Points on Rational Normal Curves and the ABCT Variety

TL;DR

This paper studies the ABCT variety, describing its geometric structure as a determinantal variety, deriving a recursive formula for its fundamental class, and connecting Schubert coefficients to Eulerian numbers.

Contribution

It provides a new determinantal description of the ABCT variety and establishes a recursive formula for its fundamental class, linking Schubert calculus to Eulerian numbers.

Findings

ABCT variety is a determinantal variety of a vector bundle morphism.

Recursive formula for the fundamental class of V(3,n).

Schubert coefficients are given by Eulerian numbers.

Abstract

The ABCT variety is defined as the closure of the image of $G(2,n)$ under the Veronese map. We realize the ABCT variety $V(3,n)$ as the determinantal variety of a vector bundle morphism. We use this to give a recursive formula for the fundamental class of $V(3,n)$. As an application, we show that special Schubert coefficients of this class are given by Eulerian numbers, matching a formula by Cachazo-He-Yuan. On the way to this, we prove that the variety of configuration of points on a common divisor on a smooth variety is reduced and irreducible, generalizing a result of Caminata-Moon-Schaffler.