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

Synopsis

• Function: associatedK3surface

• Usage:

  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 unique del Pezzo fivefold $Y$ containing the fourfold $X$
• Optional inputs:
  • Strategy => ..., default value null, associated K3 surface to a rational fourfold
  • Verbose => ..., default value false, associated K3 surface to a rational fourfold
• Outputs:
  • 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 a minimal K3 surface, the associated K3 surface to $X$.

Description

Thus, the code image last associatedK3surface X gives the (minimal) associated K3 surface to X. For more details and notation, see the paper Trisecant Flops, their associated K3 surfaces and the rationality of some Fano fourfolds.

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 (mu,U,C,f) = associatedK3surface X;
     -- used 4.87796 seconds

i4 : ? mu

o4 = 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

i5 : ? U

o5 = surface in PP^4 cut out by 9 hypersurfaces of degree 4

i6 : first C -- two disjoint lines

o6 = curve in PP^4 cut out by 5 hypersurfaces of degrees 1^1 2^4 

o6 : ProjectiveVariety, curve in PP^4 (subvariety of codimension 1 in U)

See also

• associatedK3surface(SpecialCubicFourfold) -- associated K3 surface to a rational cubic fourfold
• detectCongruence -- detect and return a congruence of secant curves to a surface