PhD course: Birational geometry and Group Actions 

We are pleased to announce a PhD course held at the University of Genova by Prof. Jarek Wiśniewski.

Schedule

The course has a total duration of 16 hours and will take place according to the following schedule:

• July 6 and 9 — 2-hour lectures

• July 13, 14, 15 — two 2.5-hour lectures and one 2-hour lecture

• July 21 and 23 — 2.5-hour lectures

All lectures will start at 13:30, to allow participants traveling from outside Genova to arrive comfortably.

The classroom(s) will be announced soon.

Provisional Topics

The actual content will depend on the background of the participants. The main topics include:

• Algebraic groups
  Algebraic group actions on affine varieties; algebraic tori and their actions; reductive groups; actions on projective varieties; linearization.

• Quotients of group actions
  Affine case and rings of invariants; projective case; stability conditions; stable and semistable points; geometric invariant theory; variation of stability conditions.

• Mori Dream Spaces and Cox rings
  MDS as quotients; torsor structures; variation of stability conditions via the decomposition of the big cone; differentials of Cox rings and potential sheaves.

• Toric varieties
  Toric varieties as quotients of affine spaces; fans; line bundles; polytopes and piecewise linear functions; toric Minimal Model Program; secondary cones.

• Birational geometry
  Birational maps via Włodarczyk–Morelli algebraic bordism; Białynicki-Birula decomposition for $\mathbb{C}^*$-actions; cobordism and Chow quotients.

• Varieties with GKM torus actions
  Graphs with parallel transport and compasses; line bundles as embedded graphs (grids); construction of toric varieties via GKM graphs.