# detectCongruence(SpecialGushelMukaiFourfold,ZZ) -- detect and return a congruence of (2e-1)-secant curves of degree e inside a del Pezzo fivefold

## Synopsis

• Function: detectCongruence
• Usage:
detectCongruence X
detectCongruence(X,e)
• Inputs:
• X, , containing a surface $S\subset Y$, where $Y$ denotes the unique del Pezzo fivefold containing the fourfold $X$
• e, an integer, a positive integer (optional but recommended)
• Optional inputs:
• Verbose => ..., default value false, request verbose feedback
• Outputs:
• , that is a function which takes a (general) point $p\in Y$ and returns the unique rational curve of degree $e$, $(2e-1)$-secant to $S$, contained in $Y$ and passing through $p$ (an error is thrown if such a curve does not exist or is not unique)

## Description

 i1 : -- A GM fourfold of discriminant 20 X = specialGushelMukaiFourfold("17",ZZ/33331); o1 : ProjectiveVariety, GM fourfold containing a surface of degree 9 and sectional genus 2 i2 : describe X o2 = 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) i3 : time f = detectCongruence(X,Verbose=>true); number lines contained in the image of the quadratic map and passing through a general point: 7 number 1-secant lines = 6 number 3-secant conics = 1 -- used 24.9438 seconds o3 : Congruence of 3-secant conics to surface in a fivefold in PP^8 i4 : Y = ambientFivefold X; -- del Pezzo fivefold containing X o4 : ProjectiveVariety, 5-dimensional subvariety of PP^8 i5 : p := point Y -- random point on Y o5 = point of coordinates [7214, -1460, 7057, -2440, 15907, -14345, -5937, 13402, 1] o5 : ProjectiveVariety, a point in PP^8 i6 : time C = f p; -- 3-secant conic to the surface -- used 0.711592 seconds o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y) i7 : S = surface X; o7 : ProjectiveVariety, surface in PP^8 (subvariety of codimension 3 in Y) i8 : assert(dim C == 1 and degree C == 2 and dim(C*S) == 0 and degree(C*S) == 3 and isSubset(p,C) and isSubset(C,Y))