Lemma22.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Mukai edge graph"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This jupyter notebook contains the functions implemented to prove the computational results presented in the article \"Cox rings and Mukai edge graphs\". In particular, here we present the compuations necessary to analyze Mukai edge graphs presented in Section 6. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\r",
      " -----    -----    -----      -      -----   \n",
      "|     |  |     |  |     |    | |    |     |  \n",
      "|     |  |        |         |   |   |     |  \n",
      "|     |   -----   |        |     |  |-----   \n",
      "|     |        |  |        |-----|  |   |    \n",
      "|     |  |     |  |     |  |     |  |    |   \n",
      " -----    -----    -----   -     -  -     -  \n",
      "\n",
      "...combining (and extending) ANTIC, GAP, Polymake and Singular\n",
      "Version\u001b[32m 0.10.0 \u001b[39m... \n",
      " ... which comes with absolutely no warranty whatsoever\n",
      "Type: '?Oscar' for more information\n",
      "(c) 2019-2022 by The Oscar Development Team\n"
     ]
    }
   ],
   "source": [
    "using Oscar\n",
    "using Combinatorics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we analyze the polytopes constructed in the paper \"Sagbi bases of Cox-Nagata rings\" by Sturmfels and Xu. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We begin with the polytopes appearing in the degenerations of the Cox ring of the blow up of five points in $\\mathbb{P}^2$. We see that in this case polytopes with $(5,2)$-Mukai edge graph have minimal number of facets. \n",
    "We also check that the $(5,2)$-Mukai edge graph is a subgraph of all others vertex-edge graphs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-------- Polytope 1\n",
      "Mukai edge graph is strictly contained\n",
      "fmpz[16, 84, 200, 253, 180, 71, 14]\n",
      "-------- Polytope 2\n",
      "Mukai edge graph is strictly contained\n",
      "fmpz[16, 87, 221, 301, 229, 94, 18]\n",
      "-------- Polytope 3\n",
      "Mukai edge graph is strictly contained\n",
      "fmpz[16, 87, 221, 301, 229, 94, 18]\n",
      "-------- Polytope 4\n",
      "Mukai edge graph is strictly contained\n",
      "fmpz[16, 87, 221, 301, 229, 94, 18]\n",
      "-------- Polytope 5\n",
      "Mukai edge graph is strictly contained\n",
      "fmpz[16, 87, 221, 301, 229, 94, 18]\n",
      "-------- Polytope 6\n",
      "The edge graph is the Mukay edge graph\n",
      "fmpz[16, 80, 180, 216, 148, 58, 12]\n"
     ]
    }
   ],
   "source": [
    "S_t, (e1,e2,e3,e4,e5,H45,H35,H34,H25,H24,H23,H15,H14,H13,H12,Q) = PolynomialRing(QQ, [\"e1\",\"e2\",\"e3\",\"e4\",\"e5\",\"H45\",\"H35\",\"H34\",\"H25\",\"H24\",\"H23\",\"H15\",\"H14\",\"H13\",\"H12\",\"Q\"])\n",
    "vars_S=S_t.([e1,e2,e3,e4,e5,H45,H35,H34,H25,H24,H23,H15,H14,H13,H12,Q])\n",
    "\n",
    "function Mukai_form_52(i,j)\n",
    "    E=deg_mon[i]\n",
    "    D=deg_mon[j]\n",
    "    return (E[1]*D[1])-(E[1]-E[2])*(D[1]-D[2])-(E[1]-E[3])*(D[1]-D[3])-(E[1]-E[4])*(D[1]-D[4])-(E[1]-E[5])*(D[1]-D[5])-(E[1]-E[6])*(D[1]-D[6]);\n",
    "end\n",
    "    \n",
    "deg_mon=[];\n",
    "\n",
    "\n",
    "for i in 1:5\n",
    "    for j in i+1:5\n",
    "        temp_deg_mon=[1, 1, 1, 1, 1, 1];\n",
    "        temp_deg_mon[i+1]=0;\n",
    "        temp_deg_mon[j+1]=0;\n",
    "        push!(deg_mon,temp_deg_mon);\n",
    "    end\n",
    "end\n",
    "\n",
    "reverse!(deg_mon)\n",
    "\n",
    "for i in [5,4,3,2,1]\n",
    "    temp=[0, 0, 0, 0, 0, 0];\n",
    "    temp[i+1]=1;\n",
    "    pushfirst!(deg_mon,temp);\n",
    "end\n",
    "\n",
    "push!(deg_mon,[2,1,1,1,1,1]);\n",
    "\n",
    "R, (x1,x2,x3,x4,x5,y1,y2,y3,y4,y5) = PolynomialRing(QQ, [\"x1\",\"x2\",\"x3\",\"x4\",\"x5\",\"y1\",\"y2\",\"y3\",\"y4\",\"y5\"]);\n",
    "x = R.([x1,x2,x3,x4,x5]);\n",
    "y = R.([y1,y2,y3,y4,y5]);\n",
    "equations=[R.([x1, x2, x3, x4, x5, y1*x2*x3, y1*x2*x4, y1*x2*x5, x1*y3*x4, x1*y3*x5, x1*y4*x5, x2*y3*x4, x2*y3*x5, x2*y4*x5, y3*x4*x5, y1*x2*y3*x4*x5]),\n",
    "R.([x1, x2, x3, x4, x5, x1*x2*y3, y1*x2*x4, y1*x2*x5, x1*y3*x4, x1*y3*x5, x1*y4*x5, x2*y3*x4, x2*y3*x5, x2*y4*x5, y3*x4*x5, y1*x2*y3*x4*x5]),\n",
    "R.([x1, x2, x3, x4, x5,y1*x2*x3, y1*x2*x4,y1*x2*x5, x1*y3*x4, x1*y3*x5, y1*x4*x5, x2*y3*x4, x2*y3*x5, x2*y4*x5, y3*x4*x5, y1*x2*y3*x4*x5]),\n",
    "R.([x1, x2, x3, x4, x5, y1*x2*x3, y1*x2*x4,y1*x2*x5, x1*y3*x4, x1*y3*x5, y1*x4*x5, x2*y3*x4, x2*y3*x5, y2*x4*x5, y3*x4*x5, y1*x2*y3*x4*x5]),\n",
    "R.([x1, x2, x3, x4, x5, y1*x2*x3, y1*x2*x4,y1*x2*x5, y1*x3*x4, y1*x3*x5, y1*x4*x5, y2*x3*x4, y2*x3*x5, y2*x4*x5, y3*x4*x5, y1*y3*x2*x4*x5]),\n",
    "R.([x1, x2, x3, x4, x5, y1*x2*x3, y1*x2*x4, y1*x2*x5, x1*y3*x4, x1*y3*x5, x1*y4*x5, x2*y3*x4,x2*y3*x5, x2*y4*x5, x3*y4*x5, y1*x2*x3*y4*x5])]\n",
    "\n",
    "for (el,cont) in zip(equations,range(1,6))\n",
    "    println(\"-------- Polytope \",cont)\n",
    "    M=zero_matrix(ZZ,length(el),10)\n",
    "    for (eq,row) in zip(el,range(1,length(el)))\n",
    "        for (var,col) in zip(R.([x1,x2,x3,x4,x5,y1,y2,y3,y4,y5]),range(1,10))\n",
    "            M[row,col]=degree(eq,var)\n",
    "        end\n",
    "    end\n",
    "\n",
    "    P=convex_hull(M);\n",
    "    EdGr=Oscar.Graphs.edgegraph(P);\n",
    "    E=Oscar.Graphs.edges(EdGr);\n",
    "\n",
    "    bol_mukai=true\n",
    "    for vert in range(1,length(vertices(P)))\n",
    "        cont_zero=0\n",
    "        cont=0\n",
    "        for vert2 in Graphs.neighbors(EdGr,vert)\n",
    "            cont=cont+1\n",
    "            if Mukai_form_52(vert,vert2)==0\n",
    "                cont_zero=cont_zero+1\n",
    "            end\n",
    "        end\n",
    "        if (cont!=cont_zero) & bol_mukai\n",
    "            println(\"Mukai edge graph is strictly contained\")\n",
    "            bol_mukai=false\n",
    "        end\n",
    "    end\n",
    "    if bol_mukai\n",
    "        println(\"The edge graph is the Mukay edge graph\")\n",
    "    end\n",
    "    println(f_vector(P))\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we look at the polytope appearing in the degeneration of the Cox ring of the blow up of six points in $\\mathbb{P}^2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "For vertex 1 we get 16/16 have scalar product zero\n",
      "For vertex 2 we get 16/16 have scalar product zero\n",
      "For vertex 3 we get 16/16 have scalar product zero\n",
      "For vertex 4 we get 16/16 have scalar product zero\n",
      "For vertex 5 we get 16/16 have scalar product zero\n",
      "For vertex 6 we get 16/16 have scalar product zero\n",
      "For vertex 7 we get 16/16 have scalar product zero\n",
      "For vertex 8 we get 16/16 have scalar product zero\n",
      "For vertex 9 we get 16/16 have scalar product zero\n",
      "For vertex 10 we get 16/16 have scalar product zero\n",
      "For vertex 11 we get 16/16 have scalar product zero\n",
      "For vertex 12 we get 16/16 have scalar product zero\n",
      "For vertex 13 we get 16/16 have scalar product zero\n",
      "For vertex 14 we get 16/16 have scalar product zero\n",
      "For vertex 15 we get 16/16 have scalar product zero\n",
      "For vertex 16 we get 16/16 have scalar product zero\n",
      "For vertex 17 we get 16/16 have scalar product zero\n",
      "For vertex 18 we get 16/16 have scalar product zero\n",
      "For vertex 19 we get 16/16 have scalar product zero\n",
      "For vertex 20 we get 16/16 have scalar product zero\n",
      "For vertex 21 we get 16/16 have scalar product zero\n",
      "For vertex 22 we get 16/16 have scalar product zero\n",
      "For vertex 23 we get 16/16 have scalar product zero\n",
      "For vertex 24 we get 16/16 have scalar product zero\n",
      "For vertex 25 we get 16/16 have scalar product zero\n",
      "For vertex 26 we get 16/16 have scalar product zero\n",
      "For vertex 27 we get 16/16 have scalar product zero\n",
      "The polytope has the mukai edge graph\n"
     ]
    }
   ],
   "source": [
