Overdetermined Systems of Equations on Toric, Spherical, and Other Algebraic Varieties

Abstract

Let $E_1,\ldots,E_k$ be a collection of linear series on an algebraic variety $X$ over $\mathbb{C}$. That is, $E_i\subset H^0(X, \mathcal{L}_i)$ is a finite dimensional subspace of the space of regular sections of line bundles $ \mathcal{L}_i$. Such a collection is called overdetermined if the generic system \[ s_1 = \ldots = s_k = 0, \] with $s_i\in E_i$ does not have any roots on $X$. In this paper we study solvable systems which are given by an overdetermined collection of linear series. Generalizing the notion of a resultant hypersurface we define a consistency variety $R\subset \prod_{i=1}^k E_i$ as the closure of the set of all systems which have at least one common root and study general properties of zero sets $Z_{\bf s}$ of a generic consistent system ${\bf s}\in R$. 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 $Z_{\bf s}$. For equivariant linear series on the torus $(\mathbb{C}^*)^n$ this strategy provides explicit calculations and generalizes the theory of Newton polyhedra.