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

Synopsis

• Usage:

  parametrize X

  parametrizeFanoFourfold(X,Strategy=>...)

• Inputs:
  • X, an embedded projective variety, a prime Fano fourfold $X$ of coindex at most 3 having degree $d$ and genus $g$ with $(d,g)\in\{(2,0),(4,1),(5,1),(12,7),(14,8),(16,9),(18,10)\}$
• Optional inputs:
  • Strategy => ..., default value 1
• Outputs:
  • a multi-rational map, a birational map from $\mathbb{P}^4$ to $X$

Description

This function 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 : K = ZZ/65521; X = GG_K(1,4) * random({{1},{1}},0_(GG_K(1,4)));

o2 : ProjectiveVariety, 4-dimensional subvariety of PP^9

i3 : ? X

o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
     1^2 2^5

i4 : time parametrizeFanoFourfold X
 -- used 2.34271s (cpu); 1.26979s (thread); 0s (gc)

o4 = multi-rational map consisting of one single rational map
     source variety: PP^4
     target variety: 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees 1^2 2^5 
     dominance: true
     degree: 1

o4 : MultirationalMap (birational map from PP^4 to X)

See also

• fanoFourfold -- random prime Fano fourfold of coindex at most 3
• parametrize(HodgeSpecialFourfold) -- rational parametrization
• unirationalParametrization -- unirational parametrization
• parametrize(MultiprojectiveVariety) -- try to get a parametrization of a multi-projective variety