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VirtualResolutions :: randomRationalCurve

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

Synopsis

Description

Given two positive integers d,e and a ring F, randomRationalCurve returns the ideal of a random curve in 1×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 1→ℙ1 and 1→ℙ2, respectively. The graph of the product of these two maps in 1×(ℙ1×ℙ2) is computed, from which a curve of bi-degree (d,e) in 1×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

Caveat

This creates a ring F[x0,0,x0,1,x1,0,x1,1,x1,2] in which the resulting ideal is defined.

Ways to use randomRationalCurve :