# 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 2020/2021 meetings take place on: Tuesday 9:30-11:00, 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. 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