# ambientFivefold -- get the ambient fivefold of the Hodge-special fourfold

## Synopsis

• Usage:
ambientFivefold X
• Inputs:
• Outputs:
• , the ambient fivefold of X

## Description

 i1 : S = surface {4,5,1}; o1 : ProjectiveVariety, surface in PP^6 i2 : V = random(3,S); o2 : ProjectiveVariety, hypersurface in PP^6 i3 : X = V * random(2,S); o3 : ProjectiveVariety, 4-dimensional subvariety of PP^6 i4 : F = specialFourfold(S,X,V); o4 : ProjectiveVariety, complete intersection of type (2,3) in PP^6 containing a surface of degree 7 and sectional genus 2 i5 : ambientFivefold F o5 = V o5 : ProjectiveVariety, hypersurface in PP^6

When $X$ is a GM fourfold, the ambient fivefold of $X$ is a fivefold $Y\subset\mathbb{P}^8$ of degree 5 such that $X\subset Y$ is a quadric hypersurface. We have that the fourfold $X$ is of ordinary type if and only if $Y$ is smooth.

 i6 : X = specialFourfold("21",ZZ/33331); o6 : ProjectiveVariety, GM fourfold containing a surface of degree 12 and sectional genus 5 i7 : describe X o7 = Special Gushel-Mukai fourfold of discriminant 26(') containing a surface in PP^8 of degree 12 and sectional genus 5 cut out by 16 hypersurfaces of degree 2 and with class in G(1,4) given by 7*s_(3,1)+5*s_(2,2) Type: ordinary (case 21 of Table 1 in arXiv:2002.07026) i8 : Y = ambientFivefold X; o8 : ProjectiveVariety, 5-dimensional subvariety of PP^8 i9 : isSubset(X,Y) o9 = true i10 : Y! dim:.................. 5 codim:................ 3 degree:............... 5 sectional genus:...... 1 generators:........... 2^5 linear normality:..... true connected components:. 1 purity:............... true dim sing. l.:......... -1

## Ways to use ambientFivefold :

• "ambientFivefold(HodgeSpecialFourfold)"

## For the programmer

The object ambientFivefold is .