Given a projective variety X over a field of characteristic 0,
and a positive integer r, we study the rth secant variety of Veronese
re-embeddings of X. In particular, I'll explain recent work which shows
that the degrees of the minimal equations (and more generally, syzygies)
defining these secant varieties can be bounded in terms of X and r
independent of the Veronese embedding. This is based on
http://arxiv.org/abs/1510.04904 and http://arxiv.org/abs/1608.01722