Classification of Numerical Semigroups

Andrew Ferdowsian, Jian ZhouAdvised by: Maksym FedorchukSupported by: BC URF program

Boston College

September 16, 2015

1/13

Numerical Semigroup

A numerical semigroup is a subset Γ ⊂ Z≥0 such thatΓ contains 0,Γ is closed under addition,the complement of Γ in Z≥0 is finite, denoted as gapsequence, and its elements are called gaps.

We can regard Γ ⊂ Z≥0 as a group under addition, but withoutthe 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}.

2/13

Numerical Semigroup

A numerical semigroup is a subset Γ ⊂ Z≥0 such thatΓ contains 0,Γ is closed under addition,the complement of Γ in Z≥0 is finite, denoted as gapsequence, and its elements are called gaps.

We can regard Γ ⊂ Z≥0 as a group under addition, but withoutthe 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}.

2/13

Numerical Semigroup

A numerical semigroup is a subset Γ ⊂ Z≥0 such thatΓ contains 0,Γ is closed under addition,the complement of Γ in Z≥0 is finite, denoted as gapsequence, and its elements are called gaps.

We can regard Γ ⊂ Z≥0 as a group under addition, but withoutthe 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}.

2/13

Generators, Genus, and Conductor

Every numerical semigroup can be represented uniquely by aset of minimal generators.Ex.1: 〈4,5〉 = {0,4,5,8,9,10,12, . . . } = Z \ {1,2,3,6,7,11}.Ex.2: 〈4,5,7〉 = {0,4,5,7, . . . } = Z \ {1,2,3,6}.

We defineThe genus g(Γ) to be the number of elements in the gapsequence.The Frobenius number F (Γ) is the largest element in thegap sequence.The conductor Cond(Γ) = F (Γ) + 1.

(Sylvester, 1884) If Γ = 〈a,b〉, where a and b are coprime, then

g(Γ) = (a− 1)(b − 1)/2F (Γ) = (a− 1)(b − 1)− 1

3/13

Generators, Genus, and Conductor

Every numerical semigroup can be represented uniquely by aset of minimal generators.Ex.1: 〈4,5〉 = {0,4,5,8,9,10,12, . . . } = Z \ {1,2,3,6,7,11}.Ex.2: 〈4,5,7〉 = {0,4,5,7, . . . } = Z \ {1,2,3,6}.

We defineThe genus g(Γ) to be the number of elements in the gapsequence.The Frobenius number F (Γ) is the largest element in thegap sequence.The conductor Cond(Γ) = F (Γ) + 1.

(Sylvester, 1884) If Γ = 〈a,b〉, where a and b are coprime, then

g(Γ) = (a− 1)(b − 1)/2F (Γ) = (a− 1)(b − 1)− 1

3/13

Symmetric Numerical Semigroup

Numerical semigroups arise in algebraic geometry in thecontext of curve singularities. Namely, to a semigroup Γ, oneassociates a unibranch curve singularity whose algebra offunctions is

C[Γ] := C[ta | a ∈ Γ].

The Γ-singularity, SpecC[t ], is called Gorenstein if Γ is asymmetric semigroup, i.e.,

n ∈ {gaps} ⇔ F (Γ)− n 6∈ {gaps}.

〈4,5〉 = Z \ {1,2,3,6,7,11} is symmetric.〈4,5,7〉 = Z \ {1,2,3,6} is non-symmetric.

4/13

Symmetric Numerical Semigroup

Numerical semigroups arise in algebraic geometry in thecontext of curve singularities. Namely, to a semigroup Γ, oneassociates a unibranch curve singularity whose algebra offunctions is

C[Γ] := C[ta | a ∈ Γ].

The Γ-singularity, SpecC[t ], is called Gorenstein if Γ is asymmetric semigroup, i.e.,

n ∈ {gaps} ⇔ F (Γ)− n 6∈ {gaps}.

〈4,5〉 = Z \ {1,2,3,6,7,11} is symmetric.〈4,5,7〉 = Z \ {1,2,3,6} is non-symmetric.

4/13

Open Problem

Classify numerical semigroups that satisfy the inequality

(2g − 1)2∑gi=1 bi

≤ 112.

In our work, we classified those satisfying a stronger condition

(2g − 1)2∑gi=1 bi

≤ 4,

which is equivalent to

g∑i=1

bi > g(g − 1).

5/13

Hyperelliptic Case

A numerical semigroup Γ is called hyperelliptic if Γ = 〈2,m〉,where m = 2k + 1 is odd.

In this case, we have{1,3, . . . ,2k − 1} as gaps, and so

g = k =m − 1

2,

g∑i=1

bi = 1 + 3 + · · ·+ (2k − 1).

=⇒g∑

i=1

bi = k2

> k(k − 1)

= g(g − 1).

