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

fromLineToGodeauxSurface -- compute a birational model of a numerical Godeaux surface from a given line

Synopsis

Description

Starting with a line in Q computed from the procedure fromPointInP3xP3xP3xP3ToLine, the function computes a surface $Y$ in $\mathbb{P}(2^2,3^4)$ which is a birational model of a numerical Godeaux surface. Note that because of size of the coefficients, the computations over the rational numbers are very time consuming.

i1 : kk=ZZ/101

o1 = kk

o1 : QuotientRing
 
i2 : H = precomputedModelInP3xP5(kk);

                                1                          1
o2 : Matrix (kk[w ..w , z ..z ])  <--- (kk[w ..w , z ..z ])
                 0   3   0   5              0   3   0   5
 
i3 : pt=findPointInP3xP5(kk)

o3 = | 2 44 16 22 45 -34 -48 -47 47 19 |

              1        10
o3 : Matrix kk  <--- kk
 
i4 : pt1=fromPointInP3xP5ToPointInP3xP3xP3xP3(pt)

o4 = | 38 13 32 1 32 -17 -24 1 5 27 -27 1 46 2 -36 1 |

              1        16
o4 : Matrix kk  <--- kk
 
i5 : line =fromPointInP3xP3xP3xP3ToLine(pt1)

o5 = | 48 32 -24 38 -33 -25 -11 -31 19 33 0 1 |
     | 10 46 -47 6  23  -44 -6  20  10 -4 1 0 |

              2        12
o5 : Matrix kk  <--- kk
 
i6 : I = fromLineToGodeauxSurface(line);

o6 : Ideal of kk[x ..x , y ..y ]
                  0   1   0   3
 
i7 : S=ring I

o7 = S

o7 : PolynomialRing
 
i8 : J=eliminate(I,{S_0,S_1});

o8 : Ideal of S
 
i9 : P3=kk[support J]

o9 = P3

o9 : PolynomialRing
 
i10 : betti(J'=sub(J,P3))

             0 1
o10 = total: 1 1
          0: 1 .
          1: . .
          2: . .
          3: . .
          4: . .
          5: . .
          6: . .
          7: . .
          8: . 1

o10 : BettiTally

See also

Ways to use fromLineToGodeauxSurface :

For the programmer

The object fromLineToGodeauxSurface is a method function.