A central open question in the study of syzygies is to describe the Betti table of \(\mathbb{P}^{n}\) under the degree \(d\) Veronese embedding. While the case when \(n=1\) is fully understood and classical — the minimal resolution is given by an Eagon-Northcott complex — in higher dimensions this problem is almost entirely open. For example, when \(n=2\), not only is a complete description unknown, but only a handful of examples have ever been computed.

The goal of this project is to use numerical linear algebra and high throughput, distributed computing to systematically gather new examples of Betti tables of Veronese embeddings of \(\mathbb{P}^n\).

In general such computations are quite difficult even for relatively small \(n\) and small \(d\). For instance, when \(n=2\) and \(d\leq 3\), one can compute the entire Betti table with paper and pencil or instantly via Macaulay2. Likewise, when \(d=4\), Macaulay2 performed the computation on a standard laptop in roughly 30 seconds. However, when \(d=5\), Macaulay2 fails to terminate (as tested on a standard laptop in 2016) , and this case was only recently computed, using alternate methods, in the 2016 work of Castryck-Cools-Demeyer-Lemmens (available here.) The central cause of difficulty in these computations is that the number of variables grows on the order of \(O(d^{n})\), and thus the computational complexity ratchets up dramatically.

Our computation is based on the synthesis of known results and the coordinated execution of many elementary steps. Instead of using symbolic Gröbner methods to compute a free resolution, we use linear algebra to compute the cohomology of the Koszul complex. Separating the Koszul complex into its multigraded strands allows us to break a (largely infeasible) Gröbner computation into a huge number of more tractable computations. (See the Background page for a more complete description.) While the theory behind this approach is not new, the only other effort at implementing on a large scale is the 2016 work Castryck-Cools-Demeyer-Lemmens. This is likely because, even for relatively simple cases, the problem remains quite complicated; the matrices quickly become massive and numerous.

The crux of our technique is the use of high-speed high-throughput computing to compute multigraded Betti numbers. This allows us to compute each multigraded Betti number in parallel, relying on numerical linear algebra algorithms, in particular an LU-decomposition algorithm. These algorithms are numerical in nature, and so rounding errors may creep in. However, our primary interest is in the testing and development of conjectures, so we do not require the precision of symbolic computation. See the Approach page for a more in-depth description of our methodology.

In addition to computing the graded and multigraded Betti numbers, we also compute the corresponding Schur functor decomposition, which is the most concise way the syzygy data data. We also include some auxiliary statistics, like the number of distinct irreducible representations for each graded Betti number.

Our belief in generating this experimental data is that it will lead to new conjectures, and verify existing ones, about the syzygies of Veroneses. See the Conjectures page for a list of conjectures and questions we have raised. (Note: Schur functors methods allow us to catch and correct small rounding errors. See Our Approach for further description.)