The function constructs a general curve C of genus 10 and bidegree (6,10) in P1*P2 with random8gonalGenus10Curve and constructs its canonical embedding in P10. By construction, if H is an hyperplane divisor in the planar model of the curve, then KC - H is a g18. The function returns a sequence (I, H1, H2), where I is the ideal of the canonically embedded curve in P10 and H1 and H2 are the hyperplanes passing through 10 points on the curve corresponding to H via the canonical embedding. A general linear combination of H1 and H2 cuts the curve in these 10 points plus 8 points corresponding to the g18. By construction, this procedure is rational and dominant.
i1 : p=32009; |
i2 : time (I, H1, H2)=randomGenus10Degree8CoverOfP1(p);
-- used 41.1145 seconds
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i3 : genus I, degree I o3 = (10, 18) o3 : Sequence |
i4 : K=I+ideal(random(ZZ)*H1+random(ZZ)*H2);
ZZ
o4 : Ideal of -----[x , x , x , x , x , x , x , x , x , x ]
32009 0 1 2 3 4 5 6 7 8 9
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i5 : codim K, degree K o5 = (9, 18) o5 : Sequence |
i6 : IH=I+ideal (H1)+ideal (H2); -- 2g-2-8=10 fixed points
ZZ
o6 : Ideal of -----[x , x , x , x , x , x , x , x , x , x ]
32009 0 1 2 3 4 5 6 7 8 9
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i7 : codim IH, degree IH o7 = (9, 10) o7 : Sequence |
i8 : KmH=K:IH; -- 8 moving points corresponding to the g^1_8
ZZ
o8 : Ideal of -----[x , x , x , x , x , x , x , x , x , x ]
32009 0 1 2 3 4 5 6 7 8 9
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i9 : codim KmH, degree KmH o9 = (9, 8) o9 : Sequence |