Equations for secant varieties via vector bundles

We introduce vector bundle techniques for finding equations of secant varieties. A test is established that determines when a secant variety is an irreducible component of the zero set of the equations found. We also prove an induction theorem for varieties that are not weakly defective, that allows one to conclude that the zero set of the equations found for s_{r-1}(X) have s_{r-1}(X) as an irreducible component when s_r(X) is an irreducible component of the equations found for it. The techniques are illustrated with examples of homogeneous varieties. We give an algorithm to decompose a general ternary quintic as the sum of seven fifth powers.