Asymptotic Syzygies in the Setting of Semi-Ample Growth

We study the asymptotic non-vanishing of syzygies for products of projective spaces. Generalizing the monomial methods of Ein, Erman, and Lazarsfeld \cite{einErmanLazarsfeld16} we give an explicit range in which the graded Betti numbers of ${\mathbb{P}}^{n_1}\times {\mathbb{P}}^{n_2}$ embedded by ${\mathcal{O}}_{{\mathbb{P}}^{n_1}\times {\mathbb{P}}^{n_2}}(d_1,d_2)$ are non-zero. These bounds provide the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-ample fashion behave in nuanced and previously unseen ways.