Abstract: Inspired by the work of Eisenbud-Koh-Stillman we compare the r-th secant variety of a Veronese variety (or more generally Veronese reembedding of a smooth variety) with the zero set of the (r+1)x (r+1) minors of catalecticats. To do this we define the r-th cactus variety of a given variety X as a closure of the sum of linear spans of 0-dimensional sub-schemes of X. As a result we obtain set-theoretical equations of the r-th secant for small r or dimension of the variety and high degree of the Veronese reembedding.
This is a joint work with Jarek Buczyński.