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]>

Certification

Version 2.7.1 of this package was accepted for publication in volume 14 of Journal of Software for Algebra and Geometry on 2024-04-14, in the article The SpecialFanoFourfolds package in Macaulay2 (DOI: 10.2140/jsag.2024.14.111). That version can be obtained from the journal or from the Macaulay2 source code repository.

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.