randomStandardResolution -- compute a random standard resolution of an S-module R

Synopsis

Usage:

F = randomStandardResolution(kk)

F = randomStandardResolution(kk,s)

F = randomStandardResolution(kk,s,n)

F = randomStandardResolution(kk,s,n,h)
Inputs:
- kk, a ring, the base field over which the computations are performed
- s, a string, the configuration string of the 4 base points which is either "1111", "211", "22" or "4"
- n, an integer, an integer between 1 and 5 describing the order of the torsion group of the surface determined by the standard resolution
- h, an integer, an integer between 0 and 2 describing the number of hyperelliptic fibers of the bicanonical fibration for torsion-free surfaces
Optional inputs:
- Height => ..., default value 19, optional argument for the random computations
- PrecomputedParametrization => ..., default value true, optional argument for using a precomputed unirational parametrization
Outputs:
- F, a chain complex, a standard resolution of an S-module R obtained from the given input

Description

Using our construction method, this procedure computes a standard resolution of an $S = k[x_0,x_1,y_0,\ldots,y_3]$-module $R$ $$ 0 \leftarrow R \leftarrow F_0 \leftarrow F_1 \leftarrow F_1^*(-17) \leftarrow F_0^*(-17) \leftarrow 0 $$ with a skew-symmetric map $d_2: F_1^*(-17) \rightarrow F_1 $.

The main steps of the construction are the choice of a (random) line in complete intersection $Q \subset \mathbb{P}^{11}$ and a (random) point in a linear solution space (depending on the chosen line). The procedure computes only modules $R$ which satisfy the ring condition. Such an module $R$ leads to a numerical Godeaux surface if the surface $Proj(R)$ has only canonical singularities. However, this condition is not checked. If an additional string $s$ is given, the procedure computes standard resolutions which lead to surfaces with the desired configuration of base points of the bicanonical system. If no string $s$ is indicated, the procedure uses the default setting "1111". Moreover, an additional number $n$ indicates the order of the torsion group of the resulting surface. Thus, $n$ must be an integer between 1 and 5. Note that any numerical Godeaux surface with a torsion group of an odd order cannot have double points. Hence, if the configuration of base points and the order of the torsion group are not compatible, an error message is printed.

i1 : kk = ZZ/101;

i2 : s = "1111";

i3 : F = randomStandardResolution(kk,s,5);

i4 : betti F

            0  1  2 3
o4 = total: 8 26 26 8
         0: 1  .  . .
         1: .  .  . .
         2: .  .  . .
         3: .  .  . .
         4: 4  .  . .
         5: 3  6  . .
         6: . 12  . .
         7: .  8  8 .
         8: .  . 12 .
         9: .  .  6 3
        10: .  .  . 4
        11: .  .  . .
        12: .  .  . .
        13: .  .  . .
        14: .  .  . 1

o4 : BettiTally

Caveat

This procedure works mainly over finite fields. Over the rational numbers the procedure may not terminate.

Ways to use `randomStandardResolution` :

"randomStandardResolution(Ring)"
"randomStandardResolution(Ring,String)"
"randomStandardResolution(Ring,String,ZZ)"
"randomStandardResolution(Ring,String,ZZ,ZZ)"

For the programmer

The object randomStandardResolution is a method function with options.