# Positive del Pezzo Geometry

**Abstract:** Real, complex, and tropical algebraic geometry join forces in a new branch of mathematical physics called positive geometry. We develop the positive geometry of del Pezzo surfaces and their moduli spaces, viewed as very affine varieties. Their connected components are derived from polyhedral spaces with Weyl group symmetries. We study their canonical forms and scattering amplitudes, and we solve the likelihood equations.

This page is ordered by the same sections as the paper it accompanies. For each section we make the code available that was used to produce examples, stepping stones in proofs or numerical verifications. This data is always labeled according to the corresponding location in the paper.

This page is work in progress, i.e. not all the code and data are available yet. In case of interest in certain computations not yet on this page, please contact us.

The 15 facets of the E6 pezzotope are displayed here: 10 associahedra and 5 cubes.

## Section 2: Blowing up Four Points

We provide code for confirming numerically using the scattering equations formula (CHY) that the scattering amplitude is equal to expression (10). The source code is in \(\verb|Mathematica|\). A notebook with code and explanations can be downloaded here: `section2_amplitude.nb`

A pdf version of the \(\verb|Mathematica|\) notebook can be viewed here: `section2_amplitude.pdf`

Code written by: Nick Early and Claudia Yun, 22/06/2023

Software used: Mathematica (Version 13.0)

System setup used: MacBook Pro with macOS Monterey 12.6.1, 2,6 GHz Quad-Core Intel Core i7, Memory 16 GB

## Section 3: Polygons

**Example 3.2**: The source code is in \(\verb|Mathematica|\). A notebook with code and explanations can be downloaded here: `Ex3.2.nb`

A pdf version of the \(\verb|Mathematica|\) notebook can be downloaded/viewed here: `Ex3.2.pdf`

Code written by: Claudia Yun, 22/06/2023

Software used: Mathematica (Version 13.0)

System setup used: MacBook Pro with macOS Monterey 12.6.1, 2,6 GHz Quad-Core Intel Core i7, Memory 16 GB

## Section 4: Euler Characteristic

Computation of the 2111 strata in \(Y(3,6)\) as displayed in Table 1.

*coming soon*

Code written by: Claudia Yun, 22/06/2023

Software used: Julia

System setup used: MacBook Pro with macOS Monterey 12.6.1, 2,6 GHz Quad-Core Intel Core i7, Memory 16 GB

## Section 5: Numerical Experiments

**Experiment 5.1**: Code and setup description for computing the Euler Characteristic of \(Y(3,6)\), \(Y(3,7)\)

Code written by: Alheydis Geiger, 05/04/2023

Software used: Julia (Version 1.9.0), HomotopyContinuation.jl

System setup used: MacBook Pro with macOS Monterey 12.6.6 2,6 GHz Quad-Core Intel Core i7, Memory 16 GB

**Experiment 5.2:** Code for computing the lower bound of the Euler Characteristic of Y(3,8) as in Theorem 4.1

The source code can be downloaded here: `y38.jl`

Code written by: Sascha Timme, 21/06/2023

Software used: Julia, HomotopyContinuation.jl (version 2.9.0 )

System setup used: MacBook Pro with M1 Pro Chip and 32 GB RAM

**Experiment 5.3:** Code on tropical critical points of \(Y(3,6)\).

`tropical_critical_points_Y(3,6).jl`

`tropical_critical_boundary_Y(3,6).jl`

Code written by: Alheydis Geiger, 05/04/2023

Software used: Julia (Version 1.9.0), HomotopyContinuation.jl

System setup used: MacBook Pro with macOS Monterey 12.6.6 2,6 GHz Quad-Core Intel Core i7, Memory 16 GB

**Experiment 5.4:** Code on tropical critical points of \(Y(3,7)\).

`tropical_critical_points_Y(3,7).jl`

`here`

`tropical_critical_boundary_Y(3,7).jl`

`here`

Code written by: Alheydis Geiger, 05/04/2023

Software used: Julia (Version 1.9.0), HomotopyContinuation.jl

System setup used: MacBook Pro with macOS Monterey 12.6.6 2,6 GHz Quad-Core Intel Core i7, Memory 16 GB

## Section 6: Weyl Groups, Roots, and their ML Degrees

**group action** \(E_6\), \(E_7\)

*coming soon*

Code written by: Claudia Yun, 22/04/2023

**Numerical verification of Prop. 6.1:** Code to compute ML degrees of \(E_6\) and \(E_7\)

Code written by: Alheydis Geiger, 05/04/2023

Software used: Julia (Version 1.9.0), HomotopyContinuation.jl

**Experiment 6.3:** Code to compute the ML degree of the Yoshida and Göpel parametrization

**Yoshida variety**can be downloaded here:

**Yoshida variety**with explanation is available here:

`polys_Y`

(40 polynomials in the 6 variables from \(\mathbb{P}^5\) that give the monomials of the map \(\mathbb{P}^{35}\to\mathbb{P}^{39}\).)`factorised_polys_Y`

