Syzygies of $mathbb{P}^{1}times mathbb{P}^{1}$: data and conjectures

We provide a number of new conjectures and questions concerning the syzygies of $mathbb{P}^1times mathbb{P}^1$. The conjectures are based on computing the graded Betti tables and related data for large number of different embeddings of $mathbb{P}^1times mathbb{P}^1$. These computations utilize linear algebra over finite fields and high-performance computing.