235 lines
6.4 KiB
Plaintext
235 lines
6.4 KiB
Plaintext
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{
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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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"source": [
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"# A demo of the vector_bundle package."
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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": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"from vector_bundle import *\n",
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"F.<x> = FunctionField(GF(101))"
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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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"source": [
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"## Constructing an indecomposable bundle on an elliptic curve\n",
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"We construct an indecomposable vector bundle of rank 5 and degree 3 on a function field of genus 1. \n",
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"This construction may be done automatically using the ```atiyah_bundle``` function but we break it down step by step."
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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": 5,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[Ideal (1) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x]\n",
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"[(1)]\n",
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"[((x/(x^2 + 1))*y)]\n"
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]
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}
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],
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"source": [
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"R.<y> = F[]\n",
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"K.<y> = F.extension(y^2 - x^3 - x)\n",
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"deg_1_bundle = VectorBundle(K, K.places_infinite()[0].divisor())\n",
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"E = deg_1_bundle\n",
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"print(E.coefficient_ideals())\n",
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"print(E.basis_finite())\n",
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"print(E.basis_infinite())"
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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": 7,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[Ideal (x^2/(x^2 + 3)) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (1) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x]\n",
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"[(1, 0), (0, 1)]\n",
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"[(1, 0), (100*x^3/(x^2 + 1), (x/(x^2 + 1))*y)]\n"
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]
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}
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],
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"source": [
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"E = E.extension_by_global_sections()\n",
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"print(E.coefficient_ideals())\n",
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"print(E.basis_finite())\n",
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"print(E.basis_infinite())"
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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": 8,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[Ideal (x^2/(x^2 + 3)) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (1) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x]\n",
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"[(1, 0), (0, 1)]\n",
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"[((x/(x^2 + 1))*y, 0), ((100*x^4/(x^4 + 2*x^2 + 1))*y, x^3/(x^2 + 1))]\n"
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]
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}
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],
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"source": [
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"E = E.tensor_product(deg_1_bundle)\n",
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"print(E.coefficient_ideals())\n",
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"print(E.basis_finite())\n",
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"print(E.basis_infinite())"
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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": 9,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[Ideal (x^2/(x^2 + 3)) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (x^2/(x^2 + 3)) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (x^2/(x^2 + 3)) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (x^2/(x^2 + 3)) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (1) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x]\n",
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"[(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)]\n",
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"[(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (100*x^3/(x^2 + 1), 0, 100/x^2*y, (x/(x^2 + 1))*y, 0), (x^6/(x^4 + 2*x^2 + 1), (100*x^4/(x^4 + 2*x^2 + 1))*y, (x/(x^2 + 1))*y, (100*x^4/(x^4 + 2*x^2 + 1))*y, x^3/(x^2 + 1))]\n"
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]
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}
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],
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"source": [
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"E = E.extension_by_global_sections()\n",
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"print(E.coefficient_ideals())\n",
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"print(E.basis_finite())\n",
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"print(E.basis_infinite())"
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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": 10,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"5\n",
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"3\n"
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]
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}
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],
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"source": [
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"print(E.rank())\n",
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"print(E.degree())"
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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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"source": [
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"We check the algebra of global endomorphisms of E to ensure that it is indecomposable."
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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": 11,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[\n",
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"[1 0 0 0 0]\n",
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"[0 1 0 0 0]\n",
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"[0 0 1 0 0]\n",
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"[0 0 0 1 0]\n",
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"[0 0 0 0 1]\n",
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"]"
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]
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},
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"execution_count": 11,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"E.end().h0()"
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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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"source": [
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"## Constructing a weakly stable vector bundle following Savin's method\n",
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"We construct a weakly stable vector bundle of rank 3 and degree 5 by successive extensions by line bundles. \n",
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"See [Sav07] in the references for details. \n",
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"This, again, may be achieved directly using the ```savin_bundle``` function."
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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": 12,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"3\n",
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"5\n",
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"[Ideal (1, 1/x*y) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (1) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (1) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x]\n",
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"[(1, 0, 0), (0, 1, 0), (0, 0, 1)]\n",
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"[(1, 0, 0), ((100*x/(x^2 + 1))*y, x^3/(x^2 + 1), 0), (0, (100*x^4/(x^4 + 2*x^2 + 1))*y, x^3/(x^2 + 1))]\n"
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]
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}
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],
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"source": [
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"F = VectorBundle(K, 2*K.places_infinite()[0].divisor())\n",
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"F1 = VectorBundle(K, K.places_finite()[0].divisor())\n",
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"E1 = F1\n",
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"E2 = F.non_trivial_extension(E1)\n",
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"E3 = F.non_trivial_extension(E2)\n",
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"print(E3.rank())\n",
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"print(E3.degree())\n",
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"print(E3.coefficient_ideals())\n",
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"print(E3.basis_finite())\n",
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"print(E3.basis_infinite())"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "SageMath 10.3.beta8",
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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": "3.11.2"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 4
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}
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