  The construction of joins and secant varieties is studied in the
combinatorial context of monomial ideals. For ideals generated by quadratic
monomials, the generators of the secant ideals are obstructions to graph
colorings, and this leads to a commutative algebra version of the Strong
Perfect Graph Theorem. Given any projective variety and any term order, we
explore whether the initial ideal of the secant ideal coincides with the secant
ideal of the initial ideal. For toric varieties, this leads to the notion of
delightful triangulations of convex polytopes.
