Example_5.9_1910.06490.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The goal of this worksheet is to justify Example 5.9 in <a href=\"https://arxiv.org/abs/1910.06490\"><kbd>arxiv:1910.06490</kbd></a>., to apperar in *St. Petersbourg Math. J*. We are going to give a plane curve of type $(4,6;6,6)$. The example has equation in a number field $K$ with primitive element $\\alpha$ with minimal polynomial $t^4 - 2 t^3 - 2 t^2 + 18 t + 21$. This extension contains the field of cubic roots of unity; the number $\\zeta:=\\frac{1}{6} \\alpha^{3} - \\frac{2}{3} \\alpha^{2} + \\frac{1}{2} \\alpha + 2$ is a primitive cubic root of unity."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x^2 + x + 1"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S.<t>=QQ[]\n",
    "pol=t^4 - 2*t^3 - 2*t^2 + 18*t + 21\n",
    "K.<alpha>=NumberField(pol)\n",
    "zeta=1/6*alpha^3 - 2/3*alpha^2 + 1/2*alpha + 2\n",
    "zeta.minpoly()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the construction, we need a nodal cubic $C_3$. We provide a parametrization."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R.<x,y,z>=K[]\n",
    "f=x*y*z+x^3-y^3\n",
    "sb={x:t,y:t^2,z:t^3-1}\n",
    "f.subs(sb)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We skip the long construction of a curve $C_4$ with has a singular point $P_0$ of type $\\mathbb{A}_3$ and such that it intersects $C_3$ at two common smooth points with intersecion number 6."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "g=(9219/250*alpha^3 - 18186/125*alpha^2 + 26649/250*alpha + 55566/125)*x^4 + (3262/125*alpha^3 - 13048/125*alpha^2 + 41034/125*alpha + 27468/125)*x^3*y - 80746/125*x^2*y^2 + (-1694/25*alpha^3 + 18256/125*alpha^2 + 5838/125*alpha - 117264/125)*x*y^3 + (-9051/250*alpha^3 + 18102/125*alpha^2 - 28161/250*alpha - 16191/25)*y^4 + (-188/5*alpha^3 + 564/5*alpha^2 - 564/5*alpha - 66276/125)*x^3*z + (-4428/125*alpha^3 + 17364/125*alpha^2 - 12588/125*alpha - 53484/125)*x^2*y*z + (-4312/125*alpha^3 + 17248/125*alpha^2 - 13632/125*alpha - 15384/25)*x*y^2*z + (-188/5*alpha^3 + 564/5*alpha^2 - 564/5*alpha - 55924/125)*y^3*z + (41/5*alpha^3 - 164/5*alpha^2 + 99/5*alpha + 762/5)*x^2*z^2 - 4676/125*x*y*z^2 + (-37/5*alpha^3 + 32*alpha^2 - 27*alpha - 432/5)*y^2*z^2 + (-8/5*alpha^3 + 4*alpha^2 - 108/5)*x*z^3 + (-4/5*alpha^3 + 16/5*alpha^2 - 36/5*alpha - 48/5)*y*z^3 + z^4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(9219/250*alpha^3 - 18186/125*alpha^2 + 26649/250*alpha + 55566/125)*x^4 + (3262/125*alpha^3 - 13048/125*alpha^2 + 41034/125*alpha + 27468/125)*x^3*y - 80746/125*x^2*y^2 + (-1694/25*alpha^3 + 18256/125*alpha^2 + 5838/125*alpha - 117264/125)*x*y^3 + (-9051/250*alpha^3 + 18102/125*alpha^2 - 28161/250*alpha - 16191/25)*y^4 + (-188/5*alpha^3 + 564/5*alpha^2 - 564/5*alpha - 66276/125)*x^3*z + (-4428/125*alpha^3 + 17364/125*alpha^2 - 12588/125*alpha - 53484/125)*x^2*y*z + (-4312/125*alpha^3 + 17248/125*alpha^2 - 13632/125*alpha - 15384/25)*x*y^2*z + (-188/5*alpha^3 + 564/5*alpha^2 - 564/5*alpha - 55924/125)*y^3*z + (41/5*alpha^3 - 164/5*alpha^2 + 99/5*alpha + 762/5)*x^2*z^2 - 4676/125*x*y*z^2 + (-37/5*alpha^3 + 32*alpha^2 - 27*alpha - 432/5)*y^2*z^2 + (-8/5*alpha^3 + 4*alpha^2 - 108/5)*x*z^3 + (-4/5*alpha^3 + 16/5*alpha^2 - 36/5*alpha - 48/5)*y*z^3 + z^4"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We check the property of intersection number."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(t^2 + (-2/15*alpha^3 + 8/15*alpha^2 - 6/5*alpha - 8/5)*t - 1/6*alpha^3 + 2/3*alpha^2 - 1/2*alpha - 2)^6"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.subs(sb).factor()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us compute the singular points"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Ideal (z, 42*x + (9*alpha^3 - 28*alpha^2 + 11*alpha + 116)*y) of Multivariate Polynomial Ring in x, y, z over Number Field in alpha with defining polynomial t^4 - 2*t^3 - 2*t^2 + 18*t + 21]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "jacobianideal=g.jacobian_ideal().minimal_associated_primes()\n",
