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



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 U' = associatedK3surface X;
 -- used 5.31583s (cpu); 1.97033s (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')

See also

Ways to use this method: