HyperplaneArrangements -- manipulating hyperplane arrangements

Description

A hyperplane arrangement is a finite set of hyperplanes in an affine or projective space. In this package, an arrangement is expressed as a list of (linear) defining equations for the hyperplanes. The tools provided allow the user to create new arrangements from old, and to compute various algebraic invariants of arrangements.

Introductions to the theory of hyperplane arrangements can be found in the following textbooks:

Alexandru Dimca, Hyperplane arrangements, Universitext,Springer, Cham, 2017. ISBN: 978-3-319-56221-6
Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der mathematischen Wissenschaften 300,Springer-Verlag, Berlin, 1992. ISBN: 978-3-662-02772-1
Richard P. Stanley, An introduction to hyperplane arrangements, in Geometric Combinatorics, 389-496, IAS/Park City Mathematics Series 13, American Mathematical Society, Providence, RI, 2007. ISBN: 978-1-4704-3912-5

Authors

Version

This documentation describes version 2.0 of HyperplaneArrangements.

Source code

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

Exports

Types
- Arrangement -- the class of all hyperplane arrangements
- CentralArrangement -- the class of all central hyperplane arrangements
- Flat -- intersection of hyperplanes
Functions and commands
- arrangement -- see arrangement(List,Ring) -- make a hyperplane arrangement
- arrangementSum -- see arrangementSum(Arrangement,Arrangement) -- make the direct sum of two arrangements
- deCone -- see deCone(CentralArrangement,RingElement) -- produce an affine arrangement from a central one
- der -- see der(CentralArrangement,List) -- compute the module of logarithmic derivations
- EPY -- see EPY(Arrangement,PolynomialRing) -- compute the Eisenbud-Popescu-Yuzvinsky module of an arrangement
- eulerRestriction -- see eulerRestriction(CentralArrangement,List,ZZ) -- form the Euler restriction of a central multiarrangement
- flat -- see flat(Arrangement,List) -- make a flat from a list of indices
- genericArrangement -- see genericArrangement(ZZ,ZZ,Ring) -- realize the uniform matroid using points on the monomial curve
- graphic -- see graphic(List,List,PolynomialRing) -- make a graphic arrangement
- isCentral -- see isCentral(Arrangement) -- test to see if a hyperplane arrangement is central
- isDecomposable -- see isDecomposable(CentralArrangement,Ring) -- whether a hyperplane arrangement decomposable in the sense of Papadima-Suciu
- lct -- see logCanonicalThreshold(CentralArrangement) -- compute the log-canonical threshold of an arrangement
- logCanonicalThreshold -- see logCanonicalThreshold(CentralArrangement) -- compute the log-canonical threshold of an arrangement
- makeEssential -- see makeEssential(CentralArrangement) -- make an essential arrangement out of an arbitrary one
- meet -- see meet(Flat,Flat) -- compute the meet operation in the intersection lattice
- multIdeal -- see multiplierIdeal(QQ,CentralArrangement,List) -- compute a multiplier ideal
- multiplierIdeal -- see multiplierIdeal(QQ,CentralArrangement,List) -- compute a multiplier ideal
- orlikSolomon -- see orlikSolomon(Arrangement,PolynomialRing) -- compute the defining ideal for the Orlik-Solomon algebra
- orlikTerao -- see orlikTerao(CentralArrangement,PolynomialRing) -- compute the defining ideal for the Orlik-Terao algebra
- randomArrangement -- see randomArrangement(ZZ,PolynomialRing,ZZ) -- generate an arrangement at random
- subArrangement -- see subArrangement(Arrangement,Flat) -- create the hyperplane arrangement containing a flat
- typeA -- see typeA(ZZ,Ring) -- make the hyperplane arrangement defined by a type $A_n$ root system
- typeB -- see typeB(ZZ,Ring) -- make the hyperplane arrangement defined by a type $B_n$ root system
- typeD -- see typeD(ZZ,Ring) -- make the hyperplane arrangement defined by a type $D_n$ root system
