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

Abstract

Let E1,…,Ek 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, Ei⊂H0(X,Li) is a finite dimensional subspace of the space of regular sections of line bundles Li. Such a collection is called overdetermined if the generic systems1=…=sk=0, with si∈Ei 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⊂∏i=1kEi as the closure of the set of all systems which have at least one common root and study general properties of zero sets Zs of a generic consistent system s∈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 Zs. For equivariant linear series on the torus (C⁎)n this strategy provides explicit calculations and generalizes the theory of Newton polyhedra.