Ideals of bounded rank symmetric tensors are generated in bounded degree

Over a field of characteristic zero, we prove that for each r, there exists a constant C® so that the prime ideal of the rth secant variety of any Veronese embedding of any projective space is generated by polynomials of degree at most C®. The main idea is to consider the coordinate ring of all of the ambient spaces of the Veronese embeddings at once by endowing it with the structure of a Hopf ring, and to show that its ideals are finitely generated. We also prove a similar statement for partial flag varieties and, in fact, arbitrary projective schemes, and we also get multi-graded versions of these results.