Given two positive integers d,e and a ring F, randomMonomialCurve 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 a monomial m of degree e in F[s,t], which is not se or te. This allows one to define two maps ℙ1→ℙ1 and ℙ1→ℙ2 given by sd,td and se,m,te, 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.
i1 : randomMonomialCurve(2,3,QQ); o1 : Ideal of QQ[x , x , x , x , x ] 0,0 0,1 1,0 1,1 1,2 |
This creates a ring F[x0,0,x0,1,x1,0,x1,1,x1,2] in which the resulting ideal is defined.