# Secant Varieties Working Group

### 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 $T \in \mathbb{P}(\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m)$ find criteria for $T$ to have border rank equal to $m$. (Arpan Pal from Texas A&M University is also working on this topic https://www.math.tamu.edu/~arpantamu/) High rank loci. For $X \in \mathbb{P}(V )$ let $W_r(X)$ be the closure of tensors in $\mathbb{P}(V)$ which have rank equal to $r$. Is there an equality between $W_k$ and $W_{k+1} + X$  for $k,  where $Y+Z$ denotes the join of $Y$ and $Z$? (answer is no - recently solved by E. Ballico, A. Bernardi, E. Ventura  For   $F \in S^d V$, where $V = \mathbb{C}^n, \; rk(F) \leq r, \; r \ll n, \; F$ is generic from $\sigma_r ( v_d( \mathbb{P}V)))$ we want to find decomposition $F= l_1^d + l_2^d + ... + l_r^d$ for $3 \leq d$. We would like to find an algorithm wich computes such a decomposition. For $T \in A \otimes B \otimes C, such \; that \; T= S_1 + S_2 + W$where $S_i$ are simple tensors with norm $n_i$ and $W$ is a random error with norm $\leq \epsilon$. Assuming $\langle S_1, S_2 \rangle \leq \epsilon_2$ we want to find the funcition $C(n_1,n_2,\epsilon_2)$ such that  $\epsilon_1 \leq C(n_1,n_2,\epsilon_2) \Rightarrow T=S_1' + S_2' + W'$ is the best rank 2 approximation and the norms $|S_1 - S_1'|, |S_2 - S_2'|$ 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.

Odśwież obrazek