With Wronskian through the Looking Glass

TL;DR

This paper explores the relationship between the Wronskian map and the Plücker map, revealing how the Wronskian polynomials relate to tau-functions in the KP hierarchy, thus connecting algebraic geometry with integrable systems.

Contribution

It establishes a precise link between the Wronskian map and the Plücker map, clarifying the nature of polynomials in the Wronskian image as initial tau-functions.

Findings

The Wronskian map is closely related to the Plücker map.

Polynomials in the Wronskian image are initial tau-functions of KP hierarchy.

The work recovers and clarifies previous results by Varchenko and Wright.

Abstract

In the work of Mukhin and Varchenko from 2002 there was introduced a Wronskian map from the variety of full flags in a finite dimensional vector space into a product of projective spaces. We establish a precise relationship between this map and the Pl\"ucker map. This allows us to recover the result of Varchenko and Wright saying that the polynomials appearing in the image of the Wronsky map are the initial values of the tau-functions for the Kadomtsev-Petviashvili hierarchy.