Some Hodge-special fourfolds are known to be rational. In this case, the function tries to obtain a birational map from $\mathbb{P}^4$ (or, e.g., from a quadric hypersurface in $\mathbb{P}^5$) to the fourfold.
i1 : X = specialFourfold surface {3,4};
o1 : ProjectiveVariety, cubic fourfold containing a surface of degree 5 and sectional genus 1
i2 : phi = parametrize X;
o2 : MultirationalMap (birational map from PP^4 to X)
i3 : describe phi
o3 = multi-rational map consisting of one single rational map
source variety: PP^4
target variety: hypersurface in PP^5 defined by a form of degree 3
base locus: surface in PP^4 cut out by 6 hypersurfaces of degree 4
dominance: true
multidegree: {1, 4, 7, 6, 3}
degree: 1
degree sequence (map 1/1): [4]
coefficient ring: ZZ/65521
i4 : Y = specialFourfold "tau-quadric";
o4 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
i5 : psi = parametrize Y;
o5 : MultirationalMap (birational map from PP^4 to Y)
i6 : describe psi
o6 = multi-rational map consisting of one single rational map
source variety: PP^4
target variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces of degree 2
base locus: surface in PP^4 cut out by 5 hypersurfaces of degrees 3^1 4^4
dominance: true
multidegree: {1, 4, 8, 10, 10}
degree: 1
degree sequence (map 1/1): [4]
coefficient ring: ZZ/65521
i7 : Z = specialFourfold "plane in PP^7";
o7 : ProjectiveVariety, complete intersection of three quadrics in PP^7 containing a surface of degree 1 and sectional genus 0
i8 : eta = parametrize Z;
o8 : MultirationalMap (birational map from PP^4 to Z)
i9 : describe eta
o9 = multi-rational map consisting of one single rational map
source variety: PP^4
target variety: 4-dimensional subvariety of PP^7 cut out by 3 hypersurfaces of degree 2
base locus: surface in PP^4 cut out by 4 hypersurfaces of degrees 3^1 4^3
dominance: true
multidegree: {1, 4, 7, 8, 8}
degree: 1
degree sequence (map 1/1): [4]
coefficient ring: ZZ/65521