This example demonstrates use of the Macaulay 2 computer algebra package. It is taken from the second chapter of the Macaulay 2 book,
Projective Geometry and Homological Algebra
David Eisenbud
A chapter in Computations in algebraic geometry with Macaulay 2
which starts with three constructions of the twisted cubic curve.
We begin by defining kk to be the finite field of characteristic 101,
and ringP1, ringP3 to be the coordinate rings of projective spaces of
dimensions 1 and 3.
kk = ZZ/101 ringP3 = kk[a..d] ringP1 = kk[s,t]
An elementary approach to constructing the ideal of the twisted cubic curve is as the kernel of a map of projective spaces:
cubicMap = map(ringP1, ringP3, {s^3, s^2*t, s*t^2, t^3})
idealCubic = kernel cubicMap
Macaulay2 also has a built-in facility for monomial curves:
idealCubic2 = monomialCurveIdeal(ringP3, {1,2,3})
As a third approach,
the ideal of the twisted cubic curve is generated by the 2 by 2 minors of
the matrix M:
M = matrix{{a,b,c},{b,c,d}}
idealCubic3 = minors(2, M)
This script produces the output:
/bin/sh: M2: command not found