  New classes of modules of equations for secant varieties of Veronese
varieties are defined using representation theory and geometry. Some old
modules of equations (catalecticant minors) are revisited to determine when
they are sufficient to give scheme-theoretic defining equations. An algorithm
to decompose a general ternary quintic as the sum of seven fifth powers is
given as an illustration of our methods. Our new equations and results about
them are put into a larger context by introducing vector bundle techniques for
finding equations of secant varieties in general. We include a few homogeneous
examples of this method.
