i1 : S = surface {3,4};
o1 : ProjectiveVariety, surface in PP^5

i2 : X = specialFourfold S  a random cubic fourfold through S
o2 = X
o2 : ProjectiveVariety, cubic fourfold containing a surface of degree 5 and sectional genus 1

i3 : describe X
o3 = Special cubic fourfold of discriminant 14
containing a (smooth) surface of degree 5 and sectional genus 1
cut out by 5 hypersurfaces of degree 2

i4 : Y = specialFourfold "tauquadric"  a random GM fourfold through a tauquadric
o4 = Y
o4 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0

i5 : describe Y
o5 = Special GushelMukai fourfold of discriminant 10(')
containing a surface in PP^8 of degree 2 and sectional genus 0
cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
and with class in G(1,4) given by s_(3,1)+s_(2,2)
Type: ordinary
(case 1 of Table 1 in arXiv:2002.07026)

i6 : T = surface {3,2};
o6 : ProjectiveVariety, surface in PP^7

i7 : Z = specialFourfold T  a random c. i. of 3 quadrics through T
o7 = Z
o7 : ProjectiveVariety, complete intersection of three quadrics in PP^7 containing a surface of degree 7 and sectional genus 1

i8 : describe Z
o8 = Complete intersection of 3 quadrics in PP^7
of discriminant 79 = det 8 7 
 7 16 
containing a smooth surface of degree 7 and sectional genus 1
cut out by 14 hypersurfaces of degree 2
