172 lines
3.1 KiB
Plaintext
172 lines
3.1 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"# Diagramas de Rudolf de curvas planas\n",
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"\n",
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"<p style=\"text-align:center\">Seminario Rubio de Francia</p>\n",
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"\n",
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"<p style=\"text-align:center\">Miguel Angel Marco Buzunariz</p>"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Geometría algebraica\n",
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"\n",
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"Considerar un sistema de ecuaciones polinómicas\n",
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"\n",
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"$$f_1(x_1,\\ldots,x_n)$$\n",
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"$$f_2(x_1,\\ldots,x_n)$$\n",
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"$$\\vdots$$\n",
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"$$f_m(x_1,\\ldots,x_m)$$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"El conjunto $V$ de sus soluciones forman una *variedad algebraica*."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Problema general de la geometría algebraica\n",
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"\n",
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"Estudiar la geometría de $V$ usando la estructura algebraica de las ecuaciones que la definen."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"Nos vamos a restringir a los casos más fáciles:"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"- $m=1, n=2$ (curvas planas)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"- $\\mathbb{K}=\\mathbb{C}$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## Resultados clásicos conocidos"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"- Si la curva es lisa: teoría de las superficies de Riemann\n",
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" - Superficies de género $g=\\frac{d(d-1)}{2}$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"**Ejemplo**: $y^2= 1-x^2$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"La proyección sobre el eje $x$ es $2:1$ excepto cuando $1-x^2=0$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"celltoolbar": "Slideshow",
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"kernelspec": {
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"display_name": "SageMath 9.1",
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"language": "sage",
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"name": "sagemath"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.7.9"
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},
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"rise": {
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"transition": "none"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}
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