Algebraic groups actions, invariants, birational geometry

University of Trento, March-April 2020

Jaroslaw Wisniewski, University of Warsaw

root system E8 source: Wikipedia

Summary: The course concerns algebraic group actions on algebraic varieties. The introductory part is about polynomial invariants of linear actions of finite groups. Next we consider algebraic actions of affine algebraic groups and their quotients. A special consideration is given to algebraic torus action and quotients. Finally, we focus on applications to birational geometry of projective varieties. Prerequisites: The course is aimed at students who completed commutative algebra lectures (ring and fields) and an introductory course to algebraic geometry.

Topics

1. Linear actions of finite groups and their invariants: Polynomial invariants, Reynolds operators, Hilbert theorem about finite generation of the ring of invariants. Poincare series and Molien's theorem. Groups generated by pseudo-reflections, the theorem of Todd, Shepherd and Chevalley. Readings: Sturmfels, Algorithms in invariant theory, sect 1 and 2.1, 2.2, 2.4.
2. Quotient singularties: Singularities and their resolution, quotient singularties, surface singularities, Du Val singularties. Readings: Wisniewski, Notes on singularties, Sect 1 and 2. Reid, The Du Val singularities
3. Actions of linear algebraic groups and quotients: Affine algebraic groups, linear groups, reductive groups, various characterisations. Geometric quotients, categorical quotients. Stability, quotient of the set of stable points is geometric. Linearization, semistable points, good quotients. Readings: Brion, Introduction to actions of algebraic groups, Part I.
4. Birational geometry of quotients: Linearizations, semistable locus. Variation of liniearization and stability condition. Case of C* action: quotients of affine space and Atiyah flip/flop. Mukai flop. Morelli-Wlodarczyk cobordism. Cox rings and Mori Dream Spaces. Eff and Nef cones. Readings: Laface, Velasco, A survey on Cox rings, Reid, What is a flip?, sect 1 and 2.

Problem and exercise sheets:

1. First set: finite group actions, action of cyclic groups, symmetric polynomials, groups acting on the plane.
2. Second set: Abelian group actions, characters, Poincare-Hilbert series.
3. Third set: Finite subgroups of isometries of the sphere.
4. Fourth set: Cyclic coverings, resolving surface singularities; remarks on solutions, solution of 2c by Alberto, solution of 2a by Federico.
5. Fifth set: Unipotent groups, quotients by C*.
6. Sixth set: Stability, geometric quotients; remarks,
7. Seventh set: Resolving surface An singularities via toric quotients.
8. Eighth set: Cox ring of a Del Pezzo surface; remarks,

1. Group actions and quotients (summary of Brion notes)
2. Linearizations of trivial bundle, good quotient of affine varieties
3. C* quotients of affine spaces, Atiyah flip and flop. Morelli-Wlodarczyk cobordism.
4. Cox rings

1. 1st set, invariants of cyclic groups.
2. 2nd set, Poincare series, Molien theorem.
3. 3rd set, finite groups of spacial symmetries.

More readings:

• Arzhantsev, Derenthal, Hause, Laface, Cox Rings, part I
• Brion, Introduction to actions of algebraic groups
• Dolgachev, Lectures on invariant theory
• Dolgachev Reflection groups in algebraic geometry
• Durfee, Fifteen characterizations of rational double points
• Laface, Velasco, A survey on Cox rings
• Reid, What is a flip?
• Stanley, Invariants of finite groups and their applications to combinatorics.
• Sturmfels, Algorithms in invariant theory
• Benson, Polynomial invariants of finite groups. London Mathematical Society Lecture Note Series, 190. Cambridge University Press, Cambridge, 1993.
• Mukai, An introduction to invariants and moduli, Cambridge Studies in Advanced Mathematics, 81. Cambridge University Press, Cambridge, 2003.