By Matej Dolezalek (UKON) and Nikhil Ken (UNIFI)
At the end of March, we finished our article on secant varieties of O(1,2) Segre-Veronese varieties. It is currently going through the submission process, and in the meantime, it is available as a preprint: arXiv:2503.21972
In the article, we prove that all secant varieties of the Segre-Veronese variety $\mathbb P^m\times\mathbb P^n$ have the expected dimension, provided n is sufficiently large with respect to m. This follows an analogous result due to Abo and Brambilla in the so-called subabundant case, where a naive dimension count suggests the secant variety should not fill the whole ambient space. In their paper, it seemed the superabundant case — i.e. when the secant should fill the ambient space — is more difficult, and there always remained a gap of parameters s for which the defectivity of the s-th secant variety was not known. In our paper, we manage to close this gap for sufficiently large n.
To achieve this, we investigate the combinatorial properties of configurations of double points and linear space that arise from specialization arguments commonly used for inductive proofs of non-defectivity of secant varieties. We dub one particular version of this specialization technique (generalized from an approach due to Brambilla and Ottaviani) the inductant construction and establish some of its general properties.
This then allows us to execute a long series of nested inductive arguments that together yield the titular result. Each level of such an induction requires some base cases to be verified, and some of these cases may be quite large. Hence we verify them with computer assistance, adapting a standard Monte-Carlo algorithm for checking defectivity of secant varieties. This essentially boils each case down to computing the rank of a large matrix over a finite field — the largest of our cases entail matrices of size roughly 70 000 by 70 000. The source code as well as certificates of these computations are available on GitHub.
To conclude, let us also mention briefly where this fits in the broader context of the study of secant varieties to Segre-Veronese varieties. Recently, all Segre-Veronese varieties with all multidegrees greater than or equal to 3 have been shown to have all secants non-defective (Abo, Brambilla, Galuppi and Oneto); with some technical care, some of these 3’s may even be replaced by 2’s. Such results usually rely on first investigating a particular multidegree or multidegrees (such as O(3,3)), and then applying an induction on degree, or an induction on the number factors. In view of these degree inductions and inductions on the number of factors, small multidegrees remain hard to investigate, since they cannot be reached by such an induction. In particular, O(1,1,1) and O(1,2) have little to no hope of being illuminated in this way, since both O(1,1) and O(2) are notoriously defective. Accordingly, only partial results are known in these two cases — our new result at least provides non-defectivity of O(1,2) when the quadratic factor is sufficiently large.