Parke-Taylor Varieties

This page consists of supplementary material for the paper Parke Taylor Varieties by Benjamin Hollering and Dmitrii Pavlov. The code can be found on this page as well as the github repository below and consists of two files. 

The Macaulay2 file ParkeTaylorToric.m2 contains functions for creating the parameterization of the toric Parke--Taylor ideal $\mathcal{I}(T_n)$ as well as the set of binomials $\mathcal{B}_n$, which appear in Theorem 4.11. We conjecture that the binomials $\mathcal{B}_n$ generate the integer kernel $\ker_\mathbb{Z} A_n$ and hence generate the Parke--Taylor toric ideal. The last line of code in this file shows that the vectors $B_n$ span the kernel for $n = 7, 8, 9$. 

The Macaulay2 file ParkeTaylor.m2 contains functions for constructing the parametrization of the Parke--Taylor variety and computing its vanishing ideal. The function ptCircIdeal also allows the user to easily create the lifts of the Plücker relations we describe in Proposition 5.6 together with a set of binomials which generate the integer kernel of $A_n$. Lastly, one may compute all quadratic relations which hold on the Parke--Taylor variety with the function ptQuadIdeal, though this method is probabilistic and uses interpolation over a finite field.