Contractible classes in toric varieties
Contractible classes in toric varieties
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复曲面品种中的可收缩类
DOI：
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发表时间：
2001
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通讯作者：
C. Casagrande
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作者：
C. Casagrande
设X是光滑的完全复曲面簇。设A1（X）是X上模数值等价的代数1-圈群，N1（X）= A1（X）<$ZQ .考虑在N1（X）中由X上的曲线类生成的锥NE（X）。这是M. Reid [13]证明了NE（X）是闭的，多面体的，由X上的不变曲线类生成。X是射影的当且仅当NE（X）是严格凸的;在这种情况下，NE（X）的1维面称为极值射线。在[13]中证明了每一条极值射线都允许对一个投射复曲面簇的收缩。我们认为A1（X）是Q -向量空间N1（X）中的一个格。假设X是射影的。对于每个极值射线R NE（X），我们选择R A1（X）中的本原类;我们称这个类为极值类。极值类集E是锥NE（X）的生成集，即NE（X）= ∑ γ∈ EQ ≥0 γ.对于许多目的，具有整系数的线性分解是有用的：例如，关于X上某个充分线丛的最小次数曲线，我们能说什么？极值类是否生成NE（X）<$A1（X）作为半群是一个公开的问题。本文引入了NE（X）<$A1（X）中类的集合C <$E，它是NE（X）<$A1（X）作为半群的生成元的集合. C中的类在几何上以“可收缩性”为特征：
Let X be a smooth, complete toric variety. Let A1(X) be the group of algebraic 1-cycles on X modulo numerical equivalence and N1(X) = A1(X)⊗ZQ . Consider inN1(X) the coneNE(X) generated by classes of curves on X . It is a well-known result due to M. Reid [13] that NE(X) is closed, polyhedral and generated by classes of invariant curves on X . The varietyX is projective if and only if NE(X) is strictly convex; in this case, a 1-dimensional face ofNE(X) is called an extremal ray. It is shown in [13] that every extremal ray admits a contraction to a projective toric variety. We think of A1(X) as a lattice in the Q -vector space N1(X). Suppose that X is projective. For every extremal ray R ⊂ NE(X), we choose the primitive class in R ∩ A1(X); we call this class an extremal class. The set E of extremal classes is a generating set for the cone NE(X), namely NE(X) = ∑ γ∈E Q≥0 γ. For many purposes it would be useful to have a linear decomposition with integral coefficients: for instance, what can we say about curves having minimal degree with respect to some ample line bundle onX? It is an open question whether extremal classes generate NE(X) ∩ A1(X) as a semigroup. In this paper we introduce a set C ⊇ E of classes in NE(X) ∩ A1(X) which is a set of generators of NE(X) ∩ A1(X) as a semigroup. Classes in C are geometrically characterized by “contractibility”: