Overdetermined systems of equations on toric, spherical, and other algebraic varieties

Let E-1, ... , E-k be a collection of linear series on an irreducible algebraic variety X over C which is not assumed to be complete or affine. That is, E-i subset of H-0 (X, L-i) is a finite dimensional subspace of the space of regular sections of line bundles L-i. Such a collection is called overdetermined if the generic system S-1 = ... = S-k = 0, with s(i) is an element of E-i, does not have any roots on X. In this paper we study consistent systems which are given by an overdetermined collection of linear series. Generalizing the notion of a resultant hypersurface we define a consistency variety R subset of Pi(k)(i)=1 E-i as the closure of the set of all systems which have at least one common root and study general properties of zero sets Z(s) of a generic consistent system s is an element of R. Then, in the case of equivariant linear series on spherical homogeneous spaces we provide a strategy for computing discrete invariants of such generic non-empty set Z(s). For equivariant linear series on the torus (C*)(n) this strategy provides explicit calculations and generalizes the theory of Newton polyhedra. Crown Copyright (C) 2020 Published by Elsevier Inc.