Fujita's freeness conjecture in terms of local cohomology
Fujita's freeness conjecture in terms of local cohomology
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发表时间：
1997
影响因子：
1.8
通讯作者：
Karen E. Smith
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文献类型：
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作者：
Karen E. Smith
设X是维数为（d？）1)在一个领域，让！表示其典型层。对于任何充分可逆O X-模L，Fujita的自由度猜想预言了！L d是由它的整体部分生成的，即相关联的线性因子系是基点自由的F]。本文的目的是将这个猜想重新解释为关于局部上同调模的陈述，并给出一个简单的证明（不依赖于消失定理），其中充分可逆层是由其截面生成的猜想。这种特殊情况，在设置光滑计划的复数，是众所周知的。然而，证明需要科代拉消失定理或其他深消失定理，失败的计划deened超过ELD的素特征。据我所知，对于特征p > 0的方案，还没有沿着这些路线的先前结果。这里的证明藤田的自由猜想的整体生成L的特征p > 0保持即使适当的消失定理失败。它对任何\F-有理型格式都是有效的，这类格式比光滑格式更一般，并且与具有有理奇异性的格式密切相关（并且经常等价）。现有的C上的证明可以适用于有理奇点的情况，因为科代拉消失定理在这种情况下成立。对于特征为零的域上的低维簇，Fujita自由度猜想是已知的。对于曲面，这是由Reider R]的工作得出的;对于三重曲面，则由Ein和Lazarsfeld EL]的工作得出。四重情形最近已由Kawamata K]证明。Demailly证明了特征为零的域上任意维数簇的双伴随线性级数D [1]的整体生成元的一致界的存在性。那个！Angehrn和Siu [AS]证明了L N（d）是dim（X）中二次函数N（d）的全局生成元.所有这些证明都需要使用在特征p > 0时失败的消失定理。
MIT Let X be a smooth projective algebraic variety of dimension (d ? 1) over a eld, and let ! denote its canonical sheaf. For any ample invertible O X-module L, Fujita's freeness conjecture predicts that ! L d is generated by its global sections, i.e. that the associated linear system of divisors is base point free F]. The purpose of this note is to reinterpret this conjecture as a statement about local cohomology modules and give a simple proof (that does not rely on vanishing theorems) of the conjecture where the ample invertible sheaf is generated by its sections. This special case, in the setting of smooth schemes over the complex numbers, is well known. However, the proofs require the Kodaira Vanishing Theorem or other deep vanishing theorems that fail for schemes deened over elds of prime characteristic. As far as I know, there have been no prior results along these lines for schemes of characteristic p > 0. The proof here of Fujita's Freeness Conjecture for globally generated L in characteristic p > 0 holds even when the appropriate vanishing theorems fail. It is valid on any \F-rational type" scheme, a class of schemes more general than smooth schemes and closely related to (and often equivalent to) the class of schemes with rational singularities. The existing proofs over C can be adapted to the case of rational singularities because the Kodaira Vanishing Theorem holds in this setting. For varieties of low dimension deened over a eld of characteristic zero, Fujita's Freeness Conjecture is known. For surfaces, this follows from the work of Reider R]; and for three-folds, from the work of Ein and Lazarsfeld EL]. The four-fold case has been recently proved by Kawamata K]. For varieties of arbitrary dimension over a eld of characteristic zero, Demailly proved the existence of uniform bounds for the global generatation of bi-adjoint linear series D]. That ! L N(d) is globally generated for a function N(d) quadratic in dim (X) has been proved by Angehrn and Siu AS]. All of these proofs require the use of vanishing theorems that fail in characteristic p > 0.