- vee -- see vee(Flat,Flat) -- compute the vee operation in the intersection lattice
Methods
- Arrangement ** Ring -- change the coefficient ring of an arrangement
- Arrangement == Arrangement -- whether two hyperplane arrangements are equal
- arrangement(Flat) -- get the hyperplane arrangement to which a flat belongs
- arrangement(List) -- see arrangement(List,Ring) -- make a hyperplane arrangement
- arrangement(List,Ring) -- make a hyperplane arrangement
- arrangement(RingElement) -- see arrangement(List,Ring) -- make a hyperplane arrangement
- arrangement(Matrix) -- see arrangement(Matrix,Ring) -- make a hyperplane arrangement
- arrangement(Matrix,Ring) -- make a hyperplane arrangement
- arrangement(String) -- see arrangement(String,Ring) -- access a database of classic hyperplane arrangements
- arrangement(String,PolynomialRing) -- see arrangement(String,Ring) -- access a database of classic hyperplane arrangements
- arrangement(String,Ring) -- access a database of classic hyperplane arrangements
- Arrangement ++ Arrangement -- see arrangementSum(Arrangement,Arrangement) -- make the direct sum of two arrangements
- arrangementSum(Arrangement,Arrangement) -- make the direct sum of two arrangements
- circuits(CentralArrangement) -- list the circuits of an arrangement
- closure(Arrangement,Ideal) -- see closure(Arrangement,List) -- closure operation in the intersection lattice
- closure(Arrangement,List) -- closure operation in the intersection lattice
- coefficients(Arrangement) -- make a matrix from the coefficients of the defining equations
- compress(Arrangement) -- extract nonzero equations
- cone(Arrangement,RingElement) -- creates an associated central hyperplane arrangement
- cone(Arrangement,Symbol) -- see cone(Arrangement,RingElement) -- creates an associated central hyperplane arrangement
- deCone(CentralArrangement,RingElement) -- produce an affine arrangement from a central one
- deCone(CentralArrangement,ZZ) -- see deCone(CentralArrangement,RingElement) -- produce an affine arrangement from a central one
- deletion(Arrangement,List) -- see deletion(Arrangement,RingElement) -- deletion of a subset of an arrangement
- deletion(Arrangement,RingElement) -- deletion of a subset of an arrangement
- deletion(Arrangement,Set) -- see deletion(Arrangement,RingElement) -- deletion of a subset of an arrangement
- deletion(Arrangement,ZZ) -- see deletion(Arrangement,RingElement) -- deletion of a subset of an arrangement
- der(CentralArrangement) -- see der(CentralArrangement,List) -- compute the module of logarithmic derivations
- der(CentralArrangement,List) -- compute the module of logarithmic derivations
- dual(CentralArrangement) -- see dual(CentralArrangement,Ring) -- the Gale dual of an arrangement
- dual(CentralArrangement,Ring) -- the Gale dual of an arrangement
- EPY(Arrangement) -- see EPY(Arrangement,PolynomialRing) -- compute the Eisenbud-Popescu-Yuzvinsky module of an arrangement
- EPY(Arrangement,PolynomialRing) -- compute the Eisenbud-Popescu-Yuzvinsky module of an arrangement
- EPY(Ideal) -- see EPY(Arrangement,PolynomialRing) -- compute the Eisenbud-Popescu-Yuzvinsky module of an arrangement
- EPY(Ideal,PolynomialRing) -- see EPY(Arrangement,PolynomialRing) -- compute the Eisenbud-Popescu-Yuzvinsky module of an arrangement
- euler(CentralArrangement) -- compute the Euler characteristic of the projective complement
- euler(Flat) -- see euler(CentralArrangement) -- compute the Euler characteristic of the projective complement
- eulerRestriction(CentralArrangement,List,ZZ) -- form the Euler restriction of a central multiarrangement
- Flat == Flat -- whether two flats are equal
- flat(Arrangement,List) -- make a flat from a list of indices
- flats(Arrangement) -- see flats(ZZ,Arrangement) -- list the flats of an arrangement of a given rank
- flats(ZZ,Arrangement) -- list the flats of an arrangement of a given rank
