TORIC VARIETIES
TORIC VARIETIES
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DOI：
10.1007/0-387-27103-1_10
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发表时间：
2010
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正如标准的N-分次多项式环产生了射影几何，多重分次多项式环产生了环面几何。本章的目的就是要弄清楚这句话的意义。我们开始通过解释几何和表示理论的阿贝尔群行动的向量空间产生了多阶化的多项式环和如何仿射同因这些行动是反映代数。然后，我们处理投射的情况下，考虑一个额外的分次Z。接下来是要点：复曲面簇的特征在于多重分次多项式环的数据和在精确意义上与多重分次相容的无平方单项式理想。通过几何不变的理论，我们将这种齐次坐标环的角度来看，更经典的建设复曲面品种从球迷和多面体。为了简单起见，我们在这里处理复数域k= C。
Just as standard N-graded polynomial rings give rise to projective geometry, multigraded polynomial rings give rise to toric geometry. The purpose of this chapter is to make sense of this statement. We begin by explaining how the geometry and representation theory of abelian group actions on vector spaces gives rise to multigradings on polynomial rings and how the affine quotients by such actions are reflected algebraically. Then we treat the projective case, which considers an additional grading by Z. The main point comes next: a toric variety is characterized by the data of a multigraded polynomial ring and a squarefree monomial ideal that is in a precise sense compatible with the multigrading. Through the geometry of invariant theory, we relate this homogeneous coordinate ring perspective to the more classical constructions of toric varieties from fans and polytopes. For simplicity, we work here over the field k= C of complex numbers.