Questions on Singularities and Adjoint Linear Systems

首席研究员计划研究几个不同难度的问题，这些问题涉及代数簇研究中的局部和全局方面。长期以来，人们一直认为，即使人们对研究光滑的几何对象感兴趣，也会自然地出现奇异的对象。因此，能够测量奇点是很重要的，这是通过各种不变量来实现的，这些不变量可以是数值的，也可以具有更复杂的结构。本提案中的项目涉及其中的一些不变量及其在代数簇研究中的应用。这些几何对象可以在两个上下文中研究：在特征零中，这是更熟悉的上下文，更接近我们通常的欧几里德空间；在正特征中，更接近算术的上下文。然而，这两个设置有着密切的联系，有时可以通过证明相应的正特征结果来证明特征为零的结果。然而，在这一点上，大多数关于代数簇的全局性质的结果只在特征零点是已知的。这项建议中的几个项目涉及将这些结果推广到正特征，并研究在这一背景下出现的更微妙的现象。这项建议中涉及的问题的共同特征是研究不同奇点不变量之间的联系，并在研究代数簇的整体性质时使用这种不变量。PI计划探索奇点的不同观点之间的关系，例如弧形空间与希尔伯特-塞缪尔重数之间的联系，以及弧形空间与分解定理之间的联系。他还计划研究具有正特征的伴随线性系统的正性性质问题。这样的结果在特征零点是已知的，证明中的基本工具是Kodaira的消失定理及其推广。将这些结果推广到正特征的主要困难是在新的背景下消失定理的失败。然而，在过去的几年里，基于Frobenius同态的新工具已经被开发出来。PI计划使用其中的一些工具(例如，他与Schwede关于正特征中的Seshadri常量的工作)来解决这种设置中的几个问题。

The principal investigator plans to work on several questions, of different degrees of difficulty, concerning local and global aspects in the study of algebraic varieties. It has been understood for a long time that even if one is interested in studying smooth geometric objects, singular ones naturally appear. It is thus important to be able to measure singularities, and this is done via various invariants that can be either numerical or can have a more complicated structure. The projects in this proposal are concerned with some of these invariants and with their applications to the study of algebraic varieties. These geometric objects can be studied in two contexts: in characteristic zero, which is the more familiar context, closer to our usual Euclidean space, and in positive characteristic, a context that is closer to arithmetic. However, these two settings have close connections, and sometimes one can prove results in characteristic zero by proving the corresponding results in positive characteristic. At this point, however, most of the results concerning the global properties of algebraic varieties are only known in characteristic zero. Several of the projects in this proposal are concerned with the extension of such results to positive characteristic and with the study of the more subtle phenomena that appear in this setting.The common feature of the questions addressed in this proposal is the study of the connections between different invariants of singularities and the use of such invariants in the study of the global properties of algebraic varieties. The PI plans to explore relations between different points of view on singularities, such as the connection between spaces of arcs and the Hilbert-Samuel multiplicity and the connection between spaces of arcs and the Decomposition Theorem. He also plans to work on questions regarding the positivity properties of adjoint linear systems in positive characteristic. Such results are already known in characteristic zero, the fundamental tool in the proofs being Kodaira's vanishing theorem and its generalizations. The main difficulty in extending these results to positive characteristic is the failure of vanishing theorems in the new setting. However, new tools based on the Frobenius homomorphism have been developed in the past few years. The PI plans to use some of these tools (for example, his work with Schwede on Seshadri constants in positive characteristic) to attack several problems in this setting.