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VirtualResolutions :: randomMonomialCurve

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



Given two positive integers d,e and a ring F, randomMonomialCurve 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 a monomial m of degree e in F[s,t], which is not se or te. This allows one to define two maps 1→ℙ1 and 1→ℙ2 given by sd,td and se,m,te, 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 : randomMonomialCurve(2,3,QQ);

o1 : Ideal of QQ[x   , x   , x   , x   , x   ]
                  0,0   0,1   1,0   1,1   1,2


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 randomMonomialCurve :