SpecialFanoFourfolds -- A package for working with Hodge-special fourfolds

Description

This package contains several tools related to the rationality problem for cubic fourfolds, Gushel-Mukai fourfolds, and some other special Fano fourfolds. See arXiv:2204.11518 for some applications.

The following are some papers that have benefited from this package.

References

M. Hoff and G. Staglianò, Explicit constructions of K3 surfaces and unirational Noether-Lefschetz divisors, available at arXiv:2110.15819 (2021).
G. Staglianò, Some new rational Gushel fourfolds, available at arXiv:2003.07809 (2020).
G. Staglianò, On some families of Gushel-Mukai fourfolds, available at arXiv:2002.07026 (2020).
M. Hoff and G. Staglianò, New examples of rational Gushel-Mukai fourfolds, available at arXiv:1910.12838 (2020).
F. Russo and G. Staglianò, Trisecant Flops, their associated K3 surfaces and the rationality of some Fano fourfolds, available at arXiv:1909.01263 (2020).
F. Russo and G. Staglianò, Explicit rationality of some cubic fourfolds, available at arXiv:1811.03502 (2019).
F. Russo and G. Staglianò, Congruences of 5-secant conics and the rationality of some admissible cubic fourfolds, available at arXiv:1707.00999 (2018).

Author

Giovanni Staglianò <[email protected]>

Version

This documentation describes version 2.7.1 of SpecialFanoFourfolds.

Source code

The source code from which this documentation is derived is in the file SpecialFanoFourfolds.m2.

Exports

Types
- CongruenceOfCurves -- the class of all congruences of secant curves to surfaces
- HodgeSpecialFourfold -- the class of all Hodge-special fourfolds
- IntersectionOfThreeQuadricsInP7 -- the class of all special intersection of three quadrics in P^7
- SpecialCubicFourfold -- the class of all special cubic fourfolds
- SpecialGushelMukaiFourfold -- the class of all special Gushel-Mukai fourfolds
Functions and commands
- ambientFivefold -- get the ambient fivefold of the Hodge-special fourfold
- associatedCastelnuovoSurface -- Castelnuovo surface associated to a rational complete intersection of three quadrics in P^7
- associatedK3surface -- K3 surface associated to a rational fourfold
- building -- see associatedK3surface -- K3 surface associated to a rational fourfold
- beauvilleMap -- construction of Beauville for complete intersections of three quadrics in P^7
- detectCongruence -- detect and return a congruence of secant curves to a surface
- fanoFourfold -- random prime Fano fourfold of coindex at most 3
- fromOrdinaryToGushel -- try to deform to a fourfold of Gushel type
- GMtables -- make examples of reducible subschemes of P^5
- isAdmissible -- whether an integer is admissible (in the sense of the theory of cubic fourfolds)
- isAdmissibleGM -- whether an integer is admissible (in the sense of the theory of GM fourfolds)
- mirrorFourfold -- associated fourfold to a rational cubic or GM fourfold
- normalSheaf -- normal sheaf
- parameterCount -- count of parameters
- parametrizeFanoFourfold -- rational parametrization of a prime Fano fourfold of coindex at most 3
- specialCubicFourfold -- make a special cubic fourfold
- specialFourfold -- make a Hodge-special fourfold
- specialGushelMukaiFourfold -- make a special Gushel-Mukai fourfold
- surface -- get the special surface contained in the fourfold
- toGrass -- Gushel morphism from a GM fourfold to GG(1,4)
- trisecantFlop -- examples of trisecant flops
- unirationalParametrization -- unirational parametrization
Methods
- ambientFivefold(HodgeSpecialFourfold) -- see ambientFivefold -- get the ambient fivefold of the Hodge-special fourfold
- associatedCastelnuovoSurface(IntersectionOfThreeQuadricsInP7) -- see associatedCastelnuovoSurface -- Castelnuovo surface associated to a rational complete intersection of three quadrics in P^7
- associatedK3surface(SpecialCubicFourfold) -- K3 surface associated to a rational cubic fourfold
- associatedK3surface(SpecialGushelMukaiFourfold) -- K3 surface associated to a rational Gushel-Mukai fourfold
- beauvilleMap(IntersectionOfThreeQuadricsInP7) -- see beauvilleMap -- construction of Beauville for complete intersections of three quadrics in P^7
- check(CongruenceOfCurves) -- see check(ZZ,CongruenceOfCurves) -- check that a congruence of curves is well-defined
- check(ZZ,CongruenceOfCurves) -- check that a congruence of curves is well-defined
- clean(HodgeSpecialFourfold) -- clean the internal information of a fourfold
- CongruenceOfCurves EmbeddedProjectiveVariety -- get the curve of a congruence passing through a point
- CongruenceOfCurves Ideal -- see CongruenceOfCurves EmbeddedProjectiveVariety -- get the curve of a congruence passing through a point
- detectCongruence(HodgeSpecialFourfold) -- see detectCongruence -- detect and return a congruence of secant curves to a surface
- detectCongruence(HodgeSpecialFourfold,ZZ) -- see detectCongruence -- detect and return a congruence of secant curves to a surface
- detectCongruence(SpecialCubicFourfold) -- see detectCongruence(SpecialCubicFourfold,ZZ) -- detect and return a congruence of (3e-1)-secant curves of degree e
- detectCongruence(SpecialCubicFourfold,ZZ) -- detect and return a congruence of (3e-1)-secant curves of degree e
- detectCongruence(SpecialGushelMukaiFourfold) -- see detectCongruence(SpecialGushelMukaiFourfold,ZZ) -- detect and return a congruence of (2e-1)-secant curves of degree e inside a del Pezzo fivefold
- detectCongruence(SpecialGushelMukaiFourfold,ZZ) -- detect and return a congruence of (2e-1)-secant curves of degree e inside a del Pezzo fivefold
- discriminant(HodgeSpecialFourfold) -- see discriminant(SpecialCubicFourfold) -- discriminant of a special cubic fourfold
- discriminant(SpecialCubicFourfold) -- discriminant of a special cubic fourfold
- discriminant(SpecialGushelMukaiFourfold) -- discriminant of a special Gushel-Mukai fourfold
- fanoFourfold(ZZ,ZZ) -- see fanoFourfold -- random prime Fano fourfold of coindex at most 3
- fromOrdinaryToGushel(SpecialGushelMukaiFourfold) -- see fromOrdinaryToGushel -- try to deform to a fourfold of Gushel type
- GMtables(ZZ) -- see GMtables -- make examples of reducible subschemes of P^5
- GMtables(ZZ,Ring) -- see GMtables -- make examples of reducible subschemes of P^5
- isAdmissible(SpecialCubicFourfold) -- see isAdmissible -- whether an integer is admissible (in the sense of the theory of cubic fourfolds)
- isAdmissible(ZZ) -- see isAdmissible -- whether an integer is admissible (in the sense of the theory of cubic fourfolds)
- isAdmissibleGM(SpecialGushelMukaiFourfold) -- see isAdmissibleGM -- whether an integer is admissible (in the sense of the theory of GM fourfolds)
- isAdmissibleGM(ZZ) -- see isAdmissibleGM -- whether an integer is admissible (in the sense of the theory of GM fourfolds)
- isMember(EmbeddedProjectiveVariety,CongruenceOfCurves) -- test membership in a congruence of curves
- map(CongruenceOfCurves) -- compute the parameter space of a congruence
- map(SpecialCubicFourfold) -- associated cubic map
- map(SpecialGushelMukaiFourfold) -- associated quadratic map
- mirrorFourfold(HodgeSpecialFourfold) -- see mirrorFourfold -- associated fourfold to a rational cubic or GM fourfold
