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