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Convex body semigroups and their applications

Convex body semigroups and their applications

Alberto Vigneron-Tenorio

• Dpto. Matem´

aticas Universidad de C´ adiz

INdAM meeting: International meeting on numerical semigroups 2014 Cortona, 8-12/9/2014 some joint works with J.I. Garc´ ıa-Garc´ ıa, A. S´ anchez-R.-Navarro and M.A. Moreno-Frias

Alberto Vigneron Tenorio Convex body semigroups and their applications

Convex body semigroups and their applications Circle and convex polygonal semigroups

Definition Let F be a subset of Rk, we call convex body semigroup generated by F to F = ∞

i=0 Fi ∩ Nk, where Fi = {iX|X ∈ F} with i ∈ N.

Definition Convex polygonal semigroup Circle semigroup P = ∞

i=0 Fi ∩ N2

S = ∞

i=0 Ci ∩ N2

INdAM meeting 2014 (Cortona, 9/2014) Convex body semigroups and their applications

Convex body semigroups and their applications Circle and convex polygonal semigroups Interesting computational property

Remark Does P belong to F? Just consider the ray τ defined by P, the set τ ∩ F = AB. To check if P ∈ F, consider {k ∈ N| d(P)

d(B) ≤ k ≤ d(P) d(A)}, if

this set is nonempty then P ∈ F.

INdAM meeting 2014 (Cortona, 9/2014) Convex body semigroups and their applications

Convex body semigroups and their applications Circle and convex polygonal semigroups Notation

Definition Given A ⊆ R2

≥ :

Define the cone generated by A ⊆ R2

≥ as the set

LQ≥(A) = p

• i=1

qiai|p ∈ N, qi ∈ Q≥, ai ∈ A

• .

Denote by τ1 and τ2 the extremal rays of LQ≥(A) (assume the slope

• f τ1 is greater than the slope of τ2).

int(A) = A \ {τ1, τ2} is called interior of A. d(P) is the distance d(P, O) where O is the zero element and P ∈ R2.

INdAM meeting 2014 (Cortona, 9/2014) Convex body semigroups and their applications

Convex body semigroups and their applications Circle and convex polygonal semigroups Computing the minimal generating set

Problem WHEN ARE THESE SEMIGROUPS FINITELY GENERATED???? COMPUTING THE MINIMAL SYSTEM OF GENERATORS???? Theorem (G-G, S-R, M-F, V-T 13) Given F a circle or a convex polygon, the semigroup F = ∞

i=0 Fi ∩ N2 is

finitely generated if and only if F ∩ τ1 and F ∩ τ2 contain rational points. Furthermore, in such case there exists an algorithm to compute the minimal system of generators of F.

Implemented: PolySGTools, http: // departamentos. uca. es/ C101/ pags-personales/ alberto. vigneron/ PolySGTools. rar Implemented: CircleSG, http: // hdl. handle. net/ 10498/ 15832

INdAM meeting 2014 (Cortona, 9/2014) Convex body semigroups and their applications

Convex body semigroups and their applications Semigroup rings

Notation Let (S, +) be a finitely generated commutative monoid. If ❦ is a field, we denote by ❦[S] the semigroup ring of S over ❦. Note that ❦[S] is equal to

m∈S ❦χm endowed with a multiplication which is ❦-linear and such

that χm · χn = χm+n, m and n ∈ S. Problem Characterize Cohen-Macaulay, Gorenstein and Buchsbaum affine convex body semigroups.

INdAM meeting 2014 (Cortona, 9/2014) Convex body semigroups and their applications

Convex body semigroups and their applications Cohen-Macaulay affine convex body semigroups

Theorem (Goto, Suzuki, Watanabe 76) Let S ⊂ Nr be the affine semigroup generated by {n1, . . . , nr, nr+1, . . . , nr+m} with r = dim(S). The following conditions are equivalent:

1

S is Cohen-Macaulay.

2

For any a, b ∈ S with a + ni = b + nj (1 ≤ i = j ≤ r), a − nj = b − ni ∈ S. Corollary Let S ⊆ N2, the following conditions are equivalent:

1

S is Cohen-Macaulay.

2

For all a ∈ C \ S, a + n1 or a + n2 does not belong to S. Lemma Let S ⊂ N2 be a simplicial affine semigroup such that int(C) \ int(S) is nonempty finite set, then S is not Cohen-Macaulay.

INdAM meeting 2014 (Cortona, 9/2014) Convex body semigroups and their applications

Convex body semigroups and their applications Cohen-Macaulay affine convex body semigroups Cohen-Macaulay affine circle semigroup

Proposition Assume that S = C, then S is a Cohen-Macaulay affine circle semigroup if and only if int(S) = int(C) and S ∩ τi = ni for i = 1, 2.

