Let $X,Y$ be smooth irreducible projective varieties over $K=\bar{K}$. I want to show that $\mathcal K(X\times_KY)=\mathcal K(X)+\mathcal K(Y)$ with the definition using rational maps $\phi_n$.
We have that $\omega_{X\times Y}=p_1^*\omega_X\otimes p_2^*\omega_Y$ and $H^0(X\times Y,\omega_{X\times Y}^n)\cong H^0(X,\omega_X^n)\otimes H^0(Y,\omega_Y^n)$ so the rational map $X\to \mathbb P^{P_n(X\times Y)-1}$ can be factorized as : $$X\to \mathbb P^{P_n(X)-1}\times\mathbb P^{P_n(Y)-1}\to \mathbb P^{P_n(X\times Y)-1}$$ where the first map is $(\phi_{nK_X},\phi_{nK_Y})$ and the second one is the Segre embedding. So we just need to consider the dimension of the first map.
Now what I don't understand is why the dimension of the first map is the sum of the dimensions.
My idea is that if in a regular setting $\phi_m: Z\to \mathbb P^{P_m(Z)-1}$ with maximal dimension, then $\phi_{km}$ also has maximal dimension $\forall k\geq 1$, then we can consider $m_1,m_2$ with $\phi_{m_1K_X},\phi_{m_2K_Y}$ with maximal dimension, and then consider $(\phi_{m_1m_2K_X},\phi_{m_1m_2K_Y})$ which will then have dimension $\mathcal K(X)+\mathcal K(Y)$.
I would try to argue by considering a factorization $X\to \mathbb P^{m-1}\to \mathbb P^{km-1}$ but I'm not sure of myself.
So I have these questions :
Is my claim above true and is the idea correct to show addivity ? If not why is the dimension of the first map the sum of dimensions ?
Is there a reference showing this in detail that you know of ?