Given two positive integers d,e and a ring F, randomRationalCurve returns the ideal of a random curve in ℙ1×ℙ2 of degree (d,e) defined over the base ring F.
This is done by randomly generating two homogenous polynomials of degree d and three homogenous polynomials of degree three in F[s,t] defining maps ℙ1→ℙ1 and ℙ1→ℙ2, respectively. The graph of the product of these two maps in ℙ1×(ℙ1×ℙ2) is computed, from which a curve of bi-degree (d,e) in ℙ1×ℙ2 over F is obtained by saturating and then eliminating.
If no base ring is specified, the computations are performed over ZZ/101.
randomRationalCurve(2,3,QQ); |
randomRationalCurve(2,3); |
This creates a ring F[x0,0,x0,1,x1,0,x1,1,x1,2] in which the resulting ideal is defined.