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VirtualResolutions :: randomCurveP1P2

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



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);

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.

Ways to use randomCurveP1P2 :

For the programmer

The object randomCurveP1P2 is a method function with options.