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   "source": [
    "## The image of $$\\Gamma = V(b x -1) \\subset \\mathbb{A}_\\mathbb{Q}^1 \\times \\mathbb{A}_\\mathbb{Q}^1 = \\operatorname{Spec} \\mathbb{Q}[b][x]$$ under projection onto the first component."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CapAndHomalg v\u001b[32m1.6.0\u001b[39m\n",
      "Imported OSCAR's components GAP and Singular_jll\n",
      "Type: ?CapAndHomalg for more information\n"
     ]
    }
   ],
   "source": [
    "using CapAndHomalg"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [],
   "source": [
    "LoadPackage( \"ZariskiFrames\" )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GAP: Q"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ℚ = HomalgFieldOfRationalsInSingular()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GAP: Q[b]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = ℚ[\"b\"]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GAP: Q[b][x]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R = B[\"x\"]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GAP: V_{Q[b][x]}( <...> )"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Γ = ClosedSubsetOfSpec( \"b*x-1\", R )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "V( <b*x-1> )\n"
     ]
    }
   ],
   "source": [
    "Display( Γ )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GAP: ( V_{Q[b]}( I1 ) \\ V_{Q[b]}( J1_1 ) )"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "im = ConstructibleProjection( Γ )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "( V( <> )\n",
      "\n",
      "\\ V( <b> ) )\n"
     ]
    }
   ],
   "source": [
    "Display( im )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
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   "source": [
    "Visualize( im )"
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