  This paper studies the dimension of secant varieties to Segre varieties. The
problem is cast both in the setting of tensor algebra and in the setting of
algebraic geometry. An inductive procedure is built around the ideas of
successive specializations of points and projections. This reduces the
calculation of the dimension of the secant variety in a high dimensional case
to a sequence of calculations of partial secant varieties in low dimensional
cases. As applications of the technique: We give a complete classification of
defective $t$-secant varieties to Segre varieties for t < 7. We generalize a
theorem of Catalisano-Geramita-Gimigliano on non-defectivity of tensor powers
of P^n. We determine the set of p for which unbalanced Segre varieties have
defective p-secant varieties. In addition, we show that the Segre varieties P^1
x P^1 x P^n x P^n and P^2 x P ^3 x P^3 are deficient and completely describe
the dimensions of their secant varieties. In the final section we propose a
series of conjectures about defective Segre varieties.
