next | previous | forward | backward | up | top | index | toc | Macaulay2 website
NumericalGodeaux :: bihomogeneousModel

bihomogeneousModel -- compute a birational model of a numerical Godeaux surface in P1xP3

Synopsis

Description

Given a numerical Godeaux surface $X$, the bi- and tricanonical map induce a birational map to a $\mathbb{P}^1 \times \mathbb{P}^3$. 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 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 ]
                  0   1   0   3
 
i4 : J = bihomogeneousModel(I);

o4 : Ideal of kk[x ..x , y ..y ]
                  0   1   0   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 ]
                  0   1   0   3
 
i9 : J = bihomogeneousModel(I);

o9 : Ideal of kk[x ..x , y ..y ]
                  0   1   0   3
 
i10 : tally degrees J

o10 = Tally{{0, 8} => 1}
            {1, 3} => 1
            {1, 5} => 1
            {2, 2} => 1

o10 : Tally

See also

Ways to use bihomogeneousModel :

For the programmer

The object bihomogeneousModel is a method function.