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 — room 713, 10:30--12:30

• July 13 and 14 — room 716, 10:30--13 

• July 15 — room 716, 10:30--12:30

• July 21 and 23 — room 713, 10:30--13

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.