    "S_t, (e1,e2,e3,e4,e5,e6,H12,H13,H14,H15,H16,H23,H24,H25,H26,H34,H35,H36,H45,H46,H56,Q6,Q5,Q4,Q3,Q2,Q1) = PolynomialRing(QQ, [\"e1\",\"e2\",\"e3\",\"e4\",\"e5\",\"e6\",\"H12\",\"H13\",\"H14\",\"H15\",\"H16\",\"H23\",\"H24\",\"H25\",\"H26\",\"H34\",\"H35\",\"H36\",\"H45\",\"H46\",\"H56\",\"Q6\",\"Q5\",\"Q4\",\"Q3\",\"Q2\",\"Q1\"])\n",
    "vars_S=S_t.([e1,e2,e3,e4,e5,e6,H12,H13,H14,H15,H16,H23,H24,H25,H26,H34,H35,H36,H45,H46,H56,Q6,Q5,Q4,Q3,Q2,Q1])\n",
    "\n",
    "function Mukai_form_62(i,j)\n",
    "    E=deg_mon[i]\n",
    "    D=deg_mon[j]\n",
    "    return (E[1]*D[1])-(E[1]-E[2])*(D[1]-D[2])-(E[1]-E[3])*(D[1]-D[3])-(E[1]-E[4])*(D[1]-D[4])-(E[1]-E[5])*(D[1]-D[5])-(E[1]-E[6])*(D[1]-D[6])-(E[1]-E[7])*(D[1]-D[7]);\n",
    "end\n",
    " \n",
    "deg_mon=[];\n",
    "\n",
    "for i in [1,2,3,4,5,6]\n",
    "    temp=[0, 0, 0, 0, 0, 0,0];\n",
    "    temp[i+1]=1;\n",
    "    push!(deg_mon,temp);\n",
    "end\n",
    "\n",
    "for i in 1:6\n",
    "    for j in i+1:6\n",
    "        temp_deg_mon=[1, 1, 1, 1, 1, 1, 1];\n",
    "        temp_deg_mon[i+1]=0;\n",
    "        temp_deg_mon[j+1]=0;\n",
    "        push!(deg_mon,temp_deg_mon);\n",
    "    end\n",
    "end\n",
    "\n",
    "for i in [6,5,4,3,2,1]\n",
    "    temp=[2,1,1,1,1,1,1];\n",
    "    temp[i+1]=2;\n",
    "    push!(deg_mon,temp);\n",
    "end\n",
    "\n",
    "R, (x1,x2,x3,x4,x5,x6,y1,y2,y3,y4,y5,y6) = PolynomialRing(QQ, [\"x1\",\"x2\",\"x3\",\"x4\",\"x5\",\"x6\",\"y1\",\"y2\",\"y3\",\"y4\",\"y5\",\"y6\"])\n",
    "x = R.([x1,x2,x3,x4,x5,x6]);\n",
    "y = R.([y1,y2,y3,y4,y5,y6]);\n",
    "equations=R.([x1, x2, x3, x4, x5, x6, y3*x4*x5*x6,y2*x4*x5*x6, x2*y3*x5*x6, y2*x3*x4*x6,\n",
    "                y2*x3*x4*x5, y1*x4*x5*x6, x1*y3*x5*x6, y1*x3*x4*x6, y1*x3*x4*x5, y1*x2*x5*x6, x1*y2*x4*x6,\n",
    "                x1*y2*x4*x5 ,y1*x2*x3*x6, y1*x2*x3*x5, x1*y2*x3*x4, x1*y2*y3*x4*x5*x6^2, x1*y2*y3*x4*x5^2*x6,\n",
    "                x1*y2*y3*x4^2*x5*x6, x1*y2*y3*x3*x4*x5*x6, x1*x2*y2*y3*x4*x5*x6, x1^2*y2*y3*x4*x5*x6])\n",
    "M=zero_matrix(ZZ,length(equations),12)\n",
    "for (eq,row) in zip(equations,range(1,length(equations)))\n",
    "    for (var,col) in zip(R.([x1,x2,x3,x4,x5,x6,y1,y2,y3,y4,y5,y6]),range(1,12))\n",
    "        M[row,col]=degree(eq,var)\n",
    "    end\n",
    "end\n",
    "P=convex_hull(M);\n",
    "EdGr=Oscar.Graphs.edgegraph(P);\n",
    "E=Oscar.Graphs.edges(EdGr);\n",
    "\n",
    "bol_mukai=true\n",
    "for vert in range(1,length(vertices(P)))\n",
    "    cont_zero=0\n",
    "    cont=0\n",
    "    for vert2 in Graphs.neighbors(EdGr,vert)\n",
    "        cont=cont+1\n",
    "        if Mukai_form_62(vert,vert2)==0\n",
    "            cont_zero=cont_zero+1\n",
    "        end\n",
    "    end\n",
    "    if cont_zero!=cont\n",
    "        bol_mukai=false\n",
    "    end\n",
    "    println(\"For vertex \",vert,\" we get \",cont_zero,\"/\",cont, \" have scalar product zero\")\n",
    "end\n",
    "\n",
    "if bol_mukai\n",
    "    println(\"The polytope has the mukai edge graph\")\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we look at the polytope appearing in the degeneration of the Cox ring of the blow up of seven points in $\\mathbb{P}^2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "For vertex 1 we get 27/27 have scalar product zero\n",
      "For vertex 2 we get 27/27 have scalar product zero\n",
      "For vertex 3 we get 27/27 have scalar product zero\n",
      "For vertex 4 we get 27/27 have scalar product zero\n",
      "For vertex 5 we get 27/27 have scalar product zero\n",
      "For vertex 6 we get 27/27 have scalar product zero\n",
      "For vertex 7 we get 27/27 have scalar product zero\n",
      "For vertex 8 we get 27/27 have scalar product zero\n",
      "For vertex 9 we get 27/27 have scalar product zero\n",
      "For vertex 10 we get 27/27 have scalar product zero\n",
      "For vertex 11 we get 27/27 have scalar product zero\n",
      "For vertex 12 we get 27/27 have scalar product zero\n",
      "For vertex 13 we get 27/27 have scalar product zero\n",
      "For vertex 14 we get 27/27 have scalar product zero\n",
