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Truncated Grassmannians, blow-ups along Schubert varieties and collineations

Evgeny Feigin

Abstract

Truncated Grassmannians are defined as closures of orbits of abelian unipotent groups acting on the degree truncations of projectivized wedge powers. We show that such truncations in a more general setup show up in the description of the blow-ups of general flag varieties along Schubert subvarieties. We work out the case of Grassmannians in detail.In particular, we show that our blow-ups are members of a larger family of varieties projecting onto Grassmannians, and describe the fibers of these projections via the spaces of collineations.

Truncated Grassmannians, blow-ups along Schubert varieties and collineations

Abstract

Truncated Grassmannians are defined as closures of orbits of abelian unipotent groups acting on the degree truncations of projectivized wedge powers. We show that such truncations in a more general setup show up in the description of the blow-ups of general flag varieties along Schubert subvarieties. We work out the case of Grassmannians in detail.In particular, we show that our blow-ups are members of a larger family of varieties projecting onto Grassmannians, and describe the fibers of these projections via the spaces of collineations.

Paper Structure

This paper contains 16 sections, 35 theorems, 66 equations.

Table of Contents

  1. The setup
  2. Lie algebras and groups
  3. Representations
  4. Cyclic quotients
  5. Grassmannians and Schubert varieties
  6. Vector bundles on Grassmannians
  7. Truncated Grassmannians and their relatives
  8. Truncated Grassmannians and blow-ups
  9. Truncated Grassmannians
  10. Fibers of the blow-ups
  11. Blow-ups of Grassmannians and ideal sheaves
  12. Mixed case and collineations
  13. Collineations
  14. Mixed truncated Grassmannians
  15. The general case
  16. ...and 1 more sections

Key Result

Theorem 1

The blow-up $\mathrm{Bl}_{S_r}\mathrm{Gr}_d(V)$ is isomorphic to the closure of the graph of the birational map $X_{r-1}\to\mathrm{Gr}_d(V)$. In particular, $\mathrm{Bl}_{S_r}\mathrm{Gr}_d(V)$ admits the action of $\exp({\mathfrak a}_d)$ with an open dense orbit. $\blacktriangleleft$$\blacktrianglel

Theorems & Definitions (79)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1.1
  • Remark 1.2
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • Corollary 1.5
  • ...and 69 more