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)
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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
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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 containing 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 5.86323 seconds
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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 , x , x , x , x ]
33331 0 1 2 3 4 5
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i5 : time C = f p -- 5-secant conic to the surface
-- used 0.180165 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 , x , x , x , x ]
33331 0 1 2 3 4 5
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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))
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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)
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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)
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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 9.97472 seconds
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i10 : Y = source map X; -- del Pezzo fivefold containing X
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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
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i12 : time C = f p -- 3-secant conic to the surface
-- used 0.37638 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
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i13 : S = sub(first ideals X,Y);
o13 : Ideal of Y
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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))
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