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

• Function: associatedK3surface

• Usage:

  U' = associatedK3surface X

• Inputs:
  • X, a special Gushel-Mukai fourfold, containing a surface $S\subset Y$ that admits a congruence of $(2e-1)$-secant curves of degree $e$ inside the ambient fivefold $Y$ of the fourfold $X$
• Optional inputs:
  • Singular => ..., default value null, whether to transfer computation to Singular
  • Strategy => ..., default value null, K3 surface associated to a rational fourfold
  • Verbose => ..., default value false, request verbose feedback
• Outputs:
  • U', the (minimal) K3 surface associated to $X$
• Consequences:

  • building U' will return the following four objects:
    • the dominant rational map $\mu:Y\dashrightarrow W$ defined by the linear system of hypersurfaces of degree $2e-1$ having points of multiplicity $e$ along $S$;
    • the surface $U\subset W$ determining the inverse map of the restriction of $\mu$ to $X$;
    • the list of the exceptional curves on the surface $U$;
    • a rational map of degree 1 from the surface $U$ to the minimal K3 surface $U'$.

i1 : X = specialGushelMukaiFourfold "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 in PP^8 of degree 2 and sectional genus 0
     cut out by 6 hypersurfaces of degrees (1,1,1,1,1,2)
     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 : time U' = associatedK3surface X;
 -- used 5.86188s (cpu); 4.08951s (thread); 0s (gc)

o3 : ProjectiveVariety, K3 surface associated to X

i4 : (mu,U,C,f) = building U';

i5 : ? mu

o5 = 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

i6 : ? U

o6 = surface in PP^4 cut out by 5 hypersurfaces of degrees 3^1 4^4 

i7 : first C -- two disjoint lines

o7 = curve in PP^4 cut out by 5 hypersurfaces of degrees 1^1 2^4 

o7 : ProjectiveVariety, curve in PP^4 (subvariety of codimension 1 in U)

i8 : assert(image f == U')