6/13

Hyperelliptic Case

A numerical semigroup Γ is called hyperelliptic if Γ = 〈2,m〉,where m = 2k + 1 is odd. In this case, we have{1,3, . . . ,2k − 1} as gaps, and so

g = k =m − 1

2,

g∑i=1

bi = 1 + 3 + · · ·+ (2k − 1).

=⇒g∑

i=1

bi = k2

> k(k − 1)

= g(g − 1).

6/13

Hyperelliptic Case

A numerical semigroup Γ is called hyperelliptic if Γ = 〈2,m〉,where m = 2k + 1 is odd. In this case, we have{1,3, . . . ,2k − 1} as gaps, and so

g = k =m − 1

2,

g∑i=1

bi = 1 + 3 + · · ·+ (2k − 1).

=⇒g∑

i=1

bi = k2

> k(k − 1)

= g(g − 1).

6/13

Key Observation

LemmaSuppose Γ is a semigroup of genus g and {b1, . . . ,bg} aregaps of Γ. Then bi ≤ 2i − 1.

Proof.If bi is a gap then every pair (k ,bi − k), for 1 ≤ k ≤ bbi/2c musthave 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 mustbe at least i gaps before bi . A contradiction!

When the i th gap satisfies bi = 2i − 1 we will refer to it as amaximal gap.

7/13

Corollaries and Results for Unibranch Case

Let n be the minimal generator of Γ.

CorollaryIf bi = 2i − 1 is maximal or bi = 2i − 2, then bi−1 = bi − n.

Corollary

Suppose Γ satisfies the inequality (2g − 1)2 ≤ 4∑g

i=1 bi , thenwe must have n ≤ 4.

TheoremThe non-hyperelliptic numerical semigroups that satisfy(2g − 1)2 ≤ 4

∑gi=1 bi are

Symmetric: 〈3,4〉, 〈3,5〉, 〈3,7〉, 〈4,5,6〉.Non-symmetric: 〈3,4,5〉, 〈3,5,7〉.

8/13

Corollaries and Results for Unibranch Case

Let n be the minimal generator of Γ.

CorollaryIf bi = 2i − 1 is maximal or bi = 2i − 2, then bi−1 = bi − n.

Corollary

Suppose Γ satisfies the inequality (2g − 1)2 ≤ 4∑g

i=1 bi , thenwe must have n ≤ 4.

TheoremThe non-hyperelliptic numerical semigroups that satisfy(2g − 1)2 ≤ 4

∑gi=1 bi are

Symmetric: 〈3,4〉, 〈3,5〉, 〈3,7〉, 〈4,5,6〉.Non-symmetric: 〈3,4,5〉, 〈3,5,7〉.

8/13

Setting up the Multibranch Case

Let S = C[t1]× C[t2]× · · · × C[tb]. Consider a C∗-action onS given by

λ · ti = λαi ti .

We define Sd as the degree d weighted piece of S, that is:

Sd ={(

c1 · td11 , ..., cb · tdb

b

)| ci ∈ C, ci 6= 0 =⇒ αi · di = d

}.

9/13

Multibranch Case

Consider a C-subalgebra R ⊂ S such thatR 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 connectedaffine curve with b branches and a good C∗-action.

If x = (c1td11 , . . . , cbtdb

b ) ∈ R is scaled by C∗-action, we call thenumber

w(x) := α1d1 = · · · = αbdb

the C∗-weight (or degree) of x . We set

Rd := R ∩ Sd

to be the subspace of R consisting of the elements ofC∗-weight d .

10/13

Multibranch Case

Consider a C-subalgebra R ⊂ S such thatR 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 connectedaffine curve with b branches and a good C∗-action.

If x = (c1td11 , . . . , cbtdb

b ) ∈ R is scaled by C∗-action, we call thenumber

w(x) := α1d1 = · · · = αbdb

the C∗-weight (or degree) of x . We set

Rd := R ∩ Sd

to be the subspace of R consisting of the elements ofC∗-weight d .

10/13

Multibranch Case

We define for RA generalized semigroup

Γ̃R := {(d , vd ) | d ∈ Z, vd = dimCRd}.

A numerical conductor

cond(R) := min{d | vn = dimC Sn ∀ n ≥ d}.

The module of Rosenlicht differentials

ωR :=

{w =

(u1

dt1tm11, . . . ,ub

dtbtmbb

)|

b∑i=1

Resti f · w = 0 ∀ f ∈ R

}.

We say that R is Gorenstein if ωR is a principal R-module.

11/13

Linking R and ωR

Let (ωR)d be the d th graded piece of the module ωR, and −Bbe 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” inthe unibranch case.

12/13

Looking forward

We define the following:The analog of “the sum of gaps”:

χ1 =B∑

d=0

dim(ωR)−B+d · (d − B) =B∑

d=0

dim Rd · (d − B)

The analog of “the sum of gaps plus (2g − 1)2”:

χ2 =2B∑

d=0

dim Rd · (d − 2B).

Open ProblemClassify multibranch Gorenstein algebras R such that

χ2

χ1≤ 5.

13/13