Mukai lifting of self-dual points in ℙ⁶
Overview
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 \(\mathrm{Gr}(2,6)\) in its Plücker 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.
On this page you can find the code and data used to prove Theorem 5.1 and the implementation of numerical symmetric tensor decomposition using eigenvalue methods in Julia. We also indicate how to verify and reproduce our computational claims. Additionally we show some visualizations for the case example of 18 points in the plane and of the saturation gaps.
Julia source code
MukaiLiftSource.zipJulia example notebook
example.ipynb
Slicing using homotopy continuation
The computational approach to slicing the Grassmannian \({\rm Gr}(2,6) \subseteq \mathbb{P}^{14}\) is detailed in section 3 of the paper.
The implementation can be found in the file Slicing.jl in the source code MukaiLiftSource.zip.
Lifting using homotopy continuation
The computational approach to the Mukai lifting problem is detailed in section 4 of the paper.
The implementation can be found in the file MukaiLiftP6/src/Lifting.jl in the source code MukaiLiftSource.zip, there is also an instructive notebook example.ipynb. You can view the notebook in your browser on the following subpage: