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WeierstrassSemigroups -- Compute smoothing families for Weierstrass semigroups

Description

This package contains methods to compute one parameter, positively graded smoothing families over QQ for certain numerical semigroup rings. By Pinkham's Theorem a semigroup is a Weierstrass semigroup if and only if the semigroup ring has a graded smoothing in negative T^1 directions.

Deformations

Finding points

Smoothness

Collecting

Checking flatness and smoothness of a database of families

non-Weierstrass semigroups

References

Pinkham, Henry C., Deformations of algebraic varieties with G_m action, Ast{'e}risque textbf{20} (1974), pp 1 - 131, Soci{'e}t{'e} Math{'e}matique de France (SMF), Paris

SeeAlso

Authors

Version

This documentation describes version 1.0 of WeierstrassSemigroups, released June 4, 2026.

Citation

If you have used this package in your research, please cite it as follows:

@misc{WeierstrassSemigroupsSource,
  title = {{WeierstrassSemigroups: Compute smoothing families for Weierstrass semigroups. Version~1.0}},
  author = {David Eisenbud and Frank-Olaf Schreyer},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
}

Exports

  • Functions and commands
    • appendFamily -- Append a one-parameter smoothing family to a data file
    • checkFlatnessOfOneParameterFamilies -- Check flatness of 1-parameter famiiles
    • checkSmoothnessOfOneParameterFamilies -- Check smoothness of 1-parameter families
    • clearDenominators -- Clear denominators
    • collectByBound -- Collect 1-parameter families filtered by bound (or congruences or range)
    • collectByCongruences -- see collectByBound -- Collect 1-parameter families filtered by bound (or congruences or range)
    • collectByRange -- see collectByBound -- Collect 1-parameter families filtered by bound (or congruences or range)
    • collectWithVersalDeformations -- Collect 1-parameter families the using versal deformations
    • displaySyzygyMatrices -- Display the syzygy matrices
    • findCompleteIntersection -- Find complete intersection defined by some of the generators
    • findPoint -- Find a kk-rational point in a variety
    • flatteningRelations -- Compute the flattening relations of an unfolding
    • getFlatFamily -- Compute the flat family depending on a subset of parameters of the universal unfolding
    • prepareInitialPositionList -- see getFlatFamily -- Compute the flat family depending on a subset of parameters of the universal unfolding
    • getFromDisk -- Read a file from the hard disk
    • getListOfIdeals -- Read a list of ideals from a dataBase
    • getOneParameterFamily -- Compute a one parameter smoothing family
    • getParameterFamily -- Compute the parametric family which uses the same terms as J
    • getRangeOfOneParameterFamily -- Compute the range of degrees of a one parameter family
    • getSmoothingFamily -- Get a smoothing family for the semigroup L
    • getSmoothingFamilyWithVersalDeformation -- Get a smoothing family using versal deformations
    • give1683Format -- Does the semigroup ideal of L has a resolution with total betti numbers 1,6,8,3?
    • depthCondition1 -- see hilbertBurchMatrices -- Check the depth conditions for the exactness of the 1,4,4,1 subcomplex of the 1,6,8,3 subcomplex
    • hasExactSubcomplex -- see hilbertBurchMatrices -- Check the depth conditions for the exactness of the 1,4,4,1 subcomplex of the 1,6,8,3 subcomplex
    • hilbertBurchConditions -- see hilbertBurchMatrices -- Check the depth conditions for the exactness of the 1,4,4,1 subcomplex of the 1,6,8,3 subcomplex
    • hilbertBurchMatrices -- Check the depth conditions for the exactness of the 1,4,4,1 subcomplex of the 1,6,8,3 subcomplex
