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

• Function: associatedK3surface

• Usage:

  U' = associatedK3surface X

• Inputs:
  • X, a special cubic fourfold, containing a surface $S\subset\mathbb{P}^5$ that admits a congruence of $(3e-1)$-secant curves of degree $e$
• 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 $\mu:\mathbb{P}^5 \dashrightarrow W$ defined by the linear system of hypersurfaces of degree $3e-1$ 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 = specialCubicFourfold "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 (smooth) surface of degree 4 and sectional genus 0
     cut out by 6 hypersurfaces of degree 2

i3 : time U' = associatedK3surface X;
 -- used 1.55574s (cpu); 0.94874s (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: PP^5
     target variety: hypersurface in PP^5 defined by a form of degree 2
     dominance: true

i6 : ? U

o6 = surface in PP^5 cut out by 7 hypersurfaces of degrees 2^1 3^6 

i7 : last C

o7 = curve in PP^5 cut out by 4 hypersurfaces of degrees 1^3 2^1 

o7 : ProjectiveVariety, curve in PP^5 (subvariety of codimension 1 in U)

i8 : assert(image f == U')