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SpecialFanoFourfolds :: associatedK3surface

associatedK3surface -- associated K3 surface to a rational cubic fourfold

Synopsis

Description

Thus, the code image last associatedK3surface X gives the ideal of 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 : SpecialCubicFourfold (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 (psi,U,C,f) = associatedK3surface(X,Verbose=>true);
-- computing the map mu from P^5 to P^5 defined by the quadrics through the surface S_14
-- computing the surface U corresponding to the fourfold X
-- computing the surface U' corresponding to another fourfold X'
-- computing the exceptional conic on U and U'
-- computing the map f from U to the minimal K3 surface of degree 14
-- computing the image of f
     -- used 6.2014 seconds
i4 : describe psi

o4 = rational map defined by forms of degree 2
     source variety: PP^5
     target variety: PP^5
     image: smooth quadric hypersurface in PP^5
     dominance: false
     birationality: false
     coefficient ring: ZZ/65521
i5 : ? U

o5 = surface of degree 10 and sectional genus 7 in PP^5 cut out by 7 hypersurfaces of degrees (2,3,3,3,3,3,3)
i6 : ? first C

o6 = irreducible conic curve in PP^5
i7 : ? image f

o7 = surface of degree 14 and sectional genus 8 in PP^8 cut out by 15 hypersurfaces of degree 2

The same method can be also applied to a special Gushel-Mukai fourfold. In this case the surface is required to admit a congruence of $(2e-1)$-secant curves of degree $e$ inside the unique del Pezzo fivefold containing the fourfold.

i8 : X = specialGushelMukaiFourfold "tau-quadric";

o8 : SpecialGushelMukaiFourfold (Gushel-Mukai fourfold containing a surface of degree 2 and sectional genus 0)
i9 : describe X

o9 = 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)
i10 : time (psi,U,C,f) = associatedK3surface X;
     -- used 1.2179 seconds
i11 : describe psi

o11 = rational map defined by forms of degree 1
      source variety: 5-dimensional variety of degree 5 in PP^8 cut out by 5 hypersurfaces of degree 2
      target variety: PP^4
      dominance: true
      coefficient ring: ZZ/65521
i12 : ? U

o12 = surface of degree 8 and sectional genus 6 in PP^4 cut out by 9 hypersurfaces of degree 4
i13 : ? first C -- two disjoint lines

o13 = smooth curve of degree 2 and genus -1 in PP^4 cut out by 5 hypersurfaces of degrees (1,2,2,2,2)

See also

Ways to use associatedK3surface :

For the programmer

The object associatedK3surface is a method function with options.