randomRationalCurve -- creates the ideal of a random rational curve of degree (d,e) in P^1xP^2

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.

i1 : randomRationalCurve(2,3,QQ);

o1 : Ideal of QQ[x   , x   , x   , x   , x   ]
                  0,0   0,1   1,0   1,1   1,2

i2 : randomRationalCurve(2,3);

               ZZ
o2 : Ideal of ---[x   , x   , x   , x   , x   ]
              101  0,0   0,1   1,0   1,1   1,2

This creates a ring F[x_0,0,x_0,1,x_1,0,x_1,1,x_1,2] in which the resulting ideal is defined.