Secant Varieties Working Group
(follow up to Secant Varieties Working Group of Varieties: Arithmetic and Transformations)
in the winter semester of 2021/2022 meetings take place on:
Wednesday 11:30-13:00, room 403
Research problems we try to address:
- Find membership test for secant variaties which do not apply for cactus varieties.
The initial work of the group in this direction is now available at arxiv: https://arxiv.org/abs/2007.16203
- For a fixed tensor find criteria for to have border rank equal to . (Arpan Pal from Texas A&M University is also working on this topic
- High rank loci. For let be the closure of tensors in which have rank equal to . Is there an equality between and for , where denotes the join of and ? (answer is no - recently solved by E. Ballico, A. Bernardi, E. Ventura
- For , where is generic from we want to find decomposition for . We would like to find an algorithm wich computes such a decomposition.
- For where are simple tensors with norm and is a random error with norm . Assuming we want to find the funcition such that is the best rank 2 approximation and the norms are bounded from above by some constant.
- Find a generic rank for a partially symmetric tensors over Segre-Veronese varieties.
- Find the dimension of a secant variety to Segre-Veronese varieties.
- Border rank of monomials. We are looking for a generalisation of the Ranestad-Schrayer bound to multiprojective spaces.
- Other variants of Comon's and Strassen's conjectures. For example, symmetric Strassen conjecture, cactus version of Comon's conjecture, border version of Comon's conjecture.
We are open to discuss any issues related to secant varieties, cactus varieties, tensor rank, Waring rank, and their border/cactus analogues, identifiability, apolarity, Hilbert schemes of points etc.