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.
