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.
Categories: math.AG math.AC