associatedCastelnuovoSurface -- Castelnuovo surface associated to a rational complete intersection of three quadrics in P^7

i1 : X = specialFourfold random({5:{1}},0_(PP_(ZZ/33331)^7));

o1 : ProjectiveVariety, complete intersection of three quadrics in PP^7 containing a surface of degree 1 and sectional genus 0

i2 : describe X

o2 = Complete intersection of 3 quadrics in PP^7
     of discriminant 31 = det| 8 1 |
                             | 1 4 |
     containing a plane
     (This is a classical example of rational fourfold)

i3 : E = associatedCastelnuovoSurface(X, Verbose=>true);
-- starting associated Castelnuovo computation
-- input: fourfold containing a plane
-- 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^7
-- computed the map μ from the fivefold in PP^7 to PP^4 defined by the hypersurfaces
-- of degree 1 with points of multiplicity 1 along the surface S of degree 1 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 4 hypersurfaces of degrees 3^1 4^3 

-- 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 4 hypersurfaces of degrees 3^1 4^3 

-- computing the top components of (U ∩ U')\{exceptional lines} via interpolation
  -- top 1, degrees: 1^1 2^3 3^3 
  -- top 2, degrees: 2^4 3^3 
  -- top 3, degrees: 2^3 3^4 
  -- top 4, degrees: 2^3 3^3 4^1 
  -- top 5, degrees: 2^3 3^3 5^1 
  -- top 6, degrees: 2^3 3^3 6^1 
-- U is already in the target space; defining f as the identity map
 ✦ associated Castelnuovo successfully completed in 1 second (cpu: 1 second)

o3 : ProjectiveVariety, Castelnuovo surface associated to X

i4 : describe X

o4 = Complete intersection of 3 quadrics in PP^7
     of discriminant 31 = det| 8 1 |
                             | 1 4 |
     containing a plane
     (This is a classical example of rational fourfold)
     Castelnuovo status: [▓▓▓▓▓ / ▓▓▓▓▓]
     Mirror fourfold: PP^4
     Surface U of degree 9, sectional genus 9, χ(O_U) = 4, cut out by 4 hypersurfaces of degrees 3^1 4^3 
     Exceptional curves: only curves of degree > 1
     Minimal Castelnuovo surface Ũ: degree 9 and sectional genus 9 in PP^4 cut out by 4 hypersurfaces of degrees 3^1 4^3 

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^7 cut out by 2 hypersurfaces of degree 2
     target variety: PP^4
     dominance: true

i7 : assert(image f == E)