1))} = \frac{1}{1 + c_1(O_{P(E)}(-1))}.$$ That is, in the language of Fulton's intersection theory, $$c(Q)$$ is the Segre class of the tautological line bundle $$O_{P(E)}(-1)$$. This ratio is interpreted by expanding it out as a power series which is legal since all the Chern classes of index $$>0$$ are nilpotent in the ring of endomorphisms of $${CH}(\mathbb{P}^n\times\mathbb{P}^m)$$ (multiplication corresponds to composition). We obtain:

$$c(Q) = 1 - c_1(O_{P(E)}(-1)) + \ldots + (-1)^m c_1(O_{P(E)}(-1))^m + \ldots,$$

and thus $$c_m(Q) = (-1)^m c_1(O_{P(E)}(-1))^m$$. So $$c_m(Q)\cap [\mathbb{P}^n\times\mathbb{P}^m] = ([\mathbb{P}^n\times\mathbb{P}^{m-1}])^m = [\mathbb{P}^n\times \{*\}].$$

The intersection product on $$\mathbb{P}^n\times\mathbb{P}^m$$ is defined so that $$c_m(Q)\cap [l\times \mathbb{P}^m] = (c_m(Q)\cap [l \times \mathbb{P}^m])\cdot [\mathbb{P}^n\times\mathbb{P}^m].$$

Then by Fulton example 8.1.6, this is $$[l\times \mathbb{P}^m]\cdot [\mathbb{P}^n\times \{*\}],$$ giving your result.

Fulton's excellent book is the end-all reference for this technology, though many results are stated in greater generality than what the situations one may most often encounter actually call for; in my experience at least, it can be a somewhat daunting task to specialize them. Chapter 8 defines the intersection product and the rigorous footing for Chern classes and their operations is developed in chapter 3.

If $$E$$ is nontrivial, then $$c(Q) = \frac{(1 + c_1(L))^{m+1}}{1 + c_1(O_{P(E)}(-1))},$$ for the pullback $$L$$ of some nontrivial line bundle on $$\mathbb{P}^n$$ to $$P(E)$$. It is messier, and I have not worked out the details, but an analogous computation as above should find $$c_m(Q)\cap [l\times \mathbb{P}^m]$$ in this case. Really, when we write $$c(Q)\cap [l\times \mathbb{P}^m]$$, this only makes sense if we are treating $$Q$$ as a bundle on $$\mathbb{P}^n\times\mathbb{P}^m$$ by applying the pullback map to it induced by the supposed isomorphism $$P(E)\cong \mathbb{P}^n\times\mathbb{P}^m$$. Then $$c(Q) = \frac{(1 + c_1(L^\prime))^{m+1}}{1 + c_1(O_{\mathbb{P}^n\times\mathbb{P}^m}(-1)) + c_1(L^\prime)},$$ where $$L^\prime$$ is the pullback of the line bundle on $$\mathbb{P}^n$$ to $$\mathbb{P}^n\times\mathbb{P}^m$$.