associatedK3surface(CubicFourfold) -- K3 surface associated to a rational cubic fourfold

i1 : X = cubicFourfold "quartic scroll";

o1 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0

i2 : describe X

o2 = Special cubic fourfold of discriminant 14
     containing a rational surface of degree 4 and sectional genus 0
     cut out by 6 hypersurfaces of degree 2

i3 : E = associatedK3surface(X, Verbose=>true);
-- starting associated K3 computation
-- input: fourfold containing a rational surface of degree 4 and sectional genus 0 cut out by 6 hypersurfaces of degree 2
-- planned steps:
-- 1. compute Fano map μ : ℙ⁵ ⇢ W
-- 2. extract surface U from the base locus of (μ|X)⁻¹ : W ⇢ X
-- 3. take a general cubic 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^5
-- computing the Fano map μ from PP^5
-- computed the map μ from PP^5 to PP^5 defined by the hypersurfaces
-- of degree 2 with points of multiplicity 1 along the surface S of degree 4 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 3
-- result: surface in PP^5 cut out by 7 hypersurfaces of degrees 2^1 3^6 

-- 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 3
-- result: surface in PP^5 cut out by 7 hypersurfaces of degrees 2^1 3^6 

-- computing the top components of (U ∩ U')\{exceptional lines} via interpolation
  -- top 1, degrees: 1^4 2^1 
  -- top 2, degrees: 1^3 2^2 
  -- top 3, degrees: 1^3 2^1 3^1 
  -- top 4, degrees: 1^3 2^1 4^1 
-- computing the map f from U to the minimal K3 surface
-- computing the image of f via 'F4' algorithm...
 ✦ associated K3 successfully completed in 2 seconds (cpu: 1 second)

o3 : ProjectiveVariety, K3 surface associated to X

i4 : describe X

o4 = Special cubic fourfold of discriminant 14
     containing a rational surface of degree 4 and sectional genus 0
     cut out by 6 hypersurfaces of degree 2
     K3 status: [▓▓▓▓▓ / ▓▓▓▓▓]
     Mirror fourfold: hypersurface in PP^5 of degree 2
     Surface U of degree 10, sectional genus 7, χ(O_U) = 2, cut out by 7 hypersurfaces of degrees 2^1 3^6 
     Exceptional curves: an irreducible conic curve
     Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by 15 hypersurfaces of degree 2

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

o6 = multi-rational map consisting of one single rational map
     source variety: PP^5
     target variety: hypersurface in PP^5 defined by a form of degree 2
     dominance: true

i7 : assert(image f == E)