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NumericalGodeaux :: tricanonicalModelInP3

tricanonicalModelInP3 -- computes the tricanonical model of a numerical Godeaux surface in P3

Synopsis

Description

Given a numerical Godeaux surface $X$, the tricanonical map induces a birational map to a $\mathbb{P}^3$. The procedure computes the birational image of the surface $Y \subset \mathbb{P}(2^2,3^4)$ (and hence of the surface $X$) under this map. The image of the tricanonical map in $\mathbb{P}^3$ is a hypersurface of degree $(3K_X)^2 -$ number of base points of $3K_X$. 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 ]
                  0   1   0   3
 
i4 : J = tricanonicalModelInP3(I);

o4 : Ideal of kk[y ..y ]
                  0   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 ]
                  0   1   0   3
 
i7 : J = tricanonicalModelInP3(I);

o7 : Ideal of kk[y ..y ]
                  0   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 ]
                   0   1   0   3
 
i11 : J = tricanonicalModelInP3(I);

o11 : Ideal of kk[y ..y ]
                   0   3
 
i12 : tally degrees J

o12 = Tally{{8} => 1}

o12 : Tally

See also

Ways to use tricanonicalModelInP3 :

For the programmer

The object tricanonicalModelInP3 is a method function.