# ag.algebraic geometry – Using principal polarisation to “cancel” Jacobian summands in isomorphism

I’m working through the sketch proof of irrationality of cubic threefolds in Huybrechts’ The geometry of cubic hypersurfaces.

Let $$J(X)$$ denote the intermediate Jacobian of a cubic threefold $$X subset mathbb{P}^4$$, and $$J(C)$$ the Jacobian of a curve $$C$$. It is true that for a blow-up $$mathrm{Bl}_Z X$$, the intermediate Jacobian decomposes as $$J(mathrm{Bl}_Z X) cong J(X) times J(Z)$$ (so note that it’s unchanged when $$Z$$ is a point). Suppose that $$X$$ is rational, i.e. there is a birational equivalence $$X simeq mathbb{P}^3$$. Then there exist smooth curves $$C_i$$ and $$D_i$$ and an isomorphism
$$J(Y) times J(D_1) times cdots times J(D_m) cong J(C_1) times cdots times J(C_n)$$
by Hironaka’s resolution of singularities, the blow-up fact we mentioned at the start, and also Corollary 3.26 from Clemens-Griffiths, which gives $$J(mathbb{P}^3) cong J(C_1)$$ for some curve $$C_1$$. The proof then goes on to claim that by “using the principal polarization”, we have that
$$J(Y) cong J(C_1) times cdots times J(C_k) .$$

Question: Why is this the case? It seems that we’re somehow cancelling the $$J(D_i)$$ out with certain $$J(C_i)$$‘s, and therefore that $$n>m$$. Are these things true, and if so how do we know that $$n>m$$? Furthermore, how can one “use the principal polarisation” to do such a thing?

Thank you.