Questions on Singularities and Adjoint Linear Systems

首席研究员计划研究几个困难程度不同的问题，涉及代数变种研究中的局部和全局方面。人们早就知道，即使一个人对研究光滑的几何物体感兴趣，奇异的物体自然会出现。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，也可以是更复杂的结构。本提案中的项目涉及其中一些不变量及其在代数变量研究中的应用。这些几何对象可以在两种情况下进行研究：在特征为零的情况下，这是更熟悉的情况，更接近于我们通常的欧几里得空间；在正特征的情况下，更接近于算术。然而，这两种设置有着密切的联系，有时可以通过证明正特征的相应结果来证明特征为零的结果。然而，在这一点上，大多数关于代数变量的整体性质的结果只知道在特征零。本提案中的几个项目涉及将这些结果扩展到积极特征，并研究在这种情况下出现的更微妙的现象。在这个建议中处理的问题的共同特征是研究奇点的不同不变量之间的联系，以及在代数变量的整体性质的研究中使用这些不变量。PI计划探索关于奇点的不同观点之间的关系，例如弧空间与Hilbert-Samuel多重性之间的联系以及弧空间与分解定理之间的联系。他还计划研究伴随线性系统正特征的正性问题。这样的结果在特征零中已经已知，证明的基本工具是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设置中消失定理的失败。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在正特征中的Seshadri常数的研究）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.