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SpecialFanoFourfolds :: parametrizeFanoFourfold

parametrizeFanoFourfold -- rational parametrization of a prime Fano fourfold of coindex at most 3

Synopsis

Description

This method is mainly based on results contained in the classical paper Algebraic varieties with canonical curve sections, by L. Roth. In some examples, more strategies are available. For instance, if $X\subset\mathbb{P}^7$ is a 4-dimensional linear section of $\mathbb{G}(1,4)\subset\mathbb{P}^9$, then by passing Strategy=>1 (which is the default choice) we get the inverse of the projection from the plane spanned by a conic contained in $X$; while with Strategy=>2 we get the projection from the unique $\sigma_{2,2}$-plane contained in $X$ (Todd's result).

i1 : G'1'4 = ideal Grass(1,4,ZZ/65521); I = G'1'4 + ideal(random(1,ring G'1'4),random(1,ring G'1'4))

                ZZ
o1 : Ideal of -----[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
              65521  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4

o2 = ideal (p   p    - p   p    + p   p   , p   p    - p   p    + p   p   , p   p    - p   p    + p   p   , p   p    - p   p    +
             2,3 1,4    1,3 2,4    1,2 3,4   2,3 0,4    0,3 2,4    0,2 3,4   1,3 0,4    0,3 1,4    0,1 3,4   1,2 0,4    0,2 1,4  
     ----------------------------------------------------------------------------------------------------------------------------
     p   p   , p   p    - p   p    + p   p   , - 32646p    - 28377p    + 26433p    - 29566p    + 3783p    + 26696p    + 8570p   
      0,1 2,4   1,2 0,3    0,2 1,3    0,1 2,3          0,1         0,2         1,2         0,3        1,3         2,3        0,4
     ----------------------------------------------------------------------------------------------------------------------------
     + 16659p    + 8444p    - 11230p   , - 22394p    - 8214p    + 23752p    - 30100p    + 5071p    + 25800p    - 20388p    -
             1,4        2,4         3,4          0,1        0,2         1,2         0,3        1,3         2,3         0,4  
     ----------------------------------------------------------------------------------------------------------------------------
     14258p    + 22523p    - 12700p   )
           1,4         2,4         3,4

                ZZ
o2 : Ideal of -----[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
              65521  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
i3 : (dim I -1, degree I,(genera I)_3)

o3 = (4, 5, 1)

o3 : Sequence
i4 : time f = parametrizeFanoFourfold I;
     -- used 1.1993 seconds

o4 : RationalMap (cubic rational map from PP^4 to 4-dimensional subvariety of PP^9)
i5 : describe f!

o5 = rational map defined by forms of degree 3
     source variety: PP^4
     target variety: 4-dimensional variety of degree 5 in PP^9 cut out by 7 hypersurfaces of degrees (1,1,2,2,2,2,2)
     dominance: true
     birationality: true (the inverse map is already calculated)
     projective degrees: {1, 3, 5, 5, 5}
     number of minimal representatives: 1
     dimension base locus: 2
     degree base locus: 4
     coefficient ring: ZZ/65521

See also

Ways to use parametrizeFanoFourfold :

For the programmer

The object parametrizeFanoFourfold is a method function with options.