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SpecialFanoFourfolds :: detectCongruence(SpecialCubicFourfold,ZZ)

detectCongruence(SpecialCubicFourfold,ZZ) -- 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 : ProjectiveVariety, 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 8.51617 seconds
i4 : p := point ambient X -- random point on P^5

o4 = point of coordinates [-7901, -15944, -7086, 2968, 3085, 1]

o4 : ProjectiveVariety, a point in PP^5
i5 : time C = f p; -- 5-secant conic to the surface
     -- used 1.15406 seconds

o5 : ProjectiveVariety, curve in PP^5
i6 : assert(dim C == 1 and degree C == 2 and dim(C * surface X) == 0 and degree(C * surface X) == 5 and isSubset(p, C))

See also