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 |
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.
The object randomCurveP1P2 is a method function with options.