Can the type of a polarization of abelian varieties jump in a family?

Question

Suppose $f: X \to Y$ is a projective family of abelian varieties, i.e. $f$ is a proper submersion of complex manifolds, which factors over $\mathbb P^N \times Y$, and each fiber $X_y = f^{-1}(y) \subset \mathbb P^N$ is an abelian variety. Then the pull-back of $\mathcal O_{\mathbb P^N}(1)$ induces a polarization on each fiber $X_y$, of some polarization type $$d(y) = (d_1(y), \dotsc, d_n(y)),$$ where $n = \dim X_y$. Is that polarization type (locally) constant in the family?