Given a numerical Godeaux surface X, the bi- and tricanonical map induce a birational map to a ℙ1 ×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 ideal J must contain an element of bidegree (0,7), for order 3 and 4 an element of bidegree (0,8) and for order 1 and 2 an element of bidegree (0,9).
i1 : kk = ZZ/nextPrime(32001); |
i2 : s = "1111"; |
i3 : I = randomGodeauxSurface(kk,s,5); o3 : Ideal of kk[x , x , y , y , y , y ] 0 1 0 1 2 3 |
i4 : J = bihomogeneousModel(I); o4 : Ideal of kk[x , x , y , y , y , y ] 0 1 0 1 2 3 |
i5 : tally degrees J o5 = Tally{{0, 7} => 1} {1, 2} => 1 {1, 5} => 1 {2, 3} => 1 o5 : Tally |
i6 : kk = ZZ/nextPrime(32001); |
i7 : t = "22"; |
i8 : I = randomGodeauxSurface(kk,t,4); o8 : Ideal of kk[x , x , y , y , y , y ] 0 1 0 1 2 3 |
i9 : J = bihomogeneousModel(I); o9 : Ideal of kk[x , x , y , y , y , y ] 0 1 0 1 2 3 |
i10 : tally degrees J o10 = Tally{{0, 8} => 1} {1, 3} => 1 {1, 5} => 1 {2, 2} => 1 o10 : Tally |