GMtables -- make examples of reducible subschemes of P^5

Synopsis

• Usage:

  GMtables(i,K)

• Inputs:
  • i, an integer, an integer between 1 and 21
  • K, a ring, the coefficient ring
• Optional inputs:
  • Verify => ..., default value true
• Outputs:
  • a triple of varieties, $(B,V,C)$, which represents a reducible subscheme of $\mathbb{P}^5$, in accordance with the 21 examples listed in Table 2 of the paper On some families of Gushel-Mukai fourfolds.

Description

i1 : (B,V,C) := GMtables(1,ZZ/33331)

o1 = (surface in PP^5 cut out by 4 hypersurfaces of degrees 1^1 2^3 , surface
     ------------------------------------------------------------------------
     in PP^5 cut out by 3 hypersurfaces of degrees 1^2 2^1 , curve in PP^5
     ------------------------------------------------------------------------
     cut out by 4 hypersurfaces of degrees 1^3 2^1 )

o1 : Sequence

i2 : B * V == C

o2 = true

The corresponding example of fourfold reported in Table 1 of the aforementioned paper can be obtained as follows.

i3 : psi = rationalMap(ideal B,Dominant=>2);

o3 : RationalMap (quadratic rational map from PP^5 to 5-dimensional subvariety of PP^8)

i4 : X = specialGushelMukaiFourfold psi ideal V;

o4 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0

This is basically the same as doing this:

i5 : specialGushelMukaiFourfold("1",ZZ/33331);

o5 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0

See also

• specialGushelMukaiFourfold(String,Ring) -- random special Gushel-Mukai fourfold of a given type
• specialGushelMukaiFourfold(Array,Array) -- construct GM fourfolds by gluing cubic or quartic scrolls to surfaces in PP^6