Secant Varieties Working Group

in the winter semester of 2021/2022 meetings take place on:
Wednesday 11:30-13:00, room 403

Research problems we try to address:

1. 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
2. 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
   https://www.math.tamu.edu/~arpantamu/)
3. 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 
4. For   , where  is generic from  we want to find decomposition  for . We would like to find an algorithm wich computes such a decomposition.
5. 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.
6. Find a generic rank for a partially symmetric tensors over  Segre-Veronese varieties.
7. Find the dimension of a secant variety to Segre-Veronese varieties.
8. Border rank of monomials. We are looking for a generalisation of the Ranestad-Schrayer bound to multiprojective spaces.
9. 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.