    "jacobianideal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "a1=K(x.reduce(jacobianideal[0])(y=1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The singular point $P_0$ is $\\left[-\\frac{3}{14} \\alpha^{3} + \\frac{2}{3} \\alpha^{2} - \\frac{11}{42} \\alpha - \\frac{58}{21}: 1: 0\\right]$. To check its type we define a $2$-variable ring (an affine chart for which the singular point is the origin). We look for the tangent cone of this point of multiplicity 2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(u + (-13/18*alpha^3 + 20/9*alpha^2 - 5/6*alpha - 28/3)*v, 2)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T.<u,v>=K[]\n",
    "p0=g(y=1,z=u,x=v+a1)\n",
    "tangentcone=sum(p0.monomial_coefficient(m)*m for m in [v^2,u*v,u^2]).factor()[0]\n",
    "tangentcone"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We perform a change of coordinates to get a more accurate Newton polygon"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 3 vertices"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2=-tangentcone[0].monomial_coefficient(v)^-1\n",
    "p0=g(y=1,z=u,x=v+a1+a2*u)\n",
    "p0.newton_polytope()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Together with the below computation, we get the confirmation that $P_0$ is of type $\\mathbb{A}_3$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "12288/78125*u^4"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newtonprincipal=sum(p0.monomial_coefficient(m)*m for m in [v^2,u^2*v,u^4])\n",
    "newtonprincipal.discriminant(v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following computation ensures that $x-a_1 y-a_2 z=0$ is the tangent line of $C_4$ at the singular point $P_0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-16/3125*z^4"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g(x=a1*y+a2*z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We perform a projective change of coordinates for which $P_0=[0:1:0]$ and the tangent line of $C_4$ at $P_0$ is $z=0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "change={x:z+a1*y+a2*x,z:x}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "c4=g.subs(change)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Ideal (z, x) of Multivariate Polynomial Ring in x, y, z over Number Field in alpha with defining polynomial t^4 - 2*t^3 - 2*t^2 + 18*t + 21]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c4.jacobian_ideal().minimal_associated_primes()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-16/3125*x^4"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c4(z=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "c3=f.subs(change)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us consider the rational $2:1$ map $\\sigma:\\mathbb{P}^2\\dashrightarrow\\mathbb{P}^2$, $\\sigma([x:y:z]):=[x z:y^2:z^2]$. The strict preimages of $C_3$ and $C_4$ are $C_6$ and $D_4$, respectively."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "d4=R(c4(x=x*z,y=y^2,z=z^2)/z^4)\n",
    "c6=R(c3(x=x*z,y=y^2,z=z^2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Ideal (z, y) of Multivariate Polynomial Ring in x, y, z over Number Field in alpha with defining polynomial t^4 - 2*t^3 - 2*t^2 + 18*t + 21,\n",
       " Ideal (y, 18*x + (13*alpha^3 - 40*alpha^2 + 15*alpha + 168)*z) of Multivariate Polynomial Ring in x, y, z over Number Field in alpha with defining polynomial t^4 - 2*t^3 - 2*t^2 + 18*t + 21]"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c6.jacobian_ideal().minimal_associated_primes()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Ideal (z, y, x) of Multivariate Polynomial Ring in x, y, z over Number Field in alpha with defining polynomial t^4 - 2*t^3 - 2*t^2 + 18*t + 21]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d4.jacobian_ideal().minimal_associated_primes()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The curve $C_6$ is not smooth, but it intersects the smooth curve $D_4$ at four points with intersection number 6."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "((-53263902703616/224482040752766143798828125*alpha^3 + 53263902703616/74827346917588714599609375*alpha^2 - 53263902703616/74827346917588714599609375*alpha - 2111493986320384/224482040752766143798828125)) * (y^4 + (85/24*alpha^3 - 85/6*alpha^2 + 35/8*alpha + 70)*y^2*z^2 + (775/48*alpha^3 - 5375/96*alpha^2 + 125/4*alpha + 6475/32)*z^4)^6"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c6.resultant(d4).factor()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is not hard to check that no conic is tangent to the curves at the four points. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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