PisaHokkaidoTV

Introduction to Schubert Varieties

Born in the 19th century to answer questions from enumerative geometry, Schubert varieties have been extensively studied since then. Lying in the intersection of geometry, combinatorics and algebra, these varieties naturally bridge problems in these different areas.

This mini-course is meant to provide an introduction to the topic, with a particular emphasis on the interaction between algebraic geometry and algebraic combinatorics.

Timetable: Mon 3rd of September 14:30--16:30; Tue 4th of September 14:30--16:30; Wed 35th of September 14:30--16:30; Thu 6th of September 14:30--16:30.

All lectures will take place in Aula Dini.

Exam: The students are asked to solve some of the proposed exercises , available here, and email them to me in pdf form. Please, do NOT scan and email me your solutions handwritten (the only exception is the Hasse diagram of the Bruhat order on S4), since I will not correct them!

Lecture notes are available here (last update 7th of Sept, h 18:40).

Diary of the lectures:

• • 3rd of Sept: Some questions in enumerative geometry; the Grassmann variety Gr(k,n); representing points in Gr(k,n) via matrices; the Pluecker embedding.

  • 4th of Sept: Pluecker relations; Schubert cells and Schubert varieties; Bruhat-Chevalley order; the case of projective spaces; defining equations for Schubert varieties.

  • 5th of Sept: Other index sets for Schubert varieties; flags, transverse flags; the Duality Theorem (proof and an application); zero-dimensional LR rule (a glimmer).

  • 6th of Sept: the flag variety; representing complete flags via matrices; Schubert varieties and cells in the flag variety; pattern avoidance smoothness criterion; sketch of generalisations of the constructions seen this week: partial flag variety, flag variety of an algebraic group, the symplectic flag variety.

If you found fun the arguments we have discussed, you might want to have a look