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

Synopsis

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, associated K3 surface 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 $U'$.

Description

For more details and notation, see the paper Explicit computations with cubic fourfolds, Gushel-Mukai fourfolds, and their associated K3 surfaces.

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 17.1826 seconds

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')

Ways to use this method: