  We study signature tensors of paths from a geometric viewpoint. The
signatures of a given class of paths parametrize an algebraic variety inside
the space of tensors, and these signature varieties provide both new tools to
investigate paths and new challenging questions about their behavior. This
paper focuses on signatures of rough paths. Their signature variety shows
surprising analogies with the Veronese variety, and our aim is to prove that
this so-called Rough Veronese is toric. The same holds for the universal
variety. Answering a question of Amendola, Friz and Sturmfels, we show that the
ideal of the universal variety does not need to be generated by quadrics.
