Mukai lifting of self-dual points in $\mathbb{P}^6$

TL;DR

This paper investigates the inverse problem of reconstructing linear spaces associated with general self-dual point sets in projective space, using numerical methods and algorithms to analyze solutions and their real instances.

Contribution

It introduces a numerical homotopy continuation approach and Julia implementation to recover linear spaces from self-dual point sets, extending Mukai's framework.

Findings

Successfully implemented an algorithm for the inverse problem

Experimentally studied real solutions to the problem

Provided computational evidence supporting the approach

Abstract

A set of $2n$ points in $\mathbb{P}^{n-1}$ is self-dual if it is invariant under the Gale transform. Motivated by Mukai's work on canonical curves, Petrakiev showed that a general self-dual set of $14$ points in $\mathbb{P}^6$ arises as the intersection of the Grassmannian ${\rm Gr}(2,6)$ in its Pl\"ucker embedding in $\mathbb{P}^{14}$ with a linear space of dimension $6$. In this paper we focus on the inverse problem of recovering such a linear space associated to a general self-dual set of points. We use numerical homotopy continuation to approach the problem and implement an algorithm in Julia to solve it. Along the way we also implement the forward problem of slicing Grassmannians and use it to experimentally study the real solutions to this problem.