Points on Rational Normal Curves and the ABCT Variety
Daniele Agostini, Lakshmi Ramesh, Dawei Shen
TL;DR
The paper analyzes ABCT varieties $V(d+1,n)$, focusing on $V(3,n)$, by realizing them as determinantal loci coming from the Veronese map on $G(2,n)$ and deriving a recursive description of their cohomology class $[V(3,n)]=[f_{n-5}]$ in $H^{2n-10}(G(3,n))$ via symmetric functions. It shows $V(3,n)$ coincides with the degeneracy locus $Z_2(3,n)$, provides explicit quartic Plücker equations for $n\ge5$, and computes degrees, including the Eulerian-number coefficient $A(n-3,1)$, which matches CHY predictions in spinor-helicity physics. The work extends the determinantal perspective to the general $Z_d(k,n)$, proving irreducibility and reducedness and establishing a broad, Coxeter-like recursion for their classes. It also develops a matroid/positroid stratification framework under Veronese maps and discusses implications for the positive geometry program, highlighting both promising structure and challenges from indeterminacy loci that may necessitate blow-ups. Overall, the results bridge algebraic geometry, combinatorics, and high-energy physics, yielding concrete tools for computing ABCT invariants and insights into the boundary structure of Veronese-embedded Grassmannians.
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.