      "For vertex 15 we get 27/27 have scalar product zero\n",
      "For vertex 16 we get 27/27 have scalar product zero\n",
      "For vertex 17 we get 27/27 have scalar product zero\n",
      "For vertex 18 we get 27/27 have scalar product zero\n",
      "For vertex 19 we get 27/27 have scalar product zero\n",
      "For vertex 20 we get 27/29 have scalar product zero\n",
      "For vertex 21 we get 27/27 have scalar product zero\n",
      "For vertex 22 we get 27/29 have scalar product zero\n",
      "For vertex 23 we get 27/27 have scalar product zero\n",
      "For vertex 24 we get 27/29 have scalar product zero\n",
      "For vertex 25 we get 27/27 have scalar product zero\n",
      "For vertex 26 we get 27/27 have scalar product zero\n",
      "For vertex 27 we get 27/27 have scalar product zero\n",
      "For vertex 28 we get 27/27 have scalar product zero\n",
      "For vertex 29 we get 27/27 have scalar product zero\n",
      "For vertex 30 we get 27/27 have scalar product zero\n",
      "For vertex 31 we get 27/27 have scalar product zero\n",
      "For vertex 32 we get 27/27 have scalar product zero\n",
      "For vertex 33 we get 27/29 have scalar product zero\n",
      "For vertex 34 we get 27/27 have scalar product zero\n",
      "For vertex 35 we get 27/29 have scalar product zero\n",
      "For vertex 36 we get 27/27 have scalar product zero\n",
      "For vertex 37 we get 27/29 have scalar product zero\n",
      "For vertex 38 we get 27/27 have scalar product zero\n",
      "For vertex 39 we get 27/27 have scalar product zero\n",
      "For vertex 40 we get 27/27 have scalar product zero\n",
      "For vertex 41 we get 27/27 have scalar product zero\n",
      "For vertex 42 we get 27/27 have scalar product zero\n",
      "For vertex 43 we get 27/27 have scalar product zero\n",
      "For vertex 44 we get 27/27 have scalar product zero\n",
      "For vertex 45 we get 27/27 have scalar product zero\n",
      "For vertex 46 we get 27/27 have scalar product zero\n",
      "For vertex 47 we get 27/27 have scalar product zero\n",
      "For vertex 48 we get 27/27 have scalar product zero\n",
      "For vertex 49 we get 27/27 have scalar product zero\n",
      "For vertex 50 we get 27/27 have scalar product zero\n",
      "For vertex 51 we get 27/27 have scalar product zero\n",
      "For vertex 52 we get 27/27 have scalar product zero\n",
      "For vertex 53 we get 27/27 have scalar product zero\n",
      "For vertex 54 we get 27/27 have scalar product zero\n",
      "For vertex 55 we get 27/27 have scalar product zero\n",
      "For vertex 56 we get 27/27 have scalar product zero\n"
     ]
    }
   ],
   "source": [
    "S_t, (e1,e2,e3,e4,e5,e6,e7,\n",
    "    H12,H13,H14,H15,H16,H17,H23,H24,H25,H26,H27,H34,H35,H36,H37,H45,H46,H47,H56,H57,H67,\n",
    "    Q67,Q57,Q56,Q47,Q46,Q45,Q37,Q36,Q35,Q34,Q27,Q26,Q25,Q24,Q23,Q17,Q16,Q15,Q14,Q13,Q12,\n",
    "    C1,C2,C3,C4,C5,C6,C7) = PolynomialRing(QQ, [\"e1\",\"e2\",\"e3\",\"e4\",\"e5\",\"e6\",\"e7\",\n",
    "        \"H12\",\"H13\",\"H14\",\"H15\",\"H16\",\"H17\",\"H23\",\"H24\",\"H25\",\"H26\",\"H27\",\"H34\",\"H35\",\"H36\",\"H37\",\"H45\",\"H46\",\"H47\",\"H56\",\"H57\",\"H67\",\n",
    "        \"Q67\",\"Q57\",\"Q56\",\"Q47\",\"Q46\",\"Q45\",\"Q37\",\"Q36\",\"Q35\",\"Q34\",\"Q27\",\"Q26\",\"Q25\",\"Q24\",\"Q23\",\"Q17\",\"Q16\",\"Q15\",\"Q14\",\"Q13\",\"Q12\",\n",
    "        \"C1\",\"C2\",\"C3\",\"C4\",\"C5\",\"C6\",\"C7\"]);\n",
    "vars_S=S_t.([e1,e2,e3,e4,e5,e6,e7,\n",
    "    H12,H13,H14,H15,H16,H17,H23,H24,H25,H26,H27,H34,H35,H36,H37,H45,H46,H47,H56,H57,H67,\n",
    "    Q67,Q57,Q56,Q47,Q46,Q45,Q37,Q36,Q35,Q34,Q27,Q26,Q25,Q24,Q23,Q17,Q16,Q15,Q14,Q13,Q12,\n",
    "    C1,C2,C3,C4,C5,C6,C7]);\n",
    "\n",
    "function Mukai_form_72(i,j)\n",
    "    E=deg_mon[i]\n",
    "    D=deg_mon[j]\n",
    "    return (E[1]*D[1])-(E[1]-E[2])*(D[1]-D[2])-(E[1]-E[3])*(D[1]-D[3])-(E[1]-E[4])*(D[1]-D[4])-(E[1]-E[5])*(D[1]-D[5])-(E[1]-E[6])*(D[1]-D[6])-(E[1]-E[7])*(D[1]-D[7])-(E[1]-E[8])*(D[1]-D[8]);\n",
    "end\n",
    " \n",
    "deg_mon=[];\n",
