  Motivated by the study of the secant variety of the Segre-Veronese variety we
propose a general framework to analyze properties of the secant varieties of
toric embeddings of affine spaces defined by simplicial complexes. We prove
that every such secant is toric, which gives a way to use combinatorial tools
to study singularities. We focus on the Segre-Veronese variety for which we
completely classify their secants that give Gorenstein or $\mathbb
Q$-Gorenstein varieties. We conclude providing the explicit description of the
singular locus.
