Questions on Singularities and Adjoint Linear Systems

首席研究员计划工作的几个问题，不同程度的困难，关于当地和全球方面的研究代数簇。很长一段时间以来，人们都知道，即使人们对研究光滑的几何物体感兴趣，奇异的物体也会自然出现。因此，能够测量奇点是很重要的，这是通过各种不变量来完成的，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种背景下研究：在特征零中，这是更熟悉的背景，更接近于我们通常的欧几里得空间;在正特征中，更接近于算术。然而，这两个设置有密切的联系，有时可以证明结果的特征零证明相应的结果在积极的特点。然而，在这一点上，大多数关于代数簇的整体性质的结果只知道在特征零。在这个建议中的几个项目涉及的扩展等结果的积极特征和研究的更微妙的现象，出现在这种setting.The共同特点的问题，在这个建议中解决的是研究不同的不变量之间的联系的奇异性和使用这种不变量的研究中的全球性质的代数簇。PI计划探索奇点的不同观点之间的关系，例如弧空间与希尔伯特-塞缪尔多重性之间的联系以及弧空间与分解定理之间的联系。他还计划工作的问题有关的积极性质的伴随线性系统的积极特征。这样的结果是已知的特征零，基本工具的证明是科代拉的消失定理及其推广。推广这些结果的主要困难是在新的设置消失定理的失败。然而，在过去的几年中，已经开发了基于Frobenius同态的新工具。PI计划使用其中的一些工具（例如，他与Schwede在Seshadri常数的正特性方面的工作）来解决这种情况下的几个问题。

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.