- flats(ZZ,CentralArrangement) -- see flats(ZZ,Arrangement) -- list the flats of an arrangement of a given rank
- genericArrangement(ZZ,ZZ) -- see genericArrangement(ZZ,ZZ,Ring) -- realize the uniform matroid using points on the monomial curve
- genericArrangement(ZZ,ZZ,Ring) -- realize the uniform matroid using points on the monomial curve
- graphic(List) -- see graphic(List,List,PolynomialRing) -- make a graphic arrangement
- graphic(List,List) -- see graphic(List,List,PolynomialRing) -- make a graphic arrangement
- graphic(List,List,PolynomialRing) -- make a graphic arrangement
- graphic(List,List,Ring) -- see graphic(List,List,PolynomialRing) -- make a graphic arrangement
- graphic(List,PolynomialRing) -- see graphic(List,List,PolynomialRing) -- make a graphic arrangement
- graphic(List,Ring) -- see graphic(List,List,PolynomialRing) -- make a graphic arrangement
- hyperplanes(Arrangement) -- the defining linear forms of an arrangement
- toList(Arrangement) -- see hyperplanes(Arrangement) -- the defining linear forms of an arrangement
- isCentral(Arrangement) -- test to see if a hyperplane arrangement is central
- isDecomposable(CentralArrangement) -- see isDecomposable(CentralArrangement,Ring) -- whether a hyperplane arrangement decomposable in the sense of Papadima-Suciu
- isDecomposable(CentralArrangement,Ring) -- whether a hyperplane arrangement decomposable in the sense of Papadima-Suciu
- lct(CentralArrangement) -- see logCanonicalThreshold(CentralArrangement) -- compute the log-canonical threshold of an arrangement
- logCanonicalThreshold(CentralArrangement) -- compute the log-canonical threshold of an arrangement
- makeEssential(CentralArrangement) -- make an essential arrangement out of an arbitrary one
- matrix(Arrangement) -- make a matrix from the defining equations
- matroid(CentralArrangement) -- get the matroid of a central arrangement
- Flat ^ Flat -- see meet(Flat,Flat) -- compute the meet operation in the intersection lattice
- meet(Flat,Flat) -- compute the meet operation in the intersection lattice
- multIdeal(QQ,CentralArrangement) -- see multiplierIdeal(QQ,CentralArrangement,List) -- compute a multiplier ideal
- multIdeal(QQ,CentralArrangement,List) -- see multiplierIdeal(QQ,CentralArrangement,List) -- compute a multiplier ideal
- multIdeal(ZZ,CentralArrangement) -- see multiplierIdeal(QQ,CentralArrangement,List) -- compute a multiplier ideal
- multIdeal(ZZ,CentralArrangement,List) -- see multiplierIdeal(QQ,CentralArrangement,List) -- compute a multiplier ideal
- multiplierIdeal(QQ,CentralArrangement) -- see multiplierIdeal(QQ,CentralArrangement,List) -- compute a multiplier ideal
- multiplierIdeal(QQ,CentralArrangement,List) -- compute a multiplier ideal
- multiplierIdeal(ZZ,CentralArrangement) -- see multiplierIdeal(QQ,CentralArrangement,List) -- compute a multiplier ideal
- multiplierIdeal(ZZ,CentralArrangement,List) -- see multiplierIdeal(QQ,CentralArrangement,List) -- compute a multiplier ideal
- orlikSolomon(Arrangement) -- see orlikSolomon(Arrangement,PolynomialRing) -- compute the defining ideal for the Orlik-Solomon algebra
- orlikSolomon(Arrangement,PolynomialRing) -- compute the defining ideal for the Orlik-Solomon algebra
- orlikSolomon(Arrangement,Ring) -- see orlikSolomon(Arrangement,PolynomialRing) -- compute the defining ideal for the Orlik-Solomon algebra
- orlikSolomon(Arrangement,Symbol) -- see orlikSolomon(Arrangement,PolynomialRing) -- compute the defining ideal for the Orlik-Solomon algebra
- orlikSolomon(CentralArrangement,PolynomialRing) -- see orlikSolomon(Arrangement,PolynomialRing) -- compute the defining ideal for the Orlik-Solomon algebra
- orlikTerao(CentralArrangement) -- see orlikTerao(CentralArrangement,PolynomialRing) -- compute the defining ideal for the Orlik-Terao algebra
- orlikTerao(CentralArrangement,PolynomialRing) -- compute the defining ideal for the Orlik-Terao algebra
