Positive Genus Pairs from Amplituhedra

This page collects supplementary computations of the article Positive Genus Pairs from Amplituhedra by Joris Koefler, Dmitrii Pavlov and Rainer Sinn. The Macaulay2 file AmplituhedronResidualCurve.m2 contains the code for Example 4.9. Up to line 35 we do the setup to obtain the algebraic boundary of the amplituhedron. In the remaining part, we verify the claims made in the example. The file delPezzoCube.m2 contains all the code and computations for obtaining the canonical form of the positive geometry from Section 6 of the paper, and its residues. Up to line 86, we set up the interpolation conditions of the cubics going through 6 fixed points in the plane, corresponding to Figure 1c. Then, up to line 161, we compute the equations of the blow-up of the plane in these six points, and the strict transforms under the blow-up of the plane cubics, see Figure 1b. In the lines from 161 to 189 we compute the unique adjoint of the semi-algebraic set, and its canonical form. In the remainder, we have the two functions residue and vertexResidues, which compute all of two-fold iterated residues, and the resiudes along the vertices, respectively. We use these funcitons to check that we recover the unique canoncial forms on the edges of our positive geometry and that the resiudes at the vertices are all the same up to sign. The Mathematica notebook delPezzoPlots.nb has four sections. In the first one, all of the equations for the six facets and the adjoint of the example in Section 6 are recorded on an affine patch of the projective space. In the second section, we compute and generate the plots of four lines on the Del Pezzo surface, and then show the plot of the surface together with all four lines bounding a curvy quadrilateral on it, as well as the elliptic curve. The penultimate section is dedicated to producing the 3D graphics of the semi-algebraic set shown in Figure 1a of our paper. Lastly, we verify, by using cylindrical algebraic decomposition, that there is only one connected component on the surface.