The following questions are about computing the pullback of the class of a hyperplane in $\mathbb{P}^?$ in chow groups for certain maps $X \to \mathbb{P}^?$.
Let $\mathbb{P}^n$ be $r$-dimensional projective space and let $(d_1,...,d_k)$ be a integers forming a partition of $r$ (i.e. $\Sigma_j d_j = r$). Then there's an obvious multilinear map given by multiplication:
$$H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(d_1)) \times \dots \times H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(d_k)) \to H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^r}(r))$$
Passing to the corresponding projective spaces we get:
$$\phi : \mathbb{P}^{\binom{n+d_1}{n} - 1} \times \dots \times \mathbb{P}^{\binom{n+d_k}{n}-1} \to \mathbb{P}^{\binom{n+r}{n}-1}$$
Denote the class of a general hyperplane by $\alpha_{j} \in A_1(\mathbb{P}^{\binom{n+d_j}{n}-1})$ (the first degree chow group graded by codimension).
I'd like to prove the following:
$$\phi^*(\alpha_r)= \alpha_1 + \alpha_2 + \dots + \alpha_k$$
How can I show this?
The fist map above being multilinear factors through the following Segre-type map:
$$H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(d_1)) \times \dots \times H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(d_k)) \to H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(d_1)) \otimes \dots \otimes H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(d_k))$$
Perhaps the pullback of a general hyperplane by this map can more easily seen to be a union of hyperplanes? Although then one must also check the map:
$$H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(d_1)) \otimes \dots \otimes H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(d_k)) \to H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^r}(r))$$
Is there a simple geometric argument hiding here somewhere?
You might find Chapter 2.1.4 (Products of Projective Spaces) of Eisenbud and Harris's (3264 & All that Intersection Theory in Algebraic Geometry) helpful.
In this case, a hyperplane section of the image of $\phi$ would correspond to a multi-homogeneous form of degree $(1,1,\ldots,1)$, which corresponds to the class $\alpha_1+\ldots+\alpha_k$.