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

Abstract

Let $E_1,\dots ,E_k$ 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 $Z_{\mathbf{s}}$ of a generic consistent system $\mathbf{s}\in R$. 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 $Z_{\mathbf{s}}$. For equivariant linear series on the torus $({\mathbb{C}}^{\ast }{)}^n$ this strategy provides explicit calculations and generalizes the theory of Newton polyhedra.

Improved exposition, minor changes. 18 pages, comments are welcome!