  We present scheme theoretic methods that apply to the study of secant
varieties. This mainly concerns finite schemes and their smoothability. The
theory generalises to the base fields of any characteristic, and even to
non-algebraically closed fields. In particular, the smoothability of finite
schemes does not depend on the embedding into a smooth variety or on base field
extensions. Independent of the base field, secant varieties to high degree
Veronese reembeddings behave well with respect to the intersection and they are
defined by minors of catalecticants whenever a suitable smoothability condition
for Gorenstein subschemes holds. The content of the article is largely
expository, although many results are presented in a stronger form than in the
literature.
