Given a numerical Godeaux surface X, the tricanonical map induces a birational map to a ℙ3. The procedure computes the birational image of the surface Y ⊂ℙ(22,34) (and hence of the surface X) under this map. The image of the tricanonical map in ℙ3 is a hypersurface of degree (3KX)2 - number of base points of 3KX. Hence, for a surface with a torsion group of order 5 the hypersurface has degree 7, for order 3 and 4 the degree is 8 and for order 1 and 2 the degree is 9.
i1 : kk = ZZ/nextPrime(32001); |
i2 : s = "1111"; |
i3 : I = randomGodeauxSurface(kk,s,1,1);
o3 : Ideal of kk[x , x , y , y , y , y ]
0 1 0 1 2 3
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i4 : J = tricanonicalModelInP3(I);
o4 : Ideal of kk[y , y , y , y ]
0 1 2 3
|
i5 : tally degrees J
o5 = Tally{{9} => 1}
o5 : Tally
|
i6 : I = randomGodeauxSurface(kk,s,5);
o6 : Ideal of kk[x , x , y , y , y , y ]
0 1 0 1 2 3
|
i7 : J = tricanonicalModelInP3(I);
o7 : Ideal of kk[y , y , y , y ]
0 1 2 3
|
i8 : tally degrees J
o8 = Tally{{7} => 1}
o8 : Tally
|
i9 : t = "22"; |
i10 : I = randomGodeauxSurface(kk,t,4);
o10 : Ideal of kk[x , x , y , y , y , y ]
0 1 0 1 2 3
|
i11 : J = tricanonicalModelInP3(I);
o11 : Ideal of kk[y , y , y , y ]
0 1 2 3
|
i12 : tally degrees J
o12 = Tally{{8} => 1}
o12 : Tally
|