  The tensor network variety is a variety of tensors associated to a graph and
a set of positive integer weights on its edges, called bond dimensions. We
determine an upper bound on the dimension of the tensor network variety. A
refined upper bound is given in cases relevant for applications such as
varieties of matrix product states and projected entangled pairs states. We
provide a range (the "supercritical range") of the parameters where the upper
bound is sharp.
