F-purity of Hypersurfaces.

F-purity;F-pure Threshold;Frobenius Morphism;Mathematics;Science;Mathematics

Hernandez, Daniel JesusZhang, Wenliang ;

University of Michigan

关键词: F-purity;    F-pure Threshold;    Frobenius Morphism;    Mathematics;    Science;    Mathematics;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/86491/dhernan_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

Recent work of Hara and Watanabe extends the classical and much-studied notion of F-purity for rings of prime characteristic to the setting of pairs. We study the conditionof F-purity for hypersurfaces (i.e.,principal ideals) in F-pure rings.In particular, much of this thesis is dedicated to the study ofF-pure thresholds, and we are strongly motivated by the connection between F-purity (defined via the Frobenius morphism) and log canonical singularities (defined via resolution of singularities) .We deduce some important properties of F-purity and F-pure thresholds, and use them to show that log canonical singularities is equivalent to dense F-pure type for very general hypersurfaces in complex affine space.We also give algorithms for computing the F-pure threshold of an arbitrary diagonal or binomial hypersurface.In the diagonal case, we compute the first non-trivial test ideal of the pair, as well as some higher jumping numbers.