Classification of Numerical Semigroups
Andrew Ferdowsian, Jian Zhou
Advised by: Maksym Fedorchuk
Supported by: BC URF program
Boston College
September 16, 2015
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Numerical Semigroup
A numerical semigroup is a subset Γ ⊂ Z≥0 such that
Γ contains 0,
Γ is closed under addition,
the complement of Γ in Z≥0 is finite, denoted as gap
sequence, and its elements are called gaps.
We can regard Γ ⊂ Z≥0 as a group under addition, but without
the property of inverses.
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Numerical Semigroup
A numerical semigroup is a subset Γ ⊂ Z≥0 such that
Γ contains 0,
Γ is closed under addition,
the complement of Γ in Z≥0 is finite, denoted as gap
sequence, and its elements are called gaps.
We can regard Γ ⊂ Z≥0 as a group under addition, but without
the property of inverses.
Example 1:
Γ = {0, 4, 5, 8, 9, 10, 12, 13, 14, . . . }.
Example 1:
Z≥0 \ Γ = {1, 2, 3, 6, 7, 11}.
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Numerical Semigroup
A numerical semigroup is a subset Γ ⊂ Z≥0 such that
Γ contains 0,
Γ is closed under addition,
the complement of Γ in Z≥0 is finite, denoted as gap
sequence, and its elements are called gaps.
We can regard Γ ⊂ Z≥0 as a group under addition, but without
the property of inverses.
Example 1:
Γ = {0, 4, 5, 8, 9, 10, 12, 13, 14, . . . }.
Example 1:
Z≥0 \ Γ = {1, 2, 3, 6, 7, 11}.
Example 2:
Γ = {0, 4, 5, 7, 8, 9, . . . }.
Example 2:
Z≥0 \ Γ = {1, 2, 3, 6}.
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Generators, Genus, and Conductor
Every numerical semigroup can be represented uniquely by a
set of minimal generators.
Ex.1: h4, 5i = {0, 4, 5, 8, 9, 10, 12, . . . } = Z \ {1, 2, 3, 6, 7, 11}.
Ex.2: h4, 5, 7i = {0, 4, 5, 7, . . . } = Z \ {1, 2, 3, 6}.
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Generators, Genus, and Conductor
Every numerical semigroup can be represented uniquely by a
set of minimal generators.
Ex.1: h4, 5i = {0, 4, 5, 8, 9, 10, 12, . . . } = Z \ {1, 2, 3, 6, 7, 11}.
Ex.2: h4, 5, 7i = {0, 4, 5, 7, . . . } = Z \ {1, 2, 3, 6}.
We define
The genus g(Γ) to be the number of elements in the gap
sequence.
The Frobenius number F (Γ) is the largest element in the
gap sequence.
The conductor Cond(Γ) = F (Γ) + 1.
(Sylvester, 1884) If Γ = ha, bi, where a and b are coprime, then
g(Γ) = (a − 1)(b − 1)/2
F (Γ) = (a − 1)(b − 1) − 1
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Symmetric Numerical Semigroup
Numerical semigroups arise in algebraic geometry in the
context of curve singularities. Namely, to a semigroup Γ, one
associates a unibranch curve singularity whose algebra of
functions is
C[Γ] := C[t a | a ∈ Γ].
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Symmetric Numerical Semigroup
Numerical semigroups arise in algebraic geometry in the
context of curve singularities. Namely, to a semigroup Γ, one
associates a unibranch curve singularity whose algebra of
functions is
C[Γ] := C[t a | a ∈ Γ].
The Γ-singularity, Spec C[t], is called Gorenstein if Γ is a
symmetric semigroup, i.e.,
n ∈ {gaps} ⇔ F (Γ) − n 6∈ {gaps}.
h4, 5i = Z \ {1, 2, 3, 6, 7, 11} is symmetric.
h4, 5, 7i = Z \ {1, 2, 3, 6} is non-symmetric.
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Open Problem
Classify numerical semigroups that satisfy the inequality
(2g − 1)2
11
.
≤
Pg
2
i=1 bi
In our work, we classified those satisfying a stronger condition
(2g − 1)2
≤ 4,
Pg
i=1 bi
which is equivalent to
g
X
bi > g(g − 1).
i=1
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Hyperelliptic Case
A numerical semigroup Γ is called hyperelliptic if Γ = h2, mi,
where m = 2k + 1 is odd.
