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   "source": [
    "# Diagramas de Rudolf de curvas planas\n",
    "\n",
    "<p style=\"text-align:center\">Seminario Rubio de Francia</p>\n",
    "\n",
    "<p style=\"text-align:center\">Miguel Angel Marco Buzunariz</p>"
   ]
  },
  {
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   },
   "source": [
    "## Geometría algebraica\n",
    "\n",
    "Considerar un sistema de ecuaciones polinómicas\n",
    "\n",
    "$$f_1(x_1,\\ldots,x_n)$$\n",
    "$$f_2(x_1,\\ldots,x_n)$$\n",
    "$$\\vdots$$\n",
    "$$f_m(x_1,\\ldots,x_m)$$"
   ]
  },
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   "source": [
    "El conjunto $V$ de sus soluciones forman una *variedad algebraica*."
   ]
  },
  {
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   "metadata": {
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    }
   },
   "source": [
    "## Problema general de la geometría algebraica\n",
    "\n",
    "Estudiar la geometría de $V$ usando la estructura algebraica de las ecuaciones que la definen."
   ]
  },
  {
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   },
   "source": [
    "Nos vamos a restringir a los casos más fáciles:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   },
   "source": [
    "- $m=1, n=2$ (curvas planas)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   },
   "source": [
    "- $\\mathbb{K}=\\mathbb{C}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Resultados clásicos conocidos"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "fragment"
    }
   },
   "source": [
    "- Si la curva es lisa: teoría de las superficies de Riemann\n",
    "  - Superficies de género $g=\\frac{d(d-1)}{2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "fragment"
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   "source": [
    "**Ejemplo**: $y^2= 1-x^2$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "La proyección sobre el eje $x$ es $2:1$ excepto cuando $1-x^2=0$"
   ]
  },
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