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.
Version History
- v. 1.0 (Oct 2019) - Initial version.
- v. 2.7.1 (May 2023) - Consolidated stable version.
- v. 2.8 (Apr 2026) - Updates and new features.
The following papers have made use of this package.
References
- F. Russo and G. Staglianò, On complete intersections of three quadrics in P^7, available at arXiv:2312.01773 (2023).
- 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 <giovanni.stagliano@unict.it>
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.
Version
This documentation describes version 2.8 of SpecialFanoFourfolds, released Apr 13, 2026.
Citation
If you have used this package in your research, please cite it as follows:
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Exports
- Types
- CongruenceOfCurves -- the class of all congruences of secant curves to surfaces
- CubicFourfold -- the class of all special cubic fourfolds
- DoublySpecialCubicFourfold -- the class of all cubic fourfolds belonging to the intersection of two Hassett divisors
- GushelMukaiFourfold -- the class of all special Gushel-Mukai fourfolds
- HodgeSpecialFourfold -- the class of all Hodge-special fourfolds
- IntersectionOfThreeQuadricsInP7 -- the class of all special intersection of three quadrics in P^7
- 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
- cubicFourfold -- make a special cubic fourfold
- 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
- gushelMukaiFourfold -- make a special Gushel-Mukai fourfold
- 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
- latticePolarization -- see polarizedK3surface -- polarized K3 surface associated to a doubly special cubic fourfold
- polarizedK3surface -- polarized K3 surface associated to a doubly special cubic fourfold
- specialFourfold -- make a Hodge-special fourfold
- surface -- get the special surface contained in the fourfold
- surfaces -- get the special surfaces contained in the cubic fourfold
- swap -- swap the order of the two special surfaces
- 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(CubicFourfold) -- K3 surface associated to a rational cubic fourfold
- associatedK3surface(GushelMukaiFourfold) -- 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
- cubicFourfold(EmbeddedProjectiveVariety,EmbeddedProjectiveVariety) -- see cubicFourfold -- make a special cubic fourfold
- cubicFourfold(Ideal,Ideal) -- see cubicFourfold -- make a special cubic fourfold
- cubicFourfold(Ideal,RingElement) -- see cubicFourfold -- make a special cubic fourfold
- cubicFourfold(EmbeddedProjectiveVariety) -- random special cubic fourfold
- cubicFourfold(Ideal) -- see cubicFourfold(EmbeddedProjectiveVariety) -- random special cubic fourfold
- cubicFourfold(String) -- see cubicFourfold(String,Ring) -- random special cubic fourfold of a given type
- cubicFourfold(String,Ring) -- random special cubic fourfold of a given type
- describe(IntersectionOfThreeQuadricsInP7) (missing documentation)
- 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(CubicFourfold) -- see detectCongruence(CubicFourfold,ZZ) -- detect and return a congruence of (3e-1)-secant curves of degree e
- detectCongruence(CubicFourfold,ZZ) -- detect and return a congruence of (3e-1)-secant curves of degree e
- detectCongruence(GushelMukaiFourfold) -- see detectCongruence(GushelMukaiFourfold,ZZ) -- detect and return a congruence of (2e-1)-secant curves of degree e inside a del Pezzo fivefold
- detectCongruence(GushelMukaiFourfold,ZZ) -- detect and return a congruence of (2e-1)-secant curves of degree e inside a del Pezzo fivefold
- discriminant(CubicFourfold) -- discriminant of a special cubic fourfold
- discriminant(HodgeSpecialFourfold) -- see discriminant(CubicFourfold) -- discriminant of a special cubic fourfold
- discriminant(GushelMukaiFourfold) -- discriminant of a special Gushel-Mukai fourfold
- EmbeddedProjectiveVariety & EmbeddedProjectiveVariety -- see DoublySpecialCubicFourfold -- the class of all cubic fourfolds belonging to the intersection of two Hassett divisors