    "\n",
    "for i in [1,2,3,4,5,6,7]\n",
    "    temp=[0, 0, 0, 0, 0, 0,0,0];\n",
    "    temp[i+1]=1;\n",
    "    push!(deg_mon,temp);\n",
    "end\n",
    "\n",
    "for i in 1:7\n",
    "    for j in i+1:7\n",
    "        temp_deg_mon=[1, 1, 1, 1, 1, 1, 1,1];\n",
    "        temp_deg_mon[i+1]=0;\n",
    "        temp_deg_mon[j+1]=0;\n",
    "        push!(deg_mon,temp_deg_mon);\n",
    "    end\n",
    "end\n",
    "\n",
    "deg_quad=[]\n",
    "for i in 1:7\n",
    "    for j in i+1:7\n",
    "        temp_deg_mon=[2, 1, 1, 1, 1, 1, 1,1];\n",
    "        temp_deg_mon[i+1]=2;\n",
    "        temp_deg_mon[j+1]=2;\n",
    "        push!(deg_quad,temp_deg_mon);\n",
    "    end\n",
    "end\n",
    "\n",
    "reverse!(deg_quad)\n",
    "for el in deg_quad \n",
    "    push!(deg_mon,el)\n",
    "end\n",
    "\n",
    "for i in range(1,7)\n",
    "    temp=[3,2,2,2,2,2,2,2];\n",
    "    temp[i+1]=1;\n",
    "    push!(deg_mon,temp);\n",
    "end\n",
    "\n",
    "R, (x1,x2,x3,x4,x5,x6,x7,y1,y2,y3,y4,y5,y6,y7) = PolynomialRing(QQ, [\"x1\",\"x2\",\"x3\",\"x4\",\"x5\",\"x6\",\"x7\",\"y1\",\"y2\",\"y3\",\"y4\",\"y5\",\"y6\",\"y7\"])\n",
    "x = R.([x1,x2,x3,x4,x5,x6,x7]);\n",
    "y = R.([y1,y2,y3,y4,y5,y6,y7]);\n",
    "equations=R.([x1, x2, x3, x4, x5, x6, x7, x3*x5*x6*x7*y4, x2*x4*x5*x6*y7, x2*x3*x5*x6*y7,\n",
    "        x2*x3*x6*x7*y4, x2*x3*x4*x5*y7, x2*x3*x5*x6*y4, x1*x4*x5*x6*y7, x1*x3*x5*x6*y7, x1*x3*x6*x7*y4,\n",
    " \n",
    "x1*x3*x4*x5*y7, x1*x3*x5*x6*y4, x1*x2*x5*x6*y7, x1*x2*x6*x7*y4, x1*x2*x4*x5*y7, x1*x2*x5*x6*y4,\n",
    "\n",
    "x1*x2*x6*x7*y3, x1*x2*x5*x7*y3, x1*x2*x5*x6*y3, x1*x2*x4*x7*y3, x1*x2*x4*x6*y3, x1*x2*x4*x5*y3,\n",
    "\n",
    "x1*x2*x3*x5*x6^2*x7*y4*y7, x1*x2*x3*x4*x5^2*x6*y7^2, x1*x2*x3*x5^2*x6^2*y4*y7, x1*x2*x3*x4*x5*x6*x7*y4*y7,\n",
    "\n",
    "x1*x2*x3*x5*x6^2*x7*y4^2, x1*x2*x3*x4*x5^2*x6*y4*y7, x1*x2*x3*x5*x6*x7^2*y3*y4, x1*x2*x3*x5*x6^2*x7*y3*y4,\n",
    "\n",
    "x1*x2*x3*x5^2*x6*x7*y3*y4, x1*x2*x3*x4*x5*x6*x7*y3*y4, x1*x2^2*x4*x5*x6*x7*y3*y7, x1*x2^2*x4*x5*x6^2*y3*y7,\n",
    "\n",
    "x1*x2^2*x4*x5^2*x6*y3*y7, x1*x2^2*x4^2*x5*x6*y3*y7, x1*x2^2*x3*x4*x5*x6*y3*y7, x1^2*x2*x4*x5*x6*x7*y3*y7,\n",
    "\n",
    "x1^2*x2*x4*x5*x6^2*y3*y7, x1^2*x2*x4*x5^2*x6*y3*y7, x1^2*x2*x4^2*x5*x6*y3*y7, x1^2*x2*x3*x4*x5*x6*y3*y7,\n",
    "\n",
    "x1^2*x2^2*x4*x5*x6*y3*y7, x1*x2^2*x3*x4*x5^2*x6^2*x7*y3*y4*y7, x1^2*x2*x3*x4*x5^2*x6^2*x7*y3*y4*y7, \n",
    "\n",
    "x1^2*x2^2*x4^2*x5^2*x6^2*y3*y7^2, x1^2*x2^2*x3*x4*x5^2*x6^2*y3*y7^2, x1^2*x2^2*x3*x4*x5*x6^2*x7*y3*y4*y7, \n",
    "\n",
    "x1^2*x2^2*x3*x4^2*x5^2*x6*y3*y7^2, x1^2*x2^2*x3*x4*x5^2*x6^2*y3*y4*y7])\n",
    "M=zero_matrix(ZZ,length(equations),14)\n",
    "for (eq,row) in zip(equations,range(1,length(equations)))\n",
    "    for (var,col) in zip(R.([x1,x2,x3,x4,x5,x6,x7,y1,y2,y3,y4,y5,y6,y7]),range(1,14))\n",
    "        M[row,col]=degree(eq,var)\n",
    "    end\n",
    "end\n",
    "P=convex_hull(M);\n",
    "EdGr=Oscar.Graphs.edgegraph(P);\n",
    "E=Oscar.Graphs.edges(EdGr);\n",
    "\n",
    "bol_mukai=true\n",
    "for vert in range(1,length(vertices(P)))\n",
    "    cont_zero=0\n",
    "    cont=0\n",
    "    for vert2 in Graphs.neighbors(EdGr,vert)\n",
    "        cont=cont+1\n",
    "        if Mukai_form_72(vert,vert2)==0\n",
    "            cont_zero=cont_zero+1\n",
    "        end\n",
    "    end\n",
    "    if cont_zero!=cont\n",
    "        bol_mukai=false\n",
    "    end\n",
    "    println(\"For vertex \",vert,\" we get \",cont_zero,\"/\",cont, \" have scalar product zero\")\n",
    "end\n",
    "\n",
    "if bol_mukai\n",
    "    println(\"The polytope has the mukai edge graph\")\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we look at the polytope appearing in the degeneration of the Cox ring of the blow up of eight points in $\\mathbb{P}^2$he case of $8$ points."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The edge graph is the Mukai-edge graph\n"
     ]
    }
   ],
   "source": [
