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SpecialFanoFourfolds :: associatedK3surface(SpecialCubicFourfold)

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

Synopsis

Description

Thus, the code image last associatedK3surface X gives the (minimal) associated K3 surface to X. For more details and notation, see the papers Trisecant Flops, their associated K3 surfaces and the rationality of some Fano fourfolds and Explicit computations with cubic fourfolds, Gushel-Mukai fourfolds, and their associated K3 surfaces.

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 (mu,U,C,f) = associatedK3surface(X,Verbose=>true);
-- the fourfold has been successfully recognized
-- computing the Fano map mu from PP^5
-- computed the map mu from PP^5 to PP^5 defined by the hypersurfaces
   of degree 2 with points of multiplicity 1 along the surface S of degree 4 and genus 0
-- computing the surface U corresponding to the fourfold X
-- computing the surface U' corresponding to another fourfold X'
-- computing the top components of (U*U')\{exceptional lines} via interpolation
top 1, degrees: 1^4 2^1 
top 2, degrees: 1^3 2^2 
top 3, degrees: 1^3 2^1 3^1 
top 4, degrees: 1^3 2^1 4^1 
-- computing the map f from U to the minimal K3 surface of degree 14
-- computing the image of f using the F4 algorithm
     -- used 1.73165 seconds
i4 : ? mu

o4 = multi-rational map consisting of one single rational map
     source variety: PP^5
     target variety: PP^5
     dominance: false
     image: hypersurface in PP^5 defined by a form of degree 2
i5 : ? U

o5 = surface in PP^5 cut out by 7 hypersurfaces of degrees 2^1 3^6 
i6 : last C

o6 = curve in PP^5 cut out by 4 hypersurfaces of degrees 1^3 2^1 

o6 : ProjectiveVariety, curve in PP^5 (subvariety of codimension 1 in U)
i7 : image f

o7 = surface in PP^8 cut out by 15 hypersurfaces of degree 2

o7 : ProjectiveVariety, surface in PP^8

See also

Ways to use this method: