Linear systems on fibrations

该奖项支持代数和几何领域的一个研究项目，这些领域是古老的和传统的数学领域，具有现代二元几何中的方法和应用。这些方法和应用程序与大多数数学分支相互作用，包括微分几何、拓扑学、数论和代数，在数学物理、宇宙学、密码学和机器人学中都很有用。从它的起源开始，几何就与体积密切相关。调查的主要对象是家庭及其产品上的材积类型差异形式。该项目解决了这些形式的多项式有限生成性以及生成元相对于维度的次数的估计以及可能的族成员的一些其他离散不变量的估计问题。在处理族上的微分形式或相对微分形式时，需要新的技巧。该项目的另一个新奇之处是基于模理论中的普适映射而不是传统上使用的上同调消失法，发展了有限生成理论。该项目研究了与纤维空间上的扭曲微分形式相关的线性约数系统的基本问题。关于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。PI分别考虑了弱对数正则族的模部分的半采样问题，纤维簇的对数正则补的构造及其在对数正则局部和全局阈值、新的β不变量和田氏α不变量中的应用，并探索了三重圆锥丛结构的判别曲线所伴随的对数正则线性系统之间同构的可能性，并应用于这些圆锥丛的二元分类.对数正则补语和Sarksov链接的严格虚构情况并不涵盖所有可能的情况，它们涵盖主要情况，其余情况例外。后一种形式通常有界族，直至两族同构。在该项目下，将开发与这些问题相关的新技术。这利用了相对环形对数奇点、b-偏振理论、由触点交错的三元组的模理论以及这些模的对应关系。定义到FLOPS的变种的模数的研究是自然的，并且符合当前二元几何的精神，其中最小模型被定义到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.