Tautological systems under the conifold transition on $G(2, 4)$

TL;DR

This paper explores the relationship between tautological systems on Grassmannians and extended GKZ systems on toric degenerations, focusing on the case of $G(2,4)$ to establish a foundational connection.

Contribution

It introduces a method to relate tautological systems on Grassmannians to extended GKZ systems on toric degenerations, validated through the specific case of $G(2,4)$.

Findings

Extended GKZ system can be viewed as a tautological system on the small resolution of the toric degeneration.

Validated the approach explicitly for the case $(k,n)=(2,4)$.

Provides a framework for studying tautological systems via toric degenerations.

Abstract

Via a natural degeneration of Grassmannian manifolds $G(k,n)$ to Gorenstein toric Fano varieties $P(k,n)$ with conifold singularities, we suggest an approach to study the relation between the tautological system on $G(k,n)$ and the extended GKZ system on the small resolution $\hat{P}(k,n)$ of $P(k,n)$. We carry out the simplest case $(k,n)=(2,4)$ to ensure its validity and show that the extended GKZ system can be regarded as a tautological system on $\hat{P}(2,4)$.