i1 : -- A general cubic fourfold of discriminant 26 X = specialCubicFourfold("Farkas-Verra C26",ZZ/33331); o1 : SpecialCubicFourfold (Cubic fourfold containing a surface of degree 7 and sectional genus 0) |
i2 : describe X o2 = Special cubic fourfold of discriminant 26 containing a 3-nodal surface of degree 7 and sectional genus 0 cut out by 13 hypersurfaces of degree 3 |
i3 : time f = detectCongruence X; S: surface of degree 7 and sectional genus 0 in PP^5 cut out by 13 hypersurfaces of degree 3 phi: cubic rational map from PP^5 to PP^12 Z=phi(P^5) number lines contained in Z and passing through the point phi(p): 8 number 2-secant lines to S passing through p: 7 number 5-secant conics to S passing through p: 1 -- used 7.05152 seconds |
i4 : p = point ring X -- random point on P^5 o4 = ideal (x - 11698x , x - 5204x , x + 2338x , x + 11586x , x - 8184x ) 4 5 3 5 2 5 1 5 0 5 ZZ o4 : Ideal of -----[x ..x ] 33331 0 5 |
i5 : time C = f p -- 5-secant conic to the surface -- used 0.244345 seconds 2 o5 = ideal (x + 3310x + 1285x + 9576x , x - 1985x - 9693x + 5568x , x + 14494x + 13817x - 16154x , x + 13279x x + 2 3 4 5 1 3 4 5 0 3 4 5 3 3 4 ---------------------------------------------------------------------------------------------------------------------------- 2 2 5235x + 5936x x - 3143x x + 3698x ) 4 3 5 4 5 5 ZZ o5 : Ideal of -----[x ..x ] 33331 0 5 |
i6 : assert(codim C == 4 and degree C == 2 and codim(C+(first ideals X)) == 5 and degree(C+(first ideals X)) == 5 and isSubset(C, p)) |
The same method can be also applied to a special Gushel-Mukai fourfold. In this case it will detect and return a congruence of (2e-1)-secant curves of degree e inside the unique del Pezzo fivefold containing the GM fourfold.
i7 : -- A general GM fourfold of discriminant 20 X = specialGushelMukaiFourfold("surface of degree 9 and genus 2",ZZ/33331); o7 : SpecialGushelMukaiFourfold (Gushel-Mukai fourfold containing a surface of degree 9 and sectional genus 2) |
i8 : describe X o8 = Special Gushel-Mukai fourfold of discriminant 20 containing a surface in PP^8 of degree 9 and sectional genus 2 cut out by 19 hypersurfaces of degree 2 and with class in G(1,4) given by 6*s_(3,1)+3*s_(2,2) Type: ordinary (case 17 of Table 1 in arXiv:2002.07026) |
i9 : time f = detectCongruence X; S: surface of degree 9 and sectional genus 2 in PP^8 cut out by 19 hypersurfaces of degree 2 phi: quadratic rational map from 5-dimensional subvariety of PP^8 to PP^13 Z=phi(del Pezzo fivefold) number lines containing in Z and passing through the point phi(p): 7 number 1-secant lines to S passing through p: 6 number 3-secant conics to S passing through p: 1 -- used 19.3376 seconds |
i10 : Y = source map X; -- del Pezzo fivefold containing X |
i11 : p = point Y -- random point on Y o11 = ideal (t + 14118t , t + 3234t , t - 16296t , t - 5674t , t - 7 8 6 8 5 8 4 8 3 ----------------------------------------------------------------------- 12127t , t - 1329t , t + 3304t , t + 779t ) 8 2 8 1 8 0 8 o11 : Ideal of Y |
i12 : time C = f p -- 3-secant conic to the surface -- used 0.347885 seconds o12 = ideal (t - 1000t + 8254t - 12393t , t + 10116t + 7449t - 15895t , 5 6 7 8 4 6 7 8 ----------------------------------------------------------------------- t + 10858t + 13401t + 13664t , t - 11215t + 13587t - 5150t , t - 3 6 7 8 2 6 7 8 1 ----------------------------------------------------------------------- 1898t + 4900t + 14451t , t - 7830t + 1802t - 14129t ) 6 7 8 0 6 7 8 o12 : Ideal of Y |
i13 : S = sub(first ideals X,Y); o13 : Ideal of Y |
i14 : assert(dim C -1 == 1 and degree C == 2 and dim(C+S)-1 == 0 and degree(C+S) == 3 and isSubset(C, p)) |
The object detectCongruence is a method function.