    • improveFamily -- Find a 1-parameter smoothing family with perhaps smaller number of terms and coefficients
    • isSmoothingFamily -- Is the family a smoothing family?
    • makeRange -- Make a range of degrees for getSmoothingFamily
    • makeUnfolding -- Makes the universal homogeneous unfolding of an ideal with positive degree parameters
    • pruneFamily -- Present the family with fewest number of variables
    • restrictedUnfolding -- Compute a restricted unfolding
    • satisfiesDegreeCondition1 -- Does the semigroup ideal of L satisfies the degree condition for the 3rd respectively 2nd syzgy matrix?
    • satisfiesDegreeCondition2 -- see satisfiesDegreeCondition1 -- Does the semigroup ideal of L satisfies the degree condition for the 3rd respectively 2nd syzgy matrix?
    • smoothnessWithReductions -- Check smoothness by using reductions to points
    • solvingFlatteningRelations -- Solving the flatttening relations over QQ
    • testBound -- Test whether b is bound for the semigroup L and compute a 1-parameter smoothing family if yes
    • testCongruences -- see testBound -- Test whether b is bound for the semigroup L and compute a 1-parameter smoothing family if yes
    • testRange -- see testBound -- Test whether b is bound for the semigroup L and compute a 1-parameter smoothing family if yes
    • toDoList -- Make a list of semigroups not previously known to be Weierstrass
  • Methods
    • appendFamily(List,Ideal,String,String) -- see appendFamily -- Append a one-parameter smoothing family to a data file
    • checkFlatnessOfOneParameterFamilies(List,String) -- see checkFlatnessOfOneParameterFamilies -- Check flatness of 1-parameter famiiles
    • checkSmoothnessOfOneParameterFamilies(List,String) -- see checkSmoothnessOfOneParameterFamilies -- Check smoothness of 1-parameter families
    • clearDenominators(Matrix) -- see clearDenominators -- Clear denominators
    • collectByBound(List,ZZ,String,String) -- see collectByBound -- Collect 1-parameter families filtered by bound (or congruences or range)
    • collectByCongruences(List,List,String,String) -- see collectByBound -- Collect 1-parameter families filtered by bound (or congruences or range)
    • collectByRange(List,List,String,String) -- see collectByBound -- Collect 1-parameter families filtered by bound (or congruences or range)
    • collectWithVersalDeformations(List,ZZ,String,String) -- see collectWithVersalDeformations -- Collect 1-parameter families the using versal deformations
    • displaySyzygyMatrices(List) -- see displaySyzygyMatrices -- Display the syzygy matrices
    • findCompleteIntersection(Ideal) -- see findCompleteIntersection -- Find complete intersection defined by some of the generators
    • findPoint(Ideal) -- see findPoint -- Find a kk-rational point in a variety
    • flatteningRelations(Ideal,Ring,Matrix) -- see flatteningRelations -- Compute the flattening relations of an unfolding
    • getFlatFamily(Ideal,Ring,Matrix) -- see getFlatFamily -- Compute the flat family depending on a subset of parameters of the universal unfolding
    • getFlatFamily(Ideal,Ring,Matrix,List) -- see getFlatFamily -- Compute the flat family depending on a subset of parameters of the universal unfolding
    • prepareInitialPositionList(List,ZZ) -- see getFlatFamily -- Compute the flat family depending on a subset of parameters of the universal unfolding
    • getFromDisk(String) -- see getFromDisk -- Read a file from the hard disk
    • getListOfIdeals(List,String) -- see getListOfIdeals -- Read a list of ideals from a dataBase
    • getOneParameterFamily(Ideal,Ideal,Matrix,ZZ) -- see getOneParameterFamily -- Compute a one parameter smoothing family
    • getParameterFamily(Ideal) -- see getParameterFamily -- Compute the parametric family which uses the same terms as J