- mirrorFourfold(IntersectionOfThreeQuadricsInP7) -- see mirrorFourfold -- associated fourfold to a rational cubic or GM fourfold
- mirrorFourfold(SpecialCubicFourfold) -- see mirrorFourfold -- associated fourfold to a rational cubic or GM fourfold
- mirrorFourfold(SpecialGushelMukaiFourfold) -- see mirrorFourfold -- associated fourfold to a rational cubic or GM fourfold
- normalSheaf(EmbeddedProjectiveVariety) -- see normalSheaf -- normal sheaf
- normalSheaf(EmbeddedProjectiveVariety,EmbeddedProjectiveVariety) -- see normalSheaf -- normal sheaf
- parameterCount(EmbeddedProjectiveVariety,EmbeddedProjectiveVariety) -- see parameterCount -- count of parameters
- parameterCount(HodgeSpecialFourfold) -- see parameterCount -- count of parameters
- parameterCount(SpecialCubicFourfold) -- count of parameters in the moduli space of GM fourfolds
- parameterCount(SpecialGushelMukaiFourfold) -- count of parameters in the moduli space of GM fourfolds
- parametrize(HodgeSpecialFourfold) -- rational parametrization
- parametrizeFanoFourfold(EmbeddedProjectiveVariety) -- see parametrizeFanoFourfold -- rational parametrization of a prime Fano fourfold of coindex at most 3
- specialCubicFourfold(EmbeddedProjectiveVariety,EmbeddedProjectiveVariety) -- see specialCubicFourfold -- make a special cubic fourfold
- specialCubicFourfold(Ideal,Ideal) -- see specialCubicFourfold -- make a special cubic fourfold
- specialCubicFourfold(Ideal,RingElement) -- see specialCubicFourfold -- make a special cubic fourfold
- specialCubicFourfold(EmbeddedProjectiveVariety) -- random special cubic fourfold
- specialCubicFourfold(Ideal) -- see specialCubicFourfold(EmbeddedProjectiveVariety) -- random special cubic fourfold
- specialCubicFourfold(String) -- see specialCubicFourfold(String,Ring) -- random special cubic fourfold of a given type
- specialCubicFourfold(String,Ring) -- random special cubic fourfold of a given type
- specialFourfold(EmbeddedProjectiveVariety) -- see specialFourfold -- make a Hodge-special fourfold
- specialFourfold(EmbeddedProjectiveVariety,EmbeddedProjectiveVariety) -- see specialFourfold -- make a Hodge-special fourfold
- specialFourfold(EmbeddedProjectiveVariety,EmbeddedProjectiveVariety,EmbeddedProjectiveVariety) -- see specialFourfold -- make a Hodge-special fourfold
- specialFourfold(Ideal) -- see specialFourfold -- make a Hodge-special fourfold
- specialFourfold(Ideal,Ideal) -- see specialFourfold -- make a Hodge-special fourfold
- specialFourfold(Ideal,RingElement) -- see specialFourfold -- make a Hodge-special fourfold
- specialFourfold(String) -- see specialFourfold -- make a Hodge-special fourfold
- specialFourfold(String,Ring) -- see specialFourfold -- make a Hodge-special fourfold
- specialFourfold(String,ZZ) -- load a prebuilt example of fourfold
- specialGushelMukaiFourfold(EmbeddedProjectiveVariety,EmbeddedProjectiveVariety) -- see specialGushelMukaiFourfold -- make a special Gushel-Mukai fourfold
- specialGushelMukaiFourfold(Ideal,Ideal) -- see specialGushelMukaiFourfold -- make a special Gushel-Mukai fourfold
- specialGushelMukaiFourfold(Array,Array) -- see specialGushelMukaiFourfold(Array,Array,String,Thing) -- construct GM fourfolds by gluing cubic or quartic scrolls to surfaces in PP^6
- specialGushelMukaiFourfold(Array,Array,String) -- see specialGushelMukaiFourfold(Array,Array,String,Thing) -- construct GM fourfolds by gluing cubic or quartic scrolls to surfaces in PP^6
- specialGushelMukaiFourfold(Array,Array,String,Thing) -- construct GM fourfolds by gluing cubic or quartic scrolls to surfaces in PP^6
- specialGushelMukaiFourfold(Array,Array,Thing) -- see specialGushelMukaiFourfold(Array,Array,String,Thing) -- construct GM fourfolds by gluing cubic or quartic scrolls to surfaces in PP^6
- specialGushelMukaiFourfold(EmbeddedProjectiveVariety) -- random special Gushel-Mukai fourfold
- specialGushelMukaiFourfold(Ideal) -- see specialGushelMukaiFourfold(EmbeddedProjectiveVariety) -- random special Gushel-Mukai fourfold
- specialGushelMukaiFourfold(String) -- see specialGushelMukaiFourfold(String,Ring) -- random special Gushel-Mukai fourfold of a given type
- specialGushelMukaiFourfold(String,Ring) -- random special Gushel-Mukai fourfold of a given type
- surface(HodgeSpecialFourfold) -- see surface -- get the special surface contained in the fourfold
- surface(List) -- get a rational surface
- surface(VisibleList,Option) -- see surface(List) -- get a rational surface
- surface(VisibleList,Option,Option) -- see surface(List) -- get a rational surface
- surface(VisibleList,Ring) -- see surface(List) -- get a rational surface
- surface(VisibleList,Ring,Option) -- see surface(List) -- get a rational surface
- surface(VisibleList,Ring,Option,Option) -- see surface(List) -- get a rational surface
- surface(MultiprojectiveVariety,MultiprojectiveVariety) -- make a Hodge-special surface
- toExternalString(HodgeSpecialFourfold) -- convert to a readable string
- toGrass(SpecialGushelMukaiFourfold) -- see toGrass -- Gushel morphism from a GM fourfold to GG(1,4)
- toGrass(EmbeddedProjectiveVariety) -- embedding of an ordinary Gushel-Mukai fourfold or a del Pezzo variety into GG(1,4)
- unirationalParametrization(HodgeSpecialFourfold) -- see unirationalParametrization -- unirational parametrization
- unirationalParametrization(SpecialCubicFourfold) -- see unirationalParametrization -- unirational parametrization
- unirationalParametrization(SpecialCubicFourfold,EmbeddedProjectiveVariety) -- see unirationalParametrization -- unirational parametrization
- unirationalParametrization(SpecialGushelMukaiFourfold) -- see unirationalParametrization -- unirational parametrization
Symbols
- Singular -- whether to transfer computation to Singular

For the programmer

The object SpecialFanoFourfolds is a package.