# 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 $\PP^1$ factor of $\PP^1\times\PP^2$
• e, an integer, degree of curve on the $\PP^2$ factor of $\PP^1\times\PP^2$
• F, a ring, base ring
• Outputs:
• an ideal, defining random rational curve in $\PP^1\times\PP^2$ 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 $\PP^1\times\PP^2$ of degree (d,e) defined over the base ring F.

This is done by randomly generating two homogeneous polynomials of degree d and three homogeneous polynomials of degree three in $F[s,t]$ defining maps $\PP^1\to\PP^1$ and $\PP^1\to\PP^2$, respectively. The graph of the product of these two maps in $\PP^1\times(\PP^1\times\PP^2)$ is computed, from which a curve of bi-degree (d,e) in $\PP^1\times\PP^2$ 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 ] 0,0 0,1 1,0 1,2 i2 : randomRationalCurve(2,3); ZZ o2 : Ideal of ---[x ..x , x ..x ] 101 0,0 0,1 1,0 1,2

## Caveat

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.

## Ways to use randomRationalCurve :

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

## For the programmer

The object randomRationalCurve is .