Elementary Transversality in the Schubert calculus in any Characteristic

TL;DR

This paper proves that general codimension-1 Schubert varieties intersect transversally in Grassmannians across all characteristics, ensuring enumerative intersection numbers are valid universally, and enhances understanding in real enumerative geometry.

Contribution

It provides a characteristic-free proof of transversality for codimension-1 Schubert varieties, addressing gaps in existing results and broadening applicability.

Findings

Transversality holds in all characteristics for these intersections.

Intersection numbers are enumerative in Chow and quantum Chow rings.

Strengthens results in real enumerative geometry.

Abstract

We give a characteristic-free proof that general codimension-1 Schubert varieties meet transversally in a Grassmannian and in some related varieties. Thus the corresponding intersection numbers computed in the Chow (and quantum Chow) rings of these varieties are enumerative in all characteristics. We show that known transversality results do not apply to these enumerative problems, emphasizing the need for additional theoretical work on transversality. The method of proof also strengthens some results in real enumerative geometry.