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.

中文翻译：

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复曲面、球面和其他代数簇的超定方程组

令 $E_1,\ldots,E_k$ 是 $\mathbb{C}$ 上代数变体 $X$ 的线性级数的集合。即$E_i\subset H^0(X, \mathcal{L}_i)$是线丛$\mathcal{L}_i$的正则截面空间的有限维子空间。如果具有 $s_i\in E_i$ 的泛型系统 \[ s_1 = \ldots = s_k = 0, \] 在 $X$ 上没有任何根，则此类集合称为超定集合。在本文中，我们研究了由超定线性系列集合给出的可解系统。概括合成超曲面的概念，我们将一致性变体 $R\subset \prod_{i=1}^k E_i$ 定义为所有系统的集合的闭包，这些系统至少具有一个公共根并研究零集的一般性质$Z_{\bf s}$ 的通用一致性系统 ${\bf s}\in R$。然后，在球齐次空间上的等变线性级数的情况下，我们提供了一种计算这种通用非空集 $Z_{\bf s}$ 的离散不变量的策略。对于圆环 $(\mathbb{C}^*)^n$ 上的等变线性系列，该策略提供了明确的计算并推广了牛顿多面体的理论。