Syzygies of \(\mathbb{P}^1 \times \mathbb{P}^1\): data and conjectures (Q2062749)

This paper deals with the question of syzygies for some of the ``simplest'' non-trivial Segre-Veronese embeddings, namely those of \({\mathbb P}^1 \times {\mathbb P}^1\) by monomials of bidegree \((d_1 , d_2)\). Syzygies for the direct images of the line bundles \({\mathcal O}_{{\mathbb P}^1 \times {\mathbb P}^1}(b_1, b_2)\) are also considered. The starting point of the paper is based on former computations and conjectures which are carefully refered. One uses Macaulay 2 and MAGMA and one works mainly over finite fields. The authors manage to organize the huge computations controlling the process via symmetries and determination of a so-called ``relevant range '' for the Betti numbers which determine al the other Betti numbers. So, they construct matrices in the corresponding ``relevant range ''. The authors succeed in finding complete results for 150 different numerical characters \(((b_1,b_2), (d_1 , d_2))\). These are also translated in terms of Schur functors decompositions. The results obtained are the basis for sets of conjectures concerning qualitative aspects of the minimal resolutions (including here conjectures relative to Schur modules) and conjectures on Boij-Söderberg coefficients. The authors give some details on the time and memory required for the high performance computations. In particular they say that ``Only a handful of cases took over a day''.