(Unofficial) Topics in Algebraic Geometry
Instructor: Karl Schwede
Text: Frobenius Splitting Methods in Geometry and Representation Theory
by Michel Brion and Shrawan Kumar (we will probably use this text 30-40% of the time).
Syllabus: Click here
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).
Please let me know if you find typos in the notes.
Notes from the lectures:
All the notes so far in one PDF file: Click here. 114 pages total.
8-26-2010 PDF 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.
8-31-2010 PDF 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.
9-2-2010 PDF We prove Fedder's criteria for F-splitting/purity We also proved some more basic facts about F-splittings and started looking at normality.
9-7-2010 PDF We study how non-normal Frobenius split varieties can be, and classify one dimensional F-split singularities.
9-9-2010 PDF We study projective varieties and when they are Frobenius split.
9-14-2010 PDF We finish up some things from the previous time and we begin our study of rational singularities.
9-17-2010 PDF We continue to study Cohen-Macaulay and rational singularities, including in the graded case.
9-21-2010 PDF We study deformations of rationality and F-purity.
9-23-2010 PDF We introduce F-rationality.
9-28-2010 PDF We continue to study F-rationality and also discuss reduction to characteristic p.
9-30-2010 PDF We prove that rational singularities are F-rational type (modulo a hard lemma) and also started talking about log canonical and log terminal singularities.
10-5-2010 PDF We continue our study of log canonical and log terminal singularities. Then we begin our study of singularities of pairs in positive characteristic.
10-7-2010 PDF We define notions of singularities of pairs in positive characteristic. We begin our description of how these change under birational maps.
10-19-2010 PDF We prove that the test ideal and the multiplier ideal coincide after reduction to characteristic p >> 0.
10-26-2010 PDF We study analogs of log canonical centers in characteristic p > 0.
10-28-2010 PDF We discuss more properties of F-pure centers, we begin our discussion on killing local cohomology with finite maps.
11-9-2010 PDF We discuss Matlis and local duality and also discuss more about F-rationality.
11-11-2010 PDF We discuss vanishing up to finite maps and also begin our discussion of tight closure.
11-16-2010 PDF We continue to discuss tight closure. We also mention Hilbert-Kunz multiplicity.
11-18-2010 PDF We finish our discussion of tight closure, and we begin our proof of Hara's surjectivity lemma.
11-23-2010 PDF We finish our proof of Hara's surjectivity lemma.
11-30-2010 PDF We talk about globally F-regular varieties and the Mehta-Ramanathan criteria for Frobenius splitting.
12-2-2010 PDF 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.
12-7-2010 PDF We talk about failure of Kodaira-type vanishing in positive characteristic.
12-9-2010 PDF We discuss Fujita's conjecture in positive characteristic.