    "R, (x1,x2,x3,x4,x5,x6,x7,x8,y1,y2,y3,y4,y5,y6,y7,y8) = PolynomialRing(QQ, [\"x1\",\"x2\",\"x3\",\"x4\",\"x5\",\"x6\",\"x7\",\"x8\",\"y1\",\"y2\",\"y3\",\"y4\",\"y5\",\"y6\",\"y7\",\"y8\"])\n",
    "x = R.([x1,x2,x3,x4,x5,x6,x7,x8]);\n",
    "y = R.([y1,y2,y3,y4,y5,y6,y7,y8]);\n",
    "equations=R.([x1, x2,x3,x4,x5, x6, x7,x8, \n",
    "x3*x4*x6*x7*x8*y5, x2*x5*x4*x7*x8*y6, \n",
    "x2*x3*x6*x7*x8*y5, x2*x3*x4*x7*x8*y6, \n",
    "x2*x3*x4*x7*x8*y5, x2*x3*x5*x4*x8*y6,\n",
    "x2*x3*x5*x4*x7*y6, x1*x5*x4*x6*x8*y7,\n",
    "x1*x3*x6*x7*x8*y5, x1*x3*x4*x6*x8*y7, \n",
    "x1*x3*x5*x4*x8*y7, x1*x3*x4*x6*x8*y5, \n",
    "x1*x3*x5*x4*x6*y7,x1*x2*x5*x6*x8*y7, \n",
    "x1*x2*x4*x7*x8*y6, x1*x2*x5*x4*x8*y7,\n",
    "x1*x2*x5*x4*x8*y6, x1*x2*x5*x4*x7*y6,\n",
    "x1*x2*x3*x6*x8*y7, x1*x2*x3*x7*x8*y5,\n",
    "x1*x2*x3*x6*x8*y5, x1*x2*x3*x5*x6*y7, \n",
    "x1*x2*x3*x4*x8*y7,x1*x2*x3*x4*x8*y6, \n",
    "x1*x2*x3*x4*x6*y7, x1*x2*x3*x4*x8*y5,\n",
    "x1*x2*x3*x5*x4*y7, x1*x2*x3*x5*x4*y6,\n",
    "x1*x2*x3*x4*x6*x7^2*x8^2*y5*y6,\n",
    "x1*x2*x3*x5*x4*x6*x7*x8^2*y5*y7, \n",
    "x1*x2*x3*x5*x4*x6*x7*x8^2*y5*y6,\n",
    "x1*x2*x3*x5*x4*x6*x7^2*x8*y5*y6, \n",
    "x1*x2*x3*x5*x4^2*x7*x8^2*y6*y7, \n",
    "x1*x2*x3*x5*x4^2*x6*x8^2*y6*y7,\n",
    "x1*x2*x3*x5*x4^2*x6*x7*x8*y6*y7,\n",
    "x1*x2*x3*x5^2*x4^2*x8^2*y6*y7,\n",
    "x1*x2*x3*x5^2*x4^2*x7*x8*y6*y7,\n",
    "x1*x2*x3*x5^2*x4^2*x6*x8*y6*y7,\n",
    "x1*x2*x3^2*x4*x6*x7*x8^2*y5*y7,\n",
    "x1*x2*x3^2*x4*x6*x7*x8^2*y5*y6,\n",
    "x1*x2*x3^2*x4*x6^2*x7*x8*y5*y7, \n",
    "x1*x2*x3^2*x4*x6*x7*x8^2*y5^2,\n",
    "x1*x2*x3^2*x5*x4*x6*x7*x8*y5*y7,\n",
    "x1*x2*x3^2*x5*x4*x6*x7*x8*y5*y6,\n",
    "x1*x2*x3^2*x5*x4^2*x8^2*y6*y7, \n",
    "x1*x2*x3^2*x4^2*x6*x7*x8*y5*y7,\n",
    "x1*x2*x3^2*x5*x4^2*x6*x8*y6*y7,\n",
    "x1*x2*x3^2*x5^2*x4^2*x8*y6*y7,\n",
    "x1*x2^2*x3*x4*x7^2*x8^2*y5*y6,\n",
    "x1*x2^2*x3*x5*x4*x6*x8^2*y6*y7,\n",
    "x1*x2^2*x3*x5*x4*x6*x7*x8*y6*y7,\n",
    "x1*x2^2*x3*x5*x4*x7*x8^2*y5*y6,\n",
    "x1*x2^2*x3*x5*x4*x7^2*x8*y5*y6,\n",
    "x1*x2^2*x3*x5^2*x4*x6*x8*y6*y7,\n",
    "x1*x2^2*x3*x5*x4^2*x8^2*y6*y7,\n",
    "x1*x2^2*x3*x5*x4^2*x7*x8*y6*y7,\n",
    "x1*x2^2*x3*x5*x4^2*x7*x8*y6^2, \n",
    "x1*x2^2*x3*x5^2*x4^2*x8*y6*y7,\n",
    "x1*x2^2*x3^2*x4*x7*x8^2*y5*y6, \n",
    "x1*x2^2*x3^2*x4*x6*x7*x8*y5*y7, \n",
    "x1*x2^2*x3^2*x5*x4*x6*x8*y6*y7,\n",
    "x1*x2^2*x3^2*x5*x4*x7*x8*y5*y6,\n",
    "x1*x2^2*x3^2*x5*x4^2*x8*y6*y7,\n",
    "x1^2*x2*x3*x5*x4*x6*x8^2*y7^2,\n",
    "x1^2*x2*x3*x4*x6*x7*x8^2*y5*y6,\n",
    "x1^2*x2*x3*x5*x4*x6^2*x8*y7^2,\n",
    "x1^2*x2*x3*x5*x4*x6*x8^2*y5*y7,\n",
    "x1^2*x2*x3*x5^2*x4*x6*x8*y7^2,\n",
    "x1^2*x2*x3*x5*x4*x6*x7*x8*y5*y6, \n",
    "x1^2*x2*x3*x5*x4^2*x8^2*y6*y7,\n",
    "x1^2*x2*x3*x5*x4^2*x6*x8*y7^2,\n",
    "x1^2*x2*x3*x5*x4^2*x6*x8*y6*y7, \n",
    "x1^2*x2*x3*x5^2*x4^2*x8*y6*y7,\n",
    "x1^2*x2*x3^2*x4*x6*x8^2*y5*y7,\n",
    "x1^2*x2*x3^2*x4*x6*x7*x8*y5*y7,\n",
    "x1^2*x2*x3^2*x4*x6^2*x8*y5*y7,\n",
    "x1^2*x2*x3^2*x5*x4*x6*x8*y5*y7,\n",
    "x1^2*x2*x3^2*x4^2*x6*x8*y5*y7,\n",
    "x1^2*x2^2*x3*x4*x7*x8^2*y5*y6,\n",
    "x1^2*x2^2*x3*x5*x4*x6*x8*y7^2, \n",
    "x1^2*x2^2*x3*x5*x4*x6*x8*y6*y7,\n",
    "x1^2*x2^2*x3*x5*x4*x7*x8*y5*y6,\n",
    "x1^2*x2^2*x3*x5*x4^2*x8*y6*y7, \n",
    "x1^2*x2^2*x3^2*x4*x6*x8*y5*y7,\n",
    "x1^2*x2^2*x3^2*x5^2*x4^2*x6*x8^2*y6*y7^2,\n",
    "x1^2*x2^2*x3^2*x5*x4^2*x6*x7*x8^2*y5*y6*y7, \n",
    "x1*x2^2*x3^2*x5*x4^2*x6*x7*x8^3*y5*y6*y7,\n",
    "x1^2*x2*x3^2*x5*x4^2*x6*x7*x8^3*y5*y6*y7,\n",
    "x1^2*x2^2*x3*x5^2*x4^2*x6*x8^3*y6*y7^2,\n",
    "x1^2*x2^2*x3^2*x4*x6*x7^2*x8^3*y5^2*y6,\n",
    "x1^2*x2^2*x3^2*x5*x4^2*x6*x8^3*y6*y7^2, \n",
    "x1^2*x2^2*x3^2*x5*x4^2*x7*x8^3*y5*y6*y7,\n",
    "x1^2*x2^2*x3^2*x5*x4^2*x6*x8^3*y5*y6*y7,\n",
    "x1*x2^2*x3^2*x5*x4^2*x6*x7^2*x8^2*y5*y6*y7,\n",
    "x1^2*x2*x3^2*x5*x4^2*x6^2*x7*x8^2*y5*y7^2,\n",
    "x1^2*x2^2*x3*x5^2*x4^2*x6*x7*x8^2*y6*y7^2, \n",
    "x1^2*x2^2*x3^2*x5*x4*x6^2*x7*x8^2*y5*y7^2, \n",
    "x1^2*x2^2*x3^2*x4^2*x6*x7^2*x8^2*y5*y6*y7, \n",
    "x1^2*x2^2*x3^2*x5*x4^2*x6*x7*x8^2*y5*y7^2, \n",
    "x1^2*x2^2*x3^2*x5*x4^2*x6*x7^2*x8*y5*y6*y7, \n",
    "x1*x2^2*x3^2*x5*x4^2*x6*x7^2*x8^2*y5*y6^2, \n",
    "x1^2*x2*x3^2*x5*x4^2*x6^2*x7*x8^2*y5*y6*y7,\n",
    "x1^2*x2^2*x3*x5^2*x4^2*x6^2*x8^2*y6*y7^2, \n",
    "x1^2*x2^2*x3^2*x5*x4*x6^2*x7*x8^2*y5*y6*y7,\n",
    "x1^2*x2^2*x3^2*x5*x4^2*x6^2*x8^2*y6*y7^2,\n",
    "x1^2*x2^2*x3^2*x5*x4^2*x6*x7*x8^2*y5*y6^2, \n",
    "x1^2*x2^2*x3^2*x5^2*x4^2*x6^2*x8*y6*y7^2, \n",
    "x1*x2^2*x3^2*x5^2*x4^2*x6*x7*x8^2*y5*y6*y7, \n",
    "x1^2*x2*x3^2*x5^2*x4^2*x6*x7*x8^2*y5*y6*y7, \n",
    "x1^2*x2^2*x3*x5^3*x4^2*x6*x8^2*y6*y7^2,\n",
    "x1^2*x2^2*x3^2*x5*x4*x6*x7^2*x8^2*y5^2*y6, \n",
    "x1^2*x2^2*x3^2*x5^2*x4^2*x7*x8^2*y5*y6*y7, \n",
    "x1^2*x2^2*x3^2*x5^2*x4^2*x6*x8^2*y5*y6*y7, \n",
    "x1^2*x2^2*x3^2*x5^3*x4^2*x6*x8*y6*y7^2, \n",
    "x1*x2^2*x3^2*x5^2*x4^3*x7*x8^2*y6^2*y7, \n",
    "x1^2*x2*x3^2*x5^2*x4^3*x6*x8^2*y6*y7^2, \n",
    "x1^2*x2^2*x3*x5^2*x4^3*x6*x8^2*y6*y7^2,\n",
    "x1^2*x2^2*x3^2*x5*x4^3*x6*x8^2*y6*y7^2,\n",
    "x1^2*x2^2*x3^2*x5^2*x4^3*x8^2*y6*y7^2, \n",
    "x1^2*x2^2*x3^2*x5^2*x4^3*x8^2*y6^2*y7,\n",
    "x1^2*x2^2*x3^2*x5^2*x4^3*x6*x8*y6*y7^2,\n",
    "x1*x2^2*x3^3*x5*x4^2*x6*x7*x8^2*y5*y6*y7, \n",
    "x1^2*x2*x3^3*x4^2*x6^2*x7*x8^2*y5^2*y7,\n",
    "x1^2*x2^2*x3^3*x4*x6^2*x7*x8^2*y5^2*y7, \n",
    "x1^2*x2^2*x3^3*x4^2*x6*x7*x8^2*y5*y6*y7, \n",
    "x1^2*x2^2*x3^3*x4^2*x6*x7*x8^2*y5^2*y7,\n",
    "x1^2*x2^2*x3^3*x5*x4^2*x6*x8^2*y5*y6*y7,\n",
    "x1^2*x2^2*x3^3*x5*x4^2*x6*x7*x8*y5*y6*y7,\n",
    "x1*x2^3*x3^2*x5*x4^2*x7^2*x8^2*y5*y6^2,\n",
    "x1^2*x2^3*x3*x5^2*x4^2*x6*x8^2*y6*y7^2, \n",
    "x1^2*x2^3*x3^2*x5*x4*x6*x7*x8^2*y5*y6*y7, \n",
    "x1^2*x2^3*x3^2*x5*x4^2*x6*x8^2*y6*y7^2,\n",
    "x1^2*x2^3*x3^2*x5*x4^2*x7*x8^2*y5*y6*y7, \n",
    "x1^2*x2^3*x3^2*x5*x4^2*x7*x8^2*y5*y6^2,\n",
    "x1^2*x2^3*x3^2*x5^2*x4^2*x6*x8*y6*y7^2,\n",
    "x1^3*x2*x3^2*x5*x4^2*x6^2*x8^2*y5*y7^2, \n",
    "x1^3*x2^2*x3*x5^2*x4^2*x6*x8^2*y6*y7^2, \n",
    "x1^3*x2^2*x3^2*x5*x4*x6^2*x8^2*y5*y7^2,\n",
    "x1^3*x2^2*x3^2*x4^2*x6*x7*x8^2*y5*y6*y7, \n",
    "x1^3*x2^2*x3^2*x5*x4^2*x6*x8^2*y5*y7^2, \n",
    "x1^3*x2^2*x3^2*x5*x4^2*x6*x8^2*y5*y6*y7, \n",
    "x1^3*x2^2*x3^2*x5*x4^2*x6*x7*x8*y5*y6*y7, \n",
    "x1^2*x2^2*x3^2*x5^2*x4^3*x6^2*x7*x8^3*y5*y6*y7^2, \n",
    "x1^2*x2^2*x3^3*x5*x4^2*x6^2*x7^2*x8^3*y5^2*y6*y7, \n",
    "x1^2*x2^2*x3^3*x5*x4^3*x6^2*x7*x8^3*y5*y6*y7^2,\n",
    "x1^2*x2^2*x3^3*x5^2*x4^3*x6*x7*x8^3*y5*y6*y7^2,\n",
    "x1^2*x2^2*x3^3*x5^2*x4^3*x6*x7*x8^3*y5*y6^2*y7,\n",
    "x1^2*x2^2*x3^3*x5^2*x4^3*x6^2*x7*x8^2*y5*y6*y7^2, \n",
    "x1^2*x2^3*x3^2*x5^2*x4^2*x6^2*x7*x8^3*y5*y6*y7^2, \n",
    "x1^2*x2^3*x3^2*x5^2*x4^3*x6*x7*x8^3*y6^2*y7^2,\n",
    "x1^2*x2^3*x3^2*x5^2*x4^3*x6*x7*x8^3*y5*y6*y7^2, \n",
    "x1^2*x2^3*x3^2*x5^3*x4^3*x6*x8^3*y6^2*y7^2, \n",
    "x1^2*x2^3*x3^2*x5^3*x4^3*x6*x7*x8^2*y6^2*y7^2, \n",
    "x1^2*x2^3*x3^3*x5*x4^2*x6^2*x7*x8^3*y5*y6*y7^2, \n",
    "x1^2*x2^3*x3^3*x5*x4^2*x6*x7^2*x8^3*y5^2*y6*y7, \n",
    "x1^2*x2^3*x3^3*x5*x4^2*x6*x7^2*x8^3*y5^2*y6^2, \n",
    "x1^2*x2^3*x3^3*x5^2*x4^2*x6^2*x7*x8^2*y5*y6*y7^2, \n",
    "x1^2*x2^3*x3^3*x5*x4^3*x6*x7*x8^3*y5*y6*y7^2, \n",
    "x1^2*x2^3*x3^3*x5^2*x4^3*x6*x8^3*y6^2*y7^2,\n",
    "x1^2*x2^3*x3^3*x5*x4^3*x6*x7^2*x8^2*y5*y6^2*y7, \n",
    "x1^2*x2^3*x3^3*x5^2*x4^3*x7*x8^3*y5*y6^2*y7,\n",
    "x1^2*x2^3*x3^3*x5^2*x4^3*x6*x7*x8^2*y5*y6*y7^2, \n",
    "x1^2*x2^3*x3^3*x5^3*x4^3*x6*x8^2*y6^2*y7^2, \n",
    "x1^3*x2^2*x3^2*x5^2*x4^2*x6^2*x7*x8^3*y5*y6*y7^2, \n",
    "x1^3*x2^2*x3^2*x5^2*x4^3*x6^2*x8^3*y6*y7^3,\n",
    "x1^3*x2^2*x3^2*x5^3*x4^3*x6*x8^3*y6*y7^3, \n",
    "x1^3*x2^2*x3^2*x5^2*x4^3*x6^2*x8^3*y5*y6*y7^2,\n",
    "x1^3*x2^2*x3^2*x5^3*x4^3*x6^2*x8^2*y6*y7^3,\n",
    "x1^3*x2^2*x3^3*x4^2*x6^2*x7^2*x8^3*y5^2*y6*y7, \n",
    "x1^3*x2^2*x3^3*x5*x4^2*x6^2*x7*x8^3*y5^2*y7^2,\n",
    "x1^3*x2^2*x3^3*x5*x4^2*x6^2*x7*x8^3*y5^2*y6*y7,\n",
    "x1^3*x2^2*x3^3*x5*x4^2*x6^2*x7^2*x8^2*y5^2*y6*y7,\n",
    "x1^3*x2^2*x3^3*x5*x4^3*x6*x7*x8^3*y5*y6*y7^2, \n",
    "x1^3*x2^2*x3^3*x5*x4^3*x6^2*x8^3*y5*y6*y7^2,\n",
    "x1^3*x2^2*x3^3*x5*x4^3*x6^2*x7*x8^2*y5*y6*y7^2, \n",
    "x1^3*x2^2*x3^3*x5^2*x4^3*x6*x8^3*y5*y6*y7^2, \n",
    "x1^3*x2^2*x3^3*x5^2*x4^3*x6*x7*x8^2*y5*y6*y7^2, \n",
