We give practical numerical methods to compute the period matrix of a plane algebraic curve (not necessarily smooth). We show how automorphisms and isomorphisms of such curves, as well as the decomposition of their Jacobians up to isogeny, can be calculated heuristically. Particular applications include the determination of (generically) non-Galois morphisms between curves and the identification of Prym varieties.

Abstract. Given a finite set of lattice points A, we consider the associated homogeneous binomial ideal IA and projective toric variety XA. We give a concise combinatorial description of all linear subspaces contained in the variety XA, or, equivalently, all solutions in linear forms to the system of binomial equations determined by IA. More precisely, we study the Fano scheme F k (XA) whose closed points correspond to k-dimensional linear spaces contained in XA. We show that the irreducible components of F k (XA) are in bijection to maximal Cayley structures for A of length at least k. We explicitly describe these irreducible components and their intersection behavior, characterize when F k (XA) is connected, and prove that if XA is smooth in dimension k, then every component of F k (XA) is smooth in its reduced structure. Furthermore, in the special case k = dim XA − 1, we describe the nonreduced structure of F k (XA).