Schubert induction

By Ravi Vakil. 2003

60 citations recorded by the source corpus.

A BSTRACT . We describe a Schubert induction theorem, a tool for analyzing intersections on a Grassmannian over an arbitrary base ring. The key ingredient in the proof is the Geometric Littlewood-Richardson rule of [V2]. Schubert problems are among the most classical problems in enumerative geometry of continuing interest. As an application of Schubert induction, we address several long-standing natural questions related to Schubert problems, including: the “reality” of solutions; effective numerical methods; solutions over algebraically closed fields of positive characteristic; solutions over finite fields; a generic smoothness (Kleiman-Bertini) theorem; and monodromy groups of Schubert problems. For example, we show that all Schubert problems for all Grassmannians are enumerative over the real numbers, completely answering the classical “reality question” for Schubert problems. These methods conjecturally extend to the flag variety.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/f3ba19327208e862801f986aef27ec6856853ab5

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Nicolas Ressayre
2. Izzet Coskun
3. P. Belkale
4. Ravi Vakil
5. Sam Payne
6. Alexander A. Klyachko
7. W. Crawley‐Boevey
8. A. Marian

In Computational Theory and Mathematics

1. Izzet Coskun
2. P. Belkale
3. Nicolas Ressayre
4. Ravi Vakil

In Discrete Mathematics and Combinatorics

1. Nicolas Ressayre
2. Izzet Coskun
3. Ravi Vakil
4. P. Belkale
5. Alexander A. Klyachko
6. W. Crawley‐Boevey
7. Shrawan Kumar

Citations in the corpus

• Geometric proofs of horn and saturation conjectures (2002, background): “oof of GH to establish transversality statements for Schubert calculus of Grassmannians in any characteristic (see Corollary 9.2). A proof of transversality also appears in a recent paper of R. Vakil [20]. Let us first introduce some notation, and then recall the statement of GH, phrasing it in a sli”
• A Littlewood–Richardson rule for two-step flag varieties (2009, methodology): “Vakil using a geometric Littlewood–Richardson rule was able to establish the reality of Schubert calculus [19].”
• Geometric invariant theory and the generalized eigenvalue problem (2007, background): “This map plays also a central role in Vakil’s work about Schubert calculus (see [42]).”
• A geometric Littlewood-Richardson rule (2003, methodology): “Further geometric applications are discussed in [V2].”