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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 5.75215 seconds
i4 : p = point ring X -- random point on P^5

o4 = ideal (x  - 11698x , x  - 5204x , x  + 2338x , x  + 11586x , x  -
             4         5   3        5   2        5   1         5   0  
     ------------------------------------------------------------------------
     8184x )
          5

                ZZ
o4 : Ideal of -----[x ..x ]
              33331  0   5
i5 : time C = f p -- 5-secant conic to the surface
     -- used 0.268178 seconds

                                                                            
o5 = ideal (x  + 3310x  + 1285x  + 9576x , x  - 1985x  - 9693x  + 5568x , x 
             2        3        4        5   1        3        4        5   0
     ------------------------------------------------------------------------
                                     2                    2             
     + 14494x  + 13817x  - 16154x , x  + 13279x x  + 5235x  + 5936x x  -
             3         4         5   3         3 4        4        3 5  
     ------------------------------------------------------------------------
                     2
     3143x x  + 3698x )
          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 contained 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 10.2887 seconds
i10 : Y = source map X; -- del Pezzo fivefold containing X
i11 : p = point Y -- random point on Y

o11 = ideal (t  + 12197t , t  - 8535t , t  + 1905t , t  - 15124t , t  -
              7         8   6        8   5        8   4         8   3  
      -----------------------------------------------------------------------
      10745t , t  - 6518t , t  - 8350t , t  - 7625t )
            8   2        8   1        8   0        8

o11 : Ideal of Y
i12 : time C = f p -- 3-secant conic to the surface
     -- used 0.507734 seconds

o12 = ideal (t  - 7651t  - 6957t  + 13958t , t  + 9389t  + 15953t  + 3179t ,
              5        6        7         8   4        6         7        8 
      -----------------------------------------------------------------------
      t  - 361t  + 5547t  - 1233t , t  - 5025t  + 2525t  - 15559t , t  -
       3       6        7        8   2        6        7         8   1  
      -----------------------------------------------------------------------
      11676t  - 5655t  + 7755t , t  - 11826t  - 14272t  + 13246t )
            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.