randomCurveP1P2 -- creates the ideal of a random curve in P^1xP^2

Synopsis

Usage:

randomCurveP1P2(d,g,F)

randomCurveP1P2(d,g)
Inputs:
- d, an integer, degree of the curve.
- g, an integer, genus of the curve.
- F, a ring, base ring.
Optional inputs:
- Attempt => ..., default value 1000, limit number of attempts for randomCurveP1P2
Outputs:
- an ideal, defining random curve $\PP^1\times\PP^2$ from a curve of degree d and genus g in $\PP^3$ over F.

Description

Given a positive integer d, a non-negative integer g, and a ring F, randomCurveP1P2 produces a random curve of bi-degree (d,d) and genus g in $\PP^1\times\PP^2$. This is done by using the curve function from the SpaceCurves package to first generate a random curve of degree d and genus g in $\PP^1\times\PP^2$, and then applying curveFromP3toP1P2 to produce a curve in $\PP^1\times\PP^2$.

Since curveFromP3toP1P2 relies on projecting from the point $[0:0:0:1]$ and the line $[0:0:s:t]$, randomCurveP1P2 attempts to find a curve in $\PP^3$, which does not intersect the base locus of these projections. If the curve did intersect the base locus the resulting curve in $\PP^1\times\PP^2$ would not have degree (d,d). The number of attempts used to try to find such curves is controlled by the randomCurveP1P2(...,Attempt=>...) option, which by default is set to 1000.

i1 : randomCurveP1P2(3,0);

               ZZ
o1 : Ideal of ---[x   ..x   , x   ..x   ]
              101  0,0   0,1   1,0   1,2

i2 : randomCurveP1P2(3,0,QQ);

o2 : Ideal of QQ[x   ..x   , x   ..x   ]
                  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 randomCurveP1P2:

randomCurveP1P2(ZZ,ZZ)
randomCurveP1P2(ZZ,ZZ,Ring)

For the programmer

The object randomCurveP1P2 is a method function with options.