Supported partially by the National Science Foundation DMS #1064485: Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).
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Notes from the lectures:
All the notes so far in one PDF file: Click here. 114 pages total.
8-26-2010PDF We introduce using Frobenius to measure singularities. We showed that regular rings have flat F_* R-modules. We also proved some basic facts about Frobenius splittings.
8-31-2010PDF We show that F_* R being flat implies that R is regular. We also begin Fedder's criteria, the result that will give us a very effective method for checking whether a ring is F-split.
9-2-2010PDF We prove Fedder's criteria for F-splitting/purity We also proved some more basic facts about F-splittings and started looking at normality.
9-7-2010PDF We study how non-normal Frobenius split varieties can be, and classify one dimensional F-split singularities.
9-9-2010PDF We study projective varieties and when they are Frobenius split.
9-14-2010PDF We finish up some things from the previous time and we begin our study of rational singularities.
9-17-2010PDF We continue to study Cohen-Macaulay and rational singularities, including in the graded case.
9-21-2010PDF We study deformations of rationality and F-purity.
9-28-2010PDF We continue to study F-rationality and also discuss reduction to characteristic p.
9-30-2010PDF We prove that rational singularities are F-rational type (modulo a hard lemma) and also started talking about log canonical and log terminal singularities.
10-5-2010PDF We continue our study of log canonical and log terminal singularities. Then we begin our study of singularities of pairs in positive characteristic.
10-7-2010PDF We define notions of singularities of pairs in positive characteristic. We begin our description of how these change under birational maps.
10-19-2010PDF We prove that the test ideal and the multiplier ideal coincide after reduction to characteristic p >> 0.
10-26-2010PDF We study analogs of log canonical centers in characteristic p > 0.
10-28-2010PDF We discuss more properties of F-pure centers, we begin our discussion on killing local cohomology with finite maps.
11-9-2010PDF We discuss Matlis and local duality and also discuss more about F-rationality.
11-11-2010PDF We discuss vanishing up to finite maps and also begin our discussion of tight closure.
11-16-2010PDF We continue to discuss tight closure. We also mention Hilbert-Kunz multiplicity.
11-18-2010PDF We finish our discussion of tight closure, and we begin our proof of Hara's surjectivity lemma.
11-23-2010PDF We finish our proof of Hara's surjectivity lemma.
11-30-2010PDF We talk about globally F-regular varieties and the Mehta-Ramanathan criteria for Frobenius splitting.
12-2-2010PDF We finish our discussion of the Mehta-Ramanathan criteria for Frobenius splitting, briefly discuss diagonally Frobenius split varieties, and also a discussion of Frobenius splittings of toric varieties.
12-7-2010PDF We talk about failure of Kodaira-type vanishing in positive characteristic.
12-9-2010PDF We discuss Fujita's conjecture in positive characteristic.