(factorization of the polynomials from above into the 36 different linear factors corresponding to the coordinates of \(\mathbb{P}^{35}\).)`factorised_polys_Yinx`

(The same polynomials as above, where each of the 36 linear factors is substituted by the corresponding variable x[i] in lexicographic ordering.)`system_for_Y`

(sum over the 40 polynomials in the 36 variables from above each multiplied by a parameter a[i])**Göpel variety**variety with explanation can be downloaded here:

`matrix_for_G`

(This file is needed to execute the code in MLdegree_G.jl)Code written by: Alheydis Geiger, 05/04/2023

Software used: Julia (Version 1.9.0), HomotopyContinuation.jl

## Section 8: Combinatorics of Pezzotopes

**Clique complex for** \(\mathcal{G}(E_6)\), \(\mathcal{G}(E_7)\)

*coming soon*

Code written by: Nick Early

**Theorem 8.2:**

*coming soon*

Code written by: Nick Early

**Theorem 8.1:** The positive tropical prevarieties of trop(\(\mathcal{Y}\)) and trop(\(\mathcal{G}\)) are simplicial fans with f-vector as in Theorem 8.1.

The \(\verb|Jupyter|\) notebook can be downloded here: `E6-Pezzotope.ipynb`

The \(\verb|Jupyter|\) notebook can be downloded here: `E7-Pezzotope.ipynb`

The \(\verb|Jupyter|\) notebook can be viewed here:

Jupyter notebook written by: Marta Panizzut and Alheydis Geiger, 23/06/2023

Software used: Julia (Version 1.9.0),

## Section 9: Geometry of Pezzotopes

**Proposition 9.1:** Code for verification is in \(\verb|Mathematica|\). A notebook with code and explanations can be downloaded here: `Prop9.1.nb`

A pdf version of the \(\verb|Mathematica|\) notebook can be downloded/viewed here: `Prop9.1.pdf`

Code written by: Claudia Yun, 23/06/2023

Software used: Mathematica (Version 13.0)

**Theorem 9.4**: Data and Code for \(u\)-equations and Gorenstein property of \(E_7\).

*coming soon*

## Section 10: Grassmannians, Positive Geometries, and Beyond

**Proposition 10.1:** In the proof of the proposition, we claim that for every polygon on \(\mathcal{S}^\circ_6\), we can find six pairwise disjoint lines in \(\mathcal{S}_6\) that are disjoint from the closure of the polygon in \(\mathcal{S}_6\). We verify this claim computationally for the list of polygons from Example 3.2.

The source code is in \(\verb|Mathematica|\). A notebook with code and explanations can be downloaded here: `Prop10.1.nb`

.

A pdf version of the \(\verb|Mathematica|\) notebook can be downloaded/viewed here : `Prop10.1.pdf`

.

Code written by: Claudia Yun, 22/06/2023

Software used: Mathematica (Version 13.0)

**Experiment 10.2**

`This file`

provides the rays (as vectors of length 20+1=21) of \(trop_+Y(3,6)\) by modifying the positive tropical Grassmannian with a single vector, due to \(q = p_{123}\cdot p_{345}\cdot p_{156}\cdot p_{246}-p_{234}\cdot p_{456}\cdot p_{126}\cdot p_{135}\). That vector is in position 1 in the list of 15 vectors in the file.

The ordering in each vector is lexicographic: {1,2,3},{1,2,4},{1,2,5},{1,2,6},{1,3,4},{1,3,5},{1,3,6},{1,4,5},{1,4,6},{1,5,6},{2,3,4},{2,3,5},{2,3,6},{2,4,5},{2,4,6},{2,5,6},{3,4,5},{3,4,6},{3,5,6},{4,5,6}.

The file also provides a list of the 2-dimensional cones, which can be used to construct the whole fan.

More information on code and used software on request. Code written by: Nick Early

**Theorem 10.3:** The bijection between the 34 rays and \(u\)-variables

*coming soon*

**Theorem 10.4**

*coming soon*

Code written by: Nick Early

Project page created: 22/06/2023

Project contributors: Nick Early, Alheydis Geiger, Marta Panizzut, Bernd Sturmfels, Claudia He Yun

Corresponding author of this page: Alheydis Geiger, geiger@mis.mpg.de

License for code of this project page: MIT License (https://spdx.org/licenses/MIT.html)

License for all other content of this project page (text, images, …): CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/)

Last updated 29/11/2023.