- fanoFourfold(ZZ,ZZ) -- see fanoFourfold -- random prime Fano fourfold of coindex at most 3
- fromOrdinaryToGushel(GushelMukaiFourfold) -- 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
- gushelMukaiFourfold(EmbeddedProjectiveVariety,EmbeddedProjectiveVariety) -- see gushelMukaiFourfold -- make a special Gushel-Mukai fourfold
- gushelMukaiFourfold(Ideal,Ideal) -- see gushelMukaiFourfold -- make a special Gushel-Mukai fourfold
- gushelMukaiFourfold(Array,Array) -- see gushelMukaiFourfold(Array,Array,String,Thing) -- construct GM fourfolds by gluing cubic or quartic scrolls to surfaces in PP^6
- gushelMukaiFourfold(Array,Array,String) -- see gushelMukaiFourfold(Array,Array,String,Thing) -- construct GM fourfolds by gluing cubic or quartic scrolls to surfaces in PP^6
- gushelMukaiFourfold(Array,Array,String,Thing) -- construct GM fourfolds by gluing cubic or quartic scrolls to surfaces in PP^6
- gushelMukaiFourfold(Array,Array,Thing) -- see gushelMukaiFourfold(Array,Array,String,Thing) -- construct GM fourfolds by gluing cubic or quartic scrolls to surfaces in PP^6
- gushelMukaiFourfold(EmbeddedProjectiveVariety) -- random special Gushel-Mukai fourfold
- gushelMukaiFourfold(Ideal) -- see gushelMukaiFourfold(EmbeddedProjectiveVariety) -- random special Gushel-Mukai fourfold
- gushelMukaiFourfold(String) -- see gushelMukaiFourfold(String,Ring) -- random special Gushel-Mukai fourfold of a given type
- gushelMukaiFourfold(String,Ring) -- random special Gushel-Mukai fourfold of a given type
- isAdmissible(CubicFourfold) -- 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(GushelMukaiFourfold) -- 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(CubicFourfold) -- associated cubic map
- map(GushelMukaiFourfold) -- associated quadratic map
- mirrorFourfold(CubicFourfold) -- see mirrorFourfold -- associated fourfold to a rational cubic or GM fourfold
- mirrorFourfold(GushelMukaiFourfold) -- see mirrorFourfold -- associated fourfold to a rational cubic or GM fourfold
- 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
- 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(CubicFourfold) -- count of parameters in the moduli space of GM fourfolds
- parameterCount(GushelMukaiFourfold) -- 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
- polarizedK3surface(DoublySpecialCubicFourfold) -- see polarizedK3surface -- polarized K3 surface associated to a doubly special cubic fourfold
- random(CubicFourfold) (missing documentation)
- random(GushelMukaiFourfold) (missing documentation)
- 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
- 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
- surfaces(DoublySpecialCubicFourfold) -- see surfaces -- get the special surfaces contained in the cubic fourfold
- swap(DoublySpecialCubicFourfold) -- see swap -- swap the order of the two special surfaces
- toExternalString(HodgeSpecialFourfold) -- convert to a readable string
- toGrass(GushelMukaiFourfold) -- 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(CubicFourfold) -- see unirationalParametrization -- unirational parametrization
- unirationalParametrization(CubicFourfold,EmbeddedProjectiveVariety) -- see unirationalParametrization -- unirational parametrization
- unirationalParametrization(GushelMukaiFourfold) -- see unirationalParametrization -- unirational parametrization
- unirationalParametrization(HodgeSpecialFourfold) -- see unirationalParametrization -- unirational parametrization
- Symbols
- FanoMapType -- see polarizedK3surface -- polarized K3 surface associated to a doubly special cubic fourfold
- Singular -- whether to transfer computation to Singular
For the programmer
The object SpecialFanoFourfolds is a package, defined in SpecialFanoFourfolds.m2, with auxiliary files in SpecialFanoFourfolds/.
The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SpecialFanoFourfolds/docs.m2:26:0.