Group Actions, Toric Varieties and Birational Geometry.
This page contains information for the course MA696W at Purdue
University.
Instructor: J. Wlodarczyk
e-mail: wlodar@math.purdue.edu
office hours: TTh , or by appointment; MATH 604
course website: www.math.purdue.edu/~wlodar/factorization/fact.html
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Summary:
Prerequisites. Basic knowledge about algebraic geometry (like
R.Hartshorne 'Algebraic Geometry' Chapter I or similar).
The purpose of this course is to give a survey on various
techniques used in birational geometry and its interactions with
invariant theory and toric geometry. We will introduce algebraic
group actions , good and geometric quotients, C^*-actions,
Bialynicki-Birula decomposition, reductive groups, geometric
invariant theory, Luna's slice theorem, birational cobordisms
(techniques inspired by topological cobordisms), and elements of Mori
theory. In the course we introduce and briefly discuss the theory of
toric varieties as the illustration of the above mentioned techniques
with particular emphasis on Mori theory , Morelli
cobordisms and $C^*$ -actions and the theory of valuations.
One of the main goals will be the the sketch of a proof of the
Weak Factorization Theorem which states that any birational map between
smooth projective varieties is a composition of blow-ups and blow-downs
along smooth centers.
The seminar should deal with the following:
• We prove the classical Bialynicki-Birula decomosition theorem
(for C^*-actions)
• We introduce birational cobordisms, and GIT for torus actions and
show the relation with birational factorization
• Introduce the reductive groups and the classical notion of
categorical, good and geometric quotients X//G for G reductive.
We extend the results on GIT for the reductive group actions .
• We prove the linear reductivity of some classical groups in
characteristic zero, and give the proof of Hilbert’s 14th problem for
reductive groups.
• We discuss the classical Luna's etale and Luna's slice Lemma for
Torus and Reductive group actions.
• We show the Hilbert-Mumford criterion for stability for the reductive
groups.
• We introduce toric varieties and illustrate the above concepts in the
toric setting
The focus of this course is to give an intuition about the interplay of
different areas of algebraic geometry.
Main texts:
M. Brion. Introduction to actions of algebraic
groups.
P.E. Newstead. Geometric Invariant Theory.
M. Bukstedt. Notes on Invariant group Theory
J.Wlodarczyk Algebraic Morse Theory and
Factorization of Birational Maps.
J.Wisniewski Toric Mori Theory and Fano
Manifolds. pdf.
Additional texts:
Bialynicki-Birula Some theorems on group actions.
Igor Dolgachev Lectures
on Invariant Theory
Jerzy Konarski
The B-B decomposition via Sumihiro Theorem
Tadao Oda Convex bodies and Toric Varieties
Kenji Matsuki Introduction to Mori Theory
V.L. Popov and E. B. Vinberg Invariant
Theory in Algebraic Geometry 4.
T.A. Springer Linear Algebraic Groups
Tentative contents of the course.
Final exam project information.