associatedK3surface(GushelMukaiFourfold) -- K3 surface associated to a rational Gushel-Mukai fourfold

i1 : X = gushelMukaiFourfold "tau-quadric";

o1 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0

i2 : describe X

o2 = Special Gushel-Mukai fourfold of discriminant 10(')
     containing a surface of degree 2 and sectional genus 0
     cut out by 6 hypersurfaces of degrees 1^5 2^1 
     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)

i3 : E = associatedK3surface(X, Verbose=>true);
-- starting associated K3 computation
-- input: fourfold containing a surface of degree 2 and sectional genus 0 cut out by 6 hypersurfaces of degrees 1^5 2^1 
-- planned steps:
-- 1. compute Fano map μ
-- 2. extract surface U from the base locus of (μ|X)⁻¹ : W ⇢ X
-- 3. take a general fourfold X' containing the surface and extract U'
-- 4. analyze the intersection U ∩ U' for exceptional curves
-- 5. compute the contraction map f : U -> Ũ
-- info: use building() to access construction data (μ, U, exceptional curves, f)

-- the fourfold has been successfully recognized
-- detected degree of the curves of the congruence: 1
-- computing power of ideal
-- computing/forcing image of map to PP^4
-- computing the Fano map μ from the fivefold in PP^8
-- computed the map μ from the fivefold in PP^8 to PP^4 defined by the hypersurfaces
-- of degree 1 with points of multiplicity 1 along the surface S of degree 2 and genus 0

-- computing the surface U corresponding to the fourfold X
-- computing inverse of the restriction of the Fano map...
-- inverse map computed: defined by forms of degree 4
-- result: surface in PP^4 cut out by 5 hypersurfaces of degrees 3^1 4^4 

-- computing the surface U' corresponding to another fourfold X'
-- computing inverse of the restriction of the Fano map...
-- inverse map computed: defined by forms of degree 4
-- result: surface in PP^4 cut out by 5 hypersurfaces of degrees 3^1 4^4 

-- computing the 2 exceptional lines as the top components of U ∩ U'
  -- top 1, degrees: 1^3 2^1 
  -- top 2, degrees: 1^1 2^6 
  -- top 3, degrees: 1^1 2^4 3^2 
  -- top 4, degrees: 1^1 2^4 4^2 
-- exceptional curves computed: obtained 2 line(s)
-- computing the map f from U to the minimal K3 surface
-- computing the image of f via 'F4' algorithm...
-- note: invariant mismatch for standard K3 surface
-- computing normalization of the surface image
 ✦ associated K3 successfully completed in 6 seconds (cpu: 6 seconds)

o3 : ProjectiveVariety, K3 surface associated to X

i4 : describe X

o4 = Special Gushel-Mukai fourfold of discriminant 10(')
     containing a surface of degree 2 and sectional genus 0
     cut out by 6 hypersurfaces of degrees 1^5 2^1 
     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)
     K3 status: [▓▓▓▓▓ / ▓▓▓▓▓]
     Mirror fourfold: PP^4
     Surface U of degree 8, sectional genus 6, χ(O_U) = 1, cut out by 5 hypersurfaces of degrees 3^1 4^4 
     Exceptional curves: 2 lines
     Minimal K3 surface Ũ: degree 10 and sectional genus 6 in PP^6 cut out by 6 hypersurfaces of degree 2

i5 : (mu,U,exCurves,f) = building E; ? mu

o6 = multi-rational map consisting of one single rational map
     source variety: 5-dimensional subvariety of PP^8 cut out by 5 hypersurfaces of degree 2
     target variety: PP^4
     dominance: true

i7 : assert(image f == E)