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

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.