Linear systems on fibrations

该奖项支持代数和几何领域的研究项目，这是数学的古老和传统领域，其方法和应用于现代几何。这些方法和应用程序与数学的大多数分支相互作用，包括微分几何、拓扑、数论和代数，并且在数学物理、宇宙学、密码学和机器人技术中非常有用。从起源上讲，几何与体积密切相关。调查的主要对象是家庭及其产品的不同体积类型。该项目解决了这些形式的多项式有限生成问题，以及关于族成员的维数和可能的其他一些离散不变量的生成器度的估计。在处理科上的微分形式或相对微分形式时，需要新的技术。本课题的另一个新颖之处是在模理论的基础上发展了基于全称映射的有限生成性理论，而不是传统上使用的上同调消失理论。该项目处理与纤维空间上扭曲微分形式相关的线性除数系统的基本问题。线性系统的标准问题是关于其基点的问题，关于其一般成员或特殊成员的奇点的问题，以及关于与其他系统的对应或同构的问题。分别考虑弱对数正则族模部分的半模性问题、对数正则补的构造以及对数正则局部阈值和全局阈值的应用、新的β不变量和Tian的α不变量。探讨了伴随判别曲线的对数正则线性系统之间同构的可能性，并将其应用于三次元不同的二次元束结构的双分分类。日志规范补充和Sarkisov链接的严格纤维化情况不包括所有可能的情况，它们涵盖了主要情况，其余的是例外情况。后者的形式通常是有界的家族，直到两族同构。在该项目下将开发一种与这些问题有关的新技术。这利用了相对环面对数奇点，b极化理论，三重组的模理论，交错的flops，以及这些模的对应关系。对定义到flops的变量的模量的研究是自然的，并且符合当前的几何精神，其中最小模型定义到flops。

This award supports a research project in the field of algebra and geometry, old and traditional areas of mathematics, with methods and applications in modern birational geometry. These methods and applications interact with most of branches of mathematics, including differential geometry, topology, number theory, and algebra, and can be useful in mathematical physics, cosmology, cryptography, and robotics. From its origin geometry is closely related to volume. The main objects under investigation are differential forms of volume type on families and their products. The project addresses problems on polynomial finite generatedness of those forms and on estimation of degrees of generators with respect to the dimension and possibly to some other discrete invariants of members of families. New techniques are needed in dealing with differential forms on families or relative differential forms. Another novelty of the project is to develop theory of the finite generatedness based on universal mappings from moduli theory instead of traditionally used vanishing of cohomologies.The project deals with fundamental problems on linear systems of divisors associated to twisted differential forms on fibered spaces. Standard problems about linear systems are problems about their base points, about singularities of their general or special members, and about correspondences with or isomorphisms to other systems. Respectively, the PI considers the semiampleness problem for moduli part of weakly log canonical families, the construction of log canonical complements for fibered varieties with applications to log canonical local and global thresholds, to a new beta invariant and to Tian's alpha invariant, and explores possibility of an isomorphism between log canonical linear systems adjoint to the discriminant curve for different conic bundles structures of a threefold with applications to birational classification of those conic bundles. Strictly fibered cases for log canonical complements and for Sarkisov's links do not cover all possible cases, they cover main cases and remaining ones are exceptional. The latter form usually bounded families up to birational isomorphisms. A new technique relevant to these problems will be developed under the project. This makes use of relative toroidal log singularities, theory of b-polarizations, theory of moduli of triples, interlaced by flops, and correspondences of those moduli. Investigation of moduli of varieties defined up to flops is natural and agrees with the spirit of current birational geometry where the minimal models are defined up to flops.