INdAM meeting 2014 (Cortona, 9/2014) Convex body semigroups and their applications

Convex body semigroups and their applications Cohen-Macaulay affine convex body semigroups Cohen-Macaulay affine convex polygonal semigroup

Lemma Suppose the extremal ray τ1 intersects F in only one point P1, denote by Vi the intersection of (iP1)(iP2) and ((i + 1)Pn)((i + 1)P1) for every i ∈ N and i ≫ 0. Every point Vi belongs to a parallel line to τ1 denoted by ν1.

INdAM meeting 2014 (Cortona, 9/2014) Convex body semigroups and their applications

Convex body semigroups and their applications Cohen-Macaulay affine convex body semigroups Cohen-Macaulay affine convex polygonal semigroup

Notation T1, the triangle {O, P1, V1 − jP1}. Υ1, the finite set

ConvexHull({O, j1P1, V1, ν1 ∩ τ2}) \

h<j1,h∈N

• (
• T1 ∪(OP1 \ {O, P1})) + hP1
• .

Υ′ = {P ∈ Υ1 ∩ N2|P + n1, P + n2 ∈ P}, Υ′′ = {P ∈ Υ2 ∩ N2|P + n1, P + n2 ∈ P} Υ = (Q + LQ≥(F)) ∩ N2 where Q = ν1 ∩ ν2.

1 Ν1 Τ1 Τ2

• Ν1

Ν2 Τ1 Τ2

INdAM meeting 2014 (Cortona, 9/2014) Convex body semigroups and their applications

Convex body semigroups and their applications Cohen-Macaulay affine convex body semigroups Cohen-Macaulay affine convex polygonal semigroup

Theorem Let P be a simplicial affine convex polygonal semigroup such that F ∩ τ1 and F ∩ τ2 are not both segments. Then

1

if int(C) = int(P), the semigroup P is Cohen-Macaulay if and only if P ∩ τi = ni for i = 1, 2,

2

if int(C) = int(P), the semigroup P is Cohen-Macaulay if and only if Υ ∪ Υ′ ∪ Υ′′ ⊂ P. Corollary Any affine polygonal semigroup generated by a triangle with rational vertices is Cohen-Macaulay.

INdAM meeting 2014 (Cortona, 9/2014) Convex body semigroups and their applications

Convex body semigroups and their applications Gorenstein affine convex body semigroups

Definition Denote by Ap(n) the set {s ∈ S|s − n / ∈ S}; this set is known as the Ap´ ery set of n. Theorem (Rosales, Garc´ ıa-S´ anchez 98) The following conditions are equivalent:

1

S is Gorenstein.

2

S is C-M and ∩r

i=1Ap(ni) has a unique maximal element (with

respect to the order defined by S).

INdAM meeting 2014 (Cortona, 9/2014) Convex body semigroups and their applications

Convex body semigroups and their applications Gorenstein affine convex body semigroups Gorenstein affine circle semigroups

Notation Let H ⊂ N2 be the set

• ConvexHull({O, n1, n2, n1 + n2}) \ {On1, On2, n1 + n2}
• ∪ {O}.

Proposition Let S be a C-M circle semigroup. S is Gorenstein if and only if there exists a unique maximal element in Ap(n1) ∩ Ap(n2) = C ∩ H.

INdAM meeting 2014 (Cortona, 9/2014) Convex body semigroups and their applications

Convex body semigroups and their applications Gorenstein affine convex body semigroups Gorenstein affine convex polygonal semigroups

Notation Define H′ = Υ′

1 ∩ {P ∈ N2|P − n1, P − n2 /

∈ P}, H′′ = Υ′

2 ∩ {P ∈ N2|P − n1, P − n2 /

∈ P} H′′′ = ConvexHull({Q, Q + n1, Q + n2, Q + n1 + n2}) ∩ {P ∈ N2|P − n1, P − n2 / ∈ P} Theorem Under the assumption that P is C-M, the semigroup P is Gorenstein if and only if there exists a unique maximal element in the finite set Ap(n1) ∩ Ap(n2) = P ∩ (H′ ∪ H′′ ∪ H′′′).

INdAM meeting 2014 (Cortona, 9/2014) Convex body semigroups and their applications

Convex body semigroups and their applications Gorenstein affine convex body semigroups Gorenstein affine convex polygonal semigroups

Example For instance we can generate a family of Gorenstein rings by using the triangles with vertex set {(4, 0), (4 + 2k, 0), (4 + k, k)}, where k is an integer greater than or equal to 2. We determine explicitly the intersection Ap(n1) ∩ Ap(n2). Ap(n1) ∩ Ap(n2) ∩ {y = 0} = {(0, 0), (5, 0), (6, 0), (7, 0)} Ap(n1) ∩ Ap(n2) ∩ {y = 1} = {(5, 1), (6, 1), (7, 1), (8, 1)} . . . . . . . . . Ap(n1) ∩ Ap(n2) ∩ {y = k − 2} = {(2 + k, k − 2), (3 + k, k − 2), (4 + k, k − 2), (5 + k, k − 2)} Ap(n1) ∩ Ap(n2) ∩ {y = k − 1} = {(3 + k, k − 1), (4 + k, k − 1), (5 + k, k − 1), (10 + k, k − 1)} Ap(n1) ∩ Ap(n2) ∩ {y ≥ k} = ∅

