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# specialGushelMukaiFourfold -- make a special Gushel-Mukai fourfold

## Synopsis

• Usage:
specialGushelMukaiFourfold(S,X)
• Inputs:
• S, , a smooth irreducible surface $S\subset\mathbb{P}^8$
• X, , a smooth prime Fano fourfold $X\subset \mathbb{P}^8$ of degree 10 and sectional genus 6, which contains the surface $S$
• Optional inputs:
• InputCheck => ..., default value 1
• Verbose => ..., default value false, request verbose feedback
• Outputs:
• , the special Gushel-Mukai fourfold corresponding to the pair $(S,X)$

## Description

In the following example, we define a Gushel-Mukai fourfold containing a so-called $\tau$-quadric.

 i1 : K = ZZ/33331; x = gens ring PP_K^8; i3 : S = projectiveVariety ideal(x_6-x_7, x_5, x_3-x_4, x_1, x_0-x_4, x_2*x_7-x_4*x_8); o3 : ProjectiveVariety, surface in PP^8 i4 : X = projectiveVariety ideal(x_4*x_6-x_3*x_7+x_1*x_8, x_4*x_5-x_2*x_7+x_0*x_8, x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8, x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7, x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8, x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2-x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4-2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8); o4 : ProjectiveVariety, 4-dimensional subvariety of PP^8 i5 : time F = specialGushelMukaiFourfold(S,X); -- used 3.0899s (cpu); 2.64607s (thread); 0s (gc) o5 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0 i6 : time describe F warning: clearing value of symbol x to allow access to subscripted variables based on it : debug with expression debug 9868 or with command line option --debug 9868 -- used 10.5867s (cpu); 6.02183s (thread); 0s (gc) o6 = Special Gushel-Mukai 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) i7 : assert(F == X)

## For the programmer

The object specialGushelMukaiFourfold is .