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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 paper Trisecant Flops, their associated K3 surfaces and the rationality of some Fano fourfolds.

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 1.70478 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