FoCM 2014 conference

Workshop C1 - Computational Algebraic Geometry - Semi-plenary talk

December 18, 14:35 ~ 15:25 - Room B12

## Cactus varieties of cubic forms

### Kristian Ranestad

### University of Oslo, Norway - ranestad@math.uio.no

The rank of a symmetric form is the length of its shortest decomposition as a sum of pure powers of linear forms, i.e. the shortest smooth apolar scheme. The cactus rank of the form is the the length of the shortest apolar scheme. The $a$-th cactus variety of cubic forms $C_{a,n}$ is the closure of the family of cubic forms of cactus rank $a$ in the projective space of cubic forms in $n+1$ variables. I shall report on recent work with Bernardi, Jelisiejew and Marques giving the dimension and a geometric characterization of the general member of $C_{a,n}$ when $1\leq a\leq 2n+2$.

Joint work with Alessandra Bernardi (Universita de Bologna, Italy), Joachim Jelisiejew (University of Warsaw, Poland) and Pedro Macias Marques (Universidade de Evora, Portugal).