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

## Synopsis

• Usage:
randomRationalCurve(d,e,F)
randomRationalCurve(d,e)
• Inputs:
• d, an integer, degree of curve on the 1 factor of 1×ℙ2
• e, an integer, degree of curve on the 2 factor of 1×ℙ2
• F, a ring, base ring
• Outputs:
• an ideal, defining random rational curve in 1×ℙ2 of degree (d,e) over F.

## 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 :

• randomRationalCurve(ZZ,ZZ)
• randomRationalCurve(ZZ,ZZ,Ring)