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Hyperelliptic Case
A numerical semigroup Γ is called hyperelliptic if Γ = h2, mi,
where m = 2k + 1 is odd. In this case, we have
{1, 3, . . . , 2k − 1} as gaps, and so
g=k =
g
X
m−1
,
2
bi = 1 + 3 + · · · + (2k − 1).
i=1
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Hyperelliptic Case
A numerical semigroup Γ is called hyperelliptic if Γ = h2, mi,
where m = 2k + 1 is odd. In this case, we have
{1, 3, . . . , 2k − 1} as gaps, and so
g=k =
g
X
m−1
,
2
bi = 1 + 3 + · · · + (2k − 1).
i=1
=⇒
g
X
bi = k 2
i=1
> k (k − 1)
= g(g − 1).
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Key Observation
Lemma
Suppose Γ is a semigroup of genus g and {b1 , . . . , bg } are
gaps of Γ. Then bi ≤ 2i − 1.
Proof.
If bi is a gap then every pair (k, bi − k ), for 1 ≤ k ≤ bbi /2c must
have at least one gap. Otherwise bi would not be a gap.
Suppose bi ≥ 2i. Then there exist at least i such pairs.
Since each such pair must contain at least one gap there must
be at least i gaps before bi . A contradiction!
When the i th gap satisfies bi = 2i − 1 we will refer to it as a
maximal gap.
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Corollaries and Results for Unibranch Case
Let n be the minimal generator of Γ.
Corollary
If bi = 2i − 1 is maximal or bi = 2i − 2, then bi−1 = bi − n.
Corollary
Suppose Γ satisfies the inequality (2g − 1)2 ≤ 4
we must have n ≤ 4.
Pg
i=1 bi ,
then
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Corollaries and Results for Unibranch Case
Let n be the minimal generator of Γ.
Corollary
If bi = 2i − 1 is maximal or bi = 2i − 2, then bi−1 = bi − n.
Corollary
Suppose Γ satisfies the inequality (2g − 1)2 ≤ 4
we must have n ≤ 4.
Pg
i=1 bi ,
then
Theorem
The non-hyperelliptic
numerical semigroups that satisfy
Pg
(2g − 1)2 ≤ 4 i=1 bi are
Symmetric: h3, 4i, h3, 5i, h3, 7i, h4, 5, 6i.
Non-symmetric: h3, 4, 5i, h3, 5, 7i.
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Setting up the Multibranch Case
Let S = C[t1 ] × C[t2 ] × · · · × C[tb ]. Consider a C∗ -action on
S given by
λ · t i = λ αi t i .
We define Sd as the degree d weighted piece of S, that is:
Sd =
n
o
c1 · t1d1 , ..., cb · tbdb | ci ∈ C, ci 6= 0 =⇒ αi · di = d .
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Multibranch Case
Consider a C-subalgebra R ⊂ S such that
R is C∗ -invariant.
The only degree 0 element of R is (1, 1, . . . , 1).
dimC (S/R) < ∞.
Geometrically, R is the algebra of functions on a connected
affine curve with b branches and a good C∗ -action.
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Multibranch Case
Consider a C-subalgebra R ⊂ S such that
R is C∗ -invariant.
The only degree 0 element of R is (1, 1, . . . , 1).
dimC (S/R) < ∞.
Geometrically, R is the algebra of functions on a connected
affine curve with b branches and a good C∗ -action.
If x = (c1 t1d1 , . . . , cb tbdb ) ∈ R is scaled by C∗ -action, we call the
number
w(x) := α1 d1 = · · · = αb db
the C∗ -weight (or degree) of x. We set
Rd := R ∩ Sd
to be the subspace of R consisting of the elements of
C∗ -weight d.
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Multibranch Case
We define for R
A generalized semigroup
Γ̃R := {(d, vd ) | d ∈ Z, vd = dimC Rd }.
A numerical conductor
cond(R) := min{d | vn = dimC Sn ∀ n ≥ d}.
The module of Rosenlicht differentials
(
!
)
b
X
dt1
dtb
|
Resti f · w = 0 ∀ f ∈ R
ωR := w = u1 m1 , . . . , ub mb
t1
tb
i=1
We say that R is Gorenstein if ωR is a principal R-module.
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Linking R and ωR
Let (ωR )d be the dth graded piece of the module ωR , and −B
be the degree of the generator of ωR .
We show the following:
dim(ωR )d = dim Sd − dim Rd
dim(ωR )−B+d = dim Rd
Putting the above two statements together this shows:
dim Rd + dim RB−d = dim Sd = dim SB−d .
The above formula is an analog of “the symmetry condition” in
the unibranch case.
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Looking forward
We define the following:
The analog of “the sum of gaps”:
χ1 =
B
X
dim(ωR )−B+d · (d − B) =
d=0
B
X
dim Rd · (d − B)
d=0
The analog of “the sum of gaps plus (2g − 1)2 ”:
χ2 =
2B
X
dim Rd · (d − 2B).
d=0
Open Problem
Classify multibranch Gorenstein algebras R such that
χ2
≤ 5.
χ1
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