INdAM meeting 2014 (Cortona, 9/2014) Convex body semigroups and their applications

Convex body semigroups and their applications Gorenstein affine convex body semigroups Gorenstein affine convex polygonal semigroups

Example Polygon= {(4, 0), (7, 3), (10, 0)} Ap(n1) ∩ Ap(n2) = {O, (5, 0), (6, 0), (7, 0), (5, 1), (6, 1), (7, 1), (8, 1), (6, 2), (7, 2), (8, 2), (13, 2)}

Figure: Example of Gorenstein convex polygonal semigroup.

INdAM meeting 2014 (Cortona, 9/2014) Convex body semigroups and their applications

Convex body semigroups and their applications Buchsbaum affine convex body semigroups

Notation Let S be the semigroup {a ∈ Nr|a + ni ∈ S, ∀i = 1, . . . , r + m} ⊂ C. Theorem (Garc´ ıa-S´ anchez, Rosales 02) The following conditions are equivalent:

1

S is Buchsbaum.

2

S is Cohen-Macaulay.

INdAM meeting 2014 (Cortona, 9/2014) Convex body semigroups and their applications

Convex body semigroups and their applications Buchsbaum affine convex body semigroups Buchsbaum affine circle semigroup

Proposition Let S ⊂ N2 be an affine circle semigroup. The semigroup S is Buchsbaum if and only if int(C) = int(S) and S ∩ τj is generated only by

• ne element for j = 1, 2.

INdAM meeting 2014 (Cortona, 9/2014) Convex body semigroups and their applications

Convex body semigroups and their applications Buchsbaum affine convex body semigroups Buchsbaum affine convex polygonal semigroups

Theorem Let P be a simplicial affine convex polygonal semigroup. Then

1

if int(C) = int(P), the semigroup P is Buchsbaum if and only if P ∩ τj is generated by only one element for j = 1, 2,

2

if int(C) = int(P), the semigroup P is Buchsbaum if and only if Υ′ = ∅ where Υ′ = {a ∈ (Υ1 ∪ Υ2) \ P| a + n′

1, a + n′ 2 ∈ P}, and

Υ ⊂ P.

Implemented: PolySGTools, http://departamentos.uca.es/C101/ pags-personales/alberto.vigneron/PolySGTools.rar

• INdAM meeting 2014 (Cortona, 9/2014)

Convex body semigroups and their applications

Convex body semigroups and their applications Buchsbaum affine convex body semigroups Buchsbaum affine convex polygonal semigroups

Corollary Every affine convex polygonal semigroup associated to a triangle with rational vertices is Buchsbaum. Corollary Let F be a convex polygon with vertices P1, . . . , P4 ∈ Q2

≥ and let P be

its associated affine convex polygonal semigroup. If P1 ∈ P ∩ τ1, P4 ∈ P ∩ τ2 and the points O, P2 and P3 are aligned, P is Buchsbaum.

INdAM meeting 2014 (Cortona, 9/2014) Convex body semigroups and their applications

Convex body semigroups and their applications References

J.I. Garc´ ıa-Garc´ ıa, M.A. Moreno-Fr´ ıas, A. S´ anchez-R.-Navarro, A. Vigneron-Tenorio. Affine convex body semigroups. Semigroup Forum vol. 87 (2) (2013), 331–350. J.I. Garc´ ıa-Garc´ ıa, A. S´ anchez-R.-Navarro, A. Vigneron-Tenorio. Affine convex body semigroups ans Buchsbaum rings. arXiv:1402.2597 [math.AC]. J.I. Garc´ ıa-Garc´ ıa, A. Vigneron-Tenorio. Computing families of Cohen-Macaulay and Gorenstein rings. Semigroup Forum 88 (2014), no. 3, 610–620. P.A. Garc´ ıa-S´ anchez, J.C. Rosales. On Buchsbaum simplicial affine semigroups. Pacific J. Math. 202 (2002), no. 2, 329–339.

• S. Goto, N. Suzuki, K. Watanabe.

On affine semigroup rings.

• Japan. J. Math. (N.S.) 2 (1976), no. 1, 1–12.

J.C. Rosales, P.A. Garc´ ıa-S´ anchez. On Cohen-Macaulay and Gorenstein simplicial affine semigroups.

• Proc. Edinburgh Math. Soc. (2) 41 (1998), no. 3, 517–537.

INdAM meeting 2014 (Cortona, 9/2014) Convex body semigroups and their applications

Convex body semigroups and their applications This is the end(?)

Thanks for your attention!

INdAM meeting 2014 (Cortona, 9/2014) Convex body semigroups and their applications