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SpecialFanoFourfolds :: detectCongruence

detectCongruence -- detect and return a congruence of (3e-1)-secant curves of degree e

Synopsis

Description

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))

See also

Ways to use detectCongruence :

For the programmer

The object detectCongruence is a method function.