Classification of Numerical Semigroups

Transcription

1 1/13 Classification of Numerical Semigroups Andrew Ferdowsian, Jian Zhou Advised by: Maksym Fedorchuk Supported by: BC URF program Boston College September 16, 2015

2 2/13 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.

3 2/13 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}.

4 2/13 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}.

5 3/13 Generators, Genus, and Conductor Every numerical semigroup can be represented uniquely by a set 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}.

6 3/13 Generators, Genus, and Conductor Every numerical semigroup can be represented uniquely by a set 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 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 = a, b, where a and b are coprime, then g(γ) = (a 1)(b 1)/2 F(Γ) = (a 1)(b 1) 1

7 4/13 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].

8 4/13 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 {gaps}. 4, 5 = Z \ {1, 2, 3, 6, 7, 11} is symmetric. 4, 5, 7 = Z \ {1, 2, 3, 6} is non-symmetric.

9 5/13 Open Problem Classify numerical semigroups that satisfy the inequality (2g 1) 2 g i=1 b i In our work, we classified those satisfying a stronger condition (2g 1) 2 g i=1 b i 4, which is equivalent to g b i > g(g 1). i=1

10 6/13 Hyperelliptic Case A numerical semigroup Γ is called hyperelliptic if Γ = 2, m, where m = 2k + 1 is odd.

11 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 b i = (2k 1). i=1

12 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 b i = (2k 1). i=1 = g b i = k 2 i=1 > k(k 1) = g(g 1).

13 7/13 Key Observation Lemma Suppose Γ is a semigroup of genus g and {b 1,..., b g } are gaps of Γ. Then b i 2i 1. Proof. If b i is a gap then every pair (k, b i k), for 1 k b i /2 must have at least one gap. Otherwise b i would not be a gap. Suppose b i 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 b i. A contradiction! When the i th gap satisfies b i = 2i 1 we will refer to it as a maximal gap.

14 8/13 Corollaries and Results for Unibranch Case Let n be the minimal generator of Γ. Corollary If b i = 2i 1 is maximal or b i = 2i 2, then b i 1 = b i n. Corollary Suppose Γ satisfies the inequality (2g 1) 2 4 g i=1 b i, then we must have n 4.

15 8/13 Corollaries and Results for Unibranch Case Let n be the minimal generator of Γ. Corollary If b i = 2i 1 is maximal or b i = 2i 2, then b i 1 = b i n. Corollary Suppose Γ satisfies the inequality (2g 1) 2 4 g i=1 b i, then we must have n 4. Theorem The non-hyperelliptic numerical semigroups that satisfy (2g 1) 2 4 g i=1 b i are Symmetric: 3, 4, 3, 5, 3, 7, 4, 5, 6. Non-symmetric: 3, 4, 5, 3, 5, 7.

16 9/13 Setting up the Multibranch Case Let S = C[t 1 ] C[t 2 ] C[t b ]. Consider a C -action on S given by λ t i = λ α i t i. We define S d as the degree d weighted piece of S, that is: { (c1 S d = t d 1 1,..., c b t d ) b b ci C, c i 0 = α i d i = d}.

17 10/13 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). dim C (S/R) <. Geometrically, R is the algebra of functions on a connected affine curve with b branches and a good C -action.

18 10/13 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). dim C (S/R) <. Geometrically, R is the algebra of functions on a connected affine curve with b branches and a good C -action. If x = (c 1 t d 1 1,..., c bt d b b ) R is scaled by C -action, we call the number w(x) := α 1 d 1 = = α b d b the C -weight (or degree) of x. We set R d := R S d to be the subspace of R consisting of the elements of C -weight d.

19 11/13 Multibranch Case We define for R A generalized semigroup Γ R := {(d, v d ) d Z, v d = dim C R d }. A numerical conductor cond(r) := min{d v n = dim C S n n d}. The module of Rosenlicht differentials { ( ) dt 1 dt ω R := w = u 1 t m,..., u b b 1 b 1 t m Res ti b b i=1 f w = 0 f R } We say that R is Gorenstein if ω R is a principal R-module.

20 12/13 Linking R and ω R Let (ω R ) d be the dth graded piece of the module ω R, and B be the degree of the generator of ω R. We show the following: dim(ω R ) d = dim S d dim R d dim(ω R ) B+d = dim R d Putting the above two statements together this shows: dim R d + dim R B d = dim S d = dim S B d. The above formula is an analog of the symmetry condition in the unibranch case.

21 13/13 Looking forward We define the following: The analog of the sum of gaps : B χ 1 = dim(ω R ) B+d (d B) = d=0 B dim R d (d B) d=0 The analog of the sum of gaps plus (2g 1) 2 : Open Problem χ 2 = 2B d=0 dim R d (d 2B). Classify multibranch Gorenstein algebras R such that χ 2 χ 1 5.