    "x1^3*x2^2*x3^3*x5^2*x4^3*x6^2*x8^2*y5*y6*y7^2, \n",
    "x1^3*x2^3*x3^2*x5^2*x4^2*x6^2*x8^3*y6*y7^3, \n",
    "x1^3*x2^3*x3^2*x5^2*x4^2*x6*x7*x8^3*y5*y6*y7^2,\n",
    "x1^3*x2^3*x3^2*x5^2*x4^2*x6^2*x8^3*y5*y6*y7^2,\n",
    "x1^3*x2^3*x3^2*x5^3*x4^2*x6^2*x8^2*y6*y7^3,\n",
    "x1^3*x2^3*x3^2*x5^2*x4^3*x6*x8^3*y6*y7^3,\n",
    "x1^3*x2^3*x3^2*x5^2*x4^3*x6*x8^3*y6^2*y7^2, \n",
    "x1^3*x2^3*x3^2*x5^2*x4^3*x6^2*x8^2*y6*y7^3, \n",
    "x1^3*x2^3*x3^2*x5^2*x4^3*x6*x8^3*y5*y6*y7^2, \n",
    "x1^3*x2^3*x3^2*x5^3*x4^3*x6*x8^2*y6*y7^3,\n",
    "x1^3*x2^3*x3^2*x5^3*x4^3*x6*x8^2*y6^2*y7^2, \n",
    "x1^3*x2^3*x3^3*x4^2*x6*x7^2*x8^3*y5^2*y6*y7,\n",
    "x1^3*x2^3*x3^3*x5*x4^2*x6^2*x8^3*y5*y6*y7^2,\n",
    "x1^3*x2^3*x3^3*x5*x4^2*x6^2*x7*x8^2*y5*y6*y7^2,x1^3*x2^3*x3^3*x5*x4^2*x6*x7*x8^3*y5^2*y6*y7,x1^3*x2^3*x3^3*x5*x4^2*x6*x7^2*x8^2*y5^2*y6*y7,x1^3*x2^3*x3^3*x5^2*x4^2*x6^2*x8^2*y5*y6*y7^2,x1^3*x2^3*x3^3*x5*x4^3*x6*x8^3*y5*y6*y7^2, x1^3*x2^3*x3^3*x5*x4^3*x6*x7*x8^2*y5*y6*y7^2, x1^3*x2^3*x3^3*x5*x4^3*x6*x7*x8^2*y5*y6^2*y7,x1^3*x2^3*x3^3*x5^2*x4^3*x6*x8^2*y5*y6*y7^2, x1^3*x2^3*x3^3*x5^2*x4^3*x6^2*x7*x8^4*y5*y6*y7^3, x1^3*x2^3*x3^3*x5^2*x4^3*x6^2*x7*x8^4*y5*y6^2*y7^2,\n",
    "x1^3*x2^3*x3^3*x5^2*x4^3*x6^3*x7*x8^3*y5*y6*y7^3,\n",
    "x1^3*x2^3*x3^3*x5^2*x4^3*x6^2*x7*x8^4*y5^2*y6*y7^2, \n",
    "x1^3*x2^3*x3^3*x5^3*x4^3*x6^2*x7*x8^3*y5*y6*y7^3,\n",
    "x1^3*x2^3*x3^3*x5^3*x4^3*x6^2*x7*x8^3*y5*y6^2*y7^2, \n",
    "x1^3*x2^3*x3^3*x5^3*x4^4*x6*x8^4*y6^2*y7^3, \n",
    "x1^3*x2^3*x3^3*x5^2*x4^4*x6^2*x7*x8^3*y5*y6*y7^3,\n",
    "x1^3*x2^3*x3^3*x5^3*x4^4*x6^2*x8^3*y6^2*y7^3, \n",
    "x1^3*x2^3*x3^3*x5^4*x4^4*x6*x8^3*y6^2*y7^3, \n",
    "x1^3*x2^3*x3^4*x5*x4^3*x6^2*x7*x8^4*y5^2*y6*y7^2, \n",
    "x1^3*x2^3*x3^4*x5*x4^3*x6^2*x7^2*x8^3*y5^2*y6*y7^2, \n",
    "x1^3*x2^3*x3^4*x5*x4^3*x6^2*x7^2*x8^3*y5^2*y6^2*y7,\n",
    "x1^3*x2^3*x3^4*x5^2*x4^3*x6^2*x7*x8^3*y5^2*y6*y7^2, x1^3*x2^3*x3^4*x5^2*x4^4*x6*x7*x8^3*y5*y6^2*y7^2, \n",
    "x1^3*x2^4*x3^3*x5^2*x4^3*x6*x7*x8^4*y5*y6^2*y7^2, x1^3*x2^4*x3^3*x5^2*x4^3*x6^2*x7*x8^3*y5*y6*y7^3, \n",
    "x1^3*x2^4*x3^3*x5^3*x4^3*x6^2*x8^3*y6^2*y7^3, x1^3*x2^4*x3^3*x5^3*x4^3*x6*x7*x8^3*y5*y6^2*y7^2, x1^3*x2^4*x3^3*x5^3*x4^4*x6*x8^3*y6^2*y7^3, \n",
    "x1^3*x2^4*x3^4*x5*x4^3*x6*x7^2*x8^3*y5^2*y6^2*y7, x1^4*x2^3*x3^3*x5^2*x4^3*x6^2*x8^4*y5*y6*y7^3, x1^4*x2^3*x3^3*x5^2*x4^3*x6^2*x7*x8^3*y5*y6*y7^3, \n",
    "x1^4*x2^3*x3^3*x5^2*x4^3*x6^3*x8^3*y5*y6*y7^3, x1^4*x2^3*x3^3*x5^3*x4^3*x6^2*x8^3*y5*y6*y7^3, \n",
    "x1^4*x2^3*x3^3*x5^2*x4^4*x6^2*x8^3*y5*y6*y7^3,x1^4*x2^3*x3^4*x5*x4^3*x6^2*x7*x8^3*y5^2*y6*y7^2,\n",
    "x1^4*x2^4*x3^3*x5^2*x4^3*x6^2*x8^3*y5*y6*y7^3,x1^3*x2^4*x3^4*x5^3*x4^4*x6^2*x7*x8^4*y5*y6^2*y7^3,\n",
    "x1^4*x2^3*x3^4*x5^2*x4^4*x6^3*x7*x8^4*y5^2*y6*y7^3,x1^4*x2^4*x3^3*x5^4*x4^4*x6^2*x8^4*y6^2*y7^4, \n",
    "x1^4*x2^4*x3^4*x5^2*x4^3*x6^3*x7*x8^4*y5^2*y6*y7^3, x1^4*x2^4*x3^4*x5^2*x4^4*x6^2*x7*x8^4*y5*y6^2*y7^3, x1^4*x2^4*x3^4*x5^2*x4^4*x6^2*x7*x8^4*y5^2*y6*y7^3, x1^4*x2^4*x3^4*x5^3*x4^4*x6^2*x8^4*y5*y6^2*y7^3,x1^4*x2^4*x3^4*x5^3*x4^4*x6^2*x7*x8^3*y5*y6^2*y7^3])\n",
    "\n",
    "M=zero_matrix(ZZ,length(equations),16)\n",
    "for (eq,row) in zip(equations,range(1,length(equations)))\n",
    "    for (var,col) in zip(R.([x1,x2,x3,x4,x5,x6,x7,x8,y1,y2,y3,y4,y5,y6,y7,y8]),range(1,16))\n",
    "        M[row,col]=degree(eq,var)\n",
    "    end\n",
    "end\n",
    "P=convex_hull(M);\n",
    "EdGr=Oscar.Graphs.edgegraph(P);\n",
    "E=Oscar.Graphs.edges(EdGr);\n",
    "\n",
    "bol_mukai=true\n",
    "for vert in range(1,length(vertices(P)))\n",
    "    cont_zero=0\n",
    "    cont=0\n",
    "    for vert2 in Graphs.neighbors(EdGr,vert)\n",
    "        cont=cont+1\n",
    "    end\n",
    "    if cont!=56\n",
    "       println(\"For vertex \",vert,\" we get \",cont_zero,\"/\",cont, \" have scalar product zero\")\n",
    "       bol_mukai=false\n",
    "    end\n",
    "end\n",
    "if bol_mukai\n",
    "    println(\"The edge graph is the Mukai-edge graph\")\n",
    "end\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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