    • getRangeOfOneParameterFamily(Ideal) -- see getRangeOfOneParameterFamily -- Compute the range of degrees of a one parameter family
    • getSmoothingFamily(List,List) -- see getSmoothingFamily -- Get a smoothing family for the semigroup L
    • getSmoothingFamily(List,ZZ) -- see getSmoothingFamily -- Get a smoothing family for the semigroup L
    • getSmoothingFamilyWithVersalDeformation(List) -- see getSmoothingFamilyWithVersalDeformation -- Get a smoothing family using versal deformations
    • give1683Format(List) -- see give1683Format -- Does the semigroup ideal of L has a resolution with total betti numbers 1,6,8,3?
    • give1683Format(List,List,List,List) -- see give1683Format -- Does the semigroup ideal of L has a resolution with total betti numbers 1,6,8,3?
    • give1683Format(ZZ,ZZ,ZZ,ZZ) -- see give1683Format -- Does the semigroup ideal of L has a resolution with total betti numbers 1,6,8,3?
    • depthCondition1(List) -- see hilbertBurchMatrices -- Check the depth conditions for the exactness of the 1,4,4,1 subcomplex of the 1,6,8,3 subcomplex
    • hasExactSubcomplex(List) -- see hilbertBurchMatrices -- Check the depth conditions for the exactness of the 1,4,4,1 subcomplex of the 1,6,8,3 subcomplex
    • hilbertBurchConditions(List) -- see hilbertBurchMatrices -- Check the depth conditions for the exactness of the 1,4,4,1 subcomplex of the 1,6,8,3 subcomplex
    • hilbertBurchMatrices(List) -- see hilbertBurchMatrices -- Check the depth conditions for the exactness of the 1,4,4,1 subcomplex of the 1,6,8,3 subcomplex
    • improveFamily(Ideal) -- see improveFamily -- Find a 1-parameter smoothing family with perhaps smaller number of terms and coefficients
    • isSmoothingFamily(List,Ideal,Matrix,Ideal) -- see isSmoothingFamily -- Is the family a smoothing family?
    • makeRange(List,List) -- see makeRange -- Make a range of degrees for getSmoothingFamily
    • makeUnfolding(Ideal) -- see makeUnfolding -- Makes the universal homogeneous unfolding of an ideal with positive degree parameters
    • makeUnfolding(List) -- see makeUnfolding -- Makes the universal homogeneous unfolding of an ideal with positive degree parameters
    • pruneFamily(Ideal,Ideal,Matrix) -- see pruneFamily -- Present the family with fewest number of variables
    • restrictedUnfolding(Ideal,List) -- see restrictedUnfolding -- Compute a restricted unfolding
    • satisfiesDegreeCondition1(List) -- see satisfiesDegreeCondition1 -- Does the semigroup ideal of L satisfies the degree condition for the 3rd respectively 2nd syzgy matrix?
    • satisfiesDegreeCondition2(List) -- see satisfiesDegreeCondition1 -- Does the semigroup ideal of L satisfies the degree condition for the 3rd respectively 2nd syzgy matrix?
    • smoothnessWithReductions(Ideal) -- see smoothnessWithReductions -- Check smoothness by using reductions to points
    • solvingFlatteningRelations(Ideal,Matrix,Ideal) -- see solvingFlatteningRelations -- Solving the flatttening relations over QQ
    • testBound(List,ZZ) -- see testBound -- Test whether b is bound for the semigroup L and compute a 1-parameter smoothing family if yes
    • testCongruences(List,List) -- see testBound -- Test whether b is bound for the semigroup L and compute a 1-parameter smoothing family if yes
    • testRange(List,List) -- see testBound -- Test whether b is bound for the semigroup L and compute a 1-parameter smoothing family if yes
    • toDoList(ZZ) -- see toDoList -- Make a list of semigroups not previously known to be Weierstrass
    • toDoList(ZZ,ZZ) -- see toDoList -- Make a list of semigroups not previously known to be Weierstrass
  • Symbols

For the programmer

The object WeierstrassSemigroups is a package, defined in WeierstrassSemigroups.m2.

The source of this document is in WeierstrassSemigroups.m2:1382:0.