Abstract: In his first papers on fibrations and rank 2 reflexive bundles in projective space in the late 1970s, Hartshorne presented a correspondence, attributed
originally to Serre, between such objects and space curves, in a way generalizing the well-known correspondence between locally free rank 1 bundles and divisors in projective varieties. This correspondence, which has become known as the Serre or Hartshorne-Serre construction, is one of the main tools for studying rank 2 reflexive bundles and their module spaces, with numerous applications in algebraic geometry and mathematical physics.

The aim of the mini-course is to present a very general version of Serre’s Construction, which covers torsion-free bundles of any rank on arbitrary projective varieties. To do this, it will be necessary to review the theory of bundles on projective varieties, especially torsion-free and reflexive bundles. Finally, I will present some applications of Serre’s Construction to the study of spaces of modules of bundles in projective space, to the construction of instanton bundles, and to the study of distributions and foliations.

 

Schedule

  1. Presentation, preliminaries on untwisted beams and reflective beams
  2. The Construction of Serre
  3. More on beams in projective varieties
  4. Stability of post beams 2
  5. Discussion of examples: rank 3 instantons and rank 2 reflective beams
  6. Applications to the study of distributions
  7. Module spaces, I
  8. Module spaces, II