- orlikTerao(CentralArrangement,Symbol) -- see orlikTerao(CentralArrangement,PolynomialRing) -- compute the defining ideal for the Orlik-Terao algebra
- poincare(Arrangement) -- compute the Poincaré polynomial of an arrangement
- poincare(CentralArrangement) -- see poincare(Arrangement) -- compute the Poincaré polynomial of an arrangement
- prune(Arrangement) -- makes a new hyperplane arrangement in a polynomial ring
- randomArrangement(ZZ,PolynomialRing,ZZ) -- generate an arrangement at random
- randomArrangement(ZZ,ZZ,ZZ) -- see randomArrangement(ZZ,PolynomialRing,ZZ) -- generate an arrangement at random
- rank(CentralArrangement) -- compute the rank of a central hyperplane arrangement
- rank(Flat) -- compute the rank of a flat
- Arrangement ^ Flat -- see restriction(Arrangement,Ideal) -- construct the restriction a hyperplane arrangement to a subspace
- restriction(Arrangement,Flat) -- see restriction(Arrangement,Ideal) -- construct the restriction a hyperplane arrangement to a subspace
- restriction(Arrangement,Ideal) -- construct the restriction a hyperplane arrangement to a subspace
- restriction(Arrangement,List) -- see restriction(Arrangement,Ideal) -- construct the restriction a hyperplane arrangement to a subspace
- restriction(Arrangement,RingElement) -- see restriction(Arrangement,Ideal) -- construct the restriction a hyperplane arrangement to a subspace
- restriction(Arrangement,Set) -- see restriction(Arrangement,Ideal) -- construct the restriction a hyperplane arrangement to a subspace
- restriction(Arrangement,ZZ) -- see restriction(Arrangement,Ideal) -- construct the restriction a hyperplane arrangement to a subspace
- restriction(Flat) -- see restriction(Arrangement,Ideal) -- construct the restriction a hyperplane arrangement to a subspace
- ring(Arrangement) -- get the underlying ring of a hyperplane arrangement
- Arrangement _ Flat -- see subArrangement(Arrangement,Flat) -- create the hyperplane arrangement containing a flat
- subArrangement(Arrangement,Flat) -- create the hyperplane arrangement containing a flat
- subArrangement(Flat) -- see subArrangement(Arrangement,Flat) -- create the hyperplane arrangement containing a flat
- Arrangement ** RingMap -- see substitute(Arrangement,RingMap) -- change the ring of an arrangement
- substitute(Arrangement,Ring) -- see substitute(Arrangement,RingMap) -- change the ring of an arrangement
- substitute(Arrangement,RingMap) -- change the ring of an arrangement
- toList(Flat) -- the indices of a flat
- trim(Arrangement) -- make a simple hyperplane arrangement
- typeA(ZZ) -- see typeA(ZZ,Ring) -- make the hyperplane arrangement defined by a type $A_n$ root system
- typeA(ZZ,PolynomialRing) -- see typeA(ZZ,Ring) -- make the hyperplane arrangement defined by a type $A_n$ root system
- typeA(ZZ,Ring) -- make the hyperplane arrangement defined by a type $A_n$ root system
- typeB(ZZ) -- see typeB(ZZ,Ring) -- make the hyperplane arrangement defined by a type $B_n$ root system
- typeB(ZZ,PolynomialRing) -- see typeB(ZZ,Ring) -- make the hyperplane arrangement defined by a type $B_n$ root system
- typeB(ZZ,Ring) -- make the hyperplane arrangement defined by a type $B_n$ root system
- typeD(ZZ) -- see typeD(ZZ,Ring) -- make the hyperplane arrangement defined by a type $D_n$ root system
- typeD(ZZ,PolynomialRing) -- see typeD(ZZ,Ring) -- make the hyperplane arrangement defined by a type $D_n$ root system
- typeD(ZZ,Ring) -- make the hyperplane arrangement defined by a type $D_n$ root system
- Flat | Flat -- see vee(Flat,Flat) -- compute the vee operation in the intersection lattice
- vee(Flat,Flat) -- compute the vee operation in the intersection lattice
Symbols
- Popescu -- see orlikSolomon(Arrangement,PolynomialRing) -- compute the defining ideal for the Orlik-Solomon algebra
Other things
- arrangementLibrary -- access a database of classic hyperplane arrangements

For the programmer

The object HyperplaneArrangements is a package.