  For an affine toric variety $\spec(A)$, we give a convex geometric
interpretation of the Gerstenhaber product $\HH^2(A)\times \HH^2(A)\to
\HH^3(A)$ between the Hochschild cohomology groups. In the case of Gorenstein
toric surfaces we prove that the Gerstenhaber product is the zero map. As an
application in commutative deformation theory we find the equations of the
versal base space (in special lattice degrees) up to second order for not
necessarily isolated toric Gorenstein singularities. Our construction reproves
and generalizes results obtained in [1] and [13].
