Classification of Numerical Semigroups Andrew Ferdowsian, Jian Zhou Advised by: Maksym Fedorchuk Supported by: BC URF program Boston College September 16, 2015 1/13 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. 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}. 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}. 2/13 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}. 3/13 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 3/13 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]. 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 6∈ {gaps}. h4, 5i = Z \ {1, 2, 3, 6, 7, 11} is symmetric. h4, 5, 7i = Z \ {1, 2, 3, 6} is non-symmetric. 4/13 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 5/13 Hyperelliptic Case A numerical semigroup Γ is called hyperelliptic if Γ = h2, mi, where m = 2k + 1 is odd. 6/13 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 6/13 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). 6/13 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. 7/13 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 8/13 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. 8/13 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 . 9/13 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). dimC (S/R) < ∞. Geometrically, R is the algebra of functions on a connected affine curve with b branches and a good C∗ -action. 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). 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. 10/13 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. 11/13 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. 12/13 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 13/13