Convex body semigroups and their applications Alberto - PowerPoint PPT Presentation

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 R k , we call convex body semigroup generated by F to F = � ∞ i =0 F i ∩ N k , where F i = { iX | X ∈ F } with i ∈ N . Definition Convex polygonal semigroup Circle semigroup P = � ∞ S = � ∞ i =0 F i ∩ N 2 i =0 C i ∩ N 2 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 ⊆ R 2 ≥ : Define the cone generated by A ⊆ R 2 ≥ as the set � p � � L Q ≥ ( A ) = q i a i | p ∈ N , q i ∈ Q ≥ , a i ∈ A . i =1 Denote by τ 1 and τ 2 the extremal rays of L Q ≥ ( A ) (assume the slope of τ 1 is greater than the slope of τ 2 ). int ( A ) = A \ { τ 1 , τ 2 } is called interior of A . d ( P ) is the distance d ( P , O ) where O is the zero element and P ∈ R 2 . 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) i =0 F i ∩ N 2 is Given F a circle or a convex polygon, the semigroup F = � ∞ finitely generated if and only if F ∩ τ 1 and F ∩ τ 2 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 m ∈ S ❦ χ m endowed with a multiplication which is ❦ -linear and such to � that χ m · χ n = χ m + 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 ⊂ N r be the affine semigroup generated by { n 1 , . . . , n r , n r +1 , . . . , n r + m } with r = dim( S ) . The following conditions are equivalent: S is Cohen-Macaulay. 1 For any a , b ∈ S with a + n i = b + n j ( 1 ≤ i � = j ≤ r), 2 a − n j = b − n i ∈ S . Corollary Let S ⊆ N 2 , the following conditions are equivalent: S is Cohen-Macaulay. 1 For all a ∈ C \ S, a + n 1 or a + n 2 does not belong to S. 2 Lemma Let S ⊂ N 2 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 ∩ τ i = � n i � 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 τ 1 intersects F in only one point P 1 , denote by V i the intersection of ( iP 1 )( iP 2 ) and (( i + 1) P n )(( i + 1) P 1 ) for every i ∈ N and i ≫ 0 . Every point V i belongs to a parallel line to τ 1 denoted by ν 1 . 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 T 1 , the triangle { O , P 1 , V 1 − jP 1 } . Υ 1 , the finite set ◦ ConvexHull ( { O , j 1 P 1 , V 1 , ν 1 ∩ τ 2 } ) \ � � ( T 1 ∪ ( OP 1 \ { O , P 1 } )) + hP 1 � . h < j 1 , h ∈ N Υ ′ = { P ∈ Υ 1 ∩ N 2 | P + n 1 , P + n 2 ∈ P} , Υ ′′ = { P ∈ Υ 2 ∩ N 2 | P + n 1 , P + n 2 ∈ P} L Q ≥ ( F )) ∩ N 2 where Q = ν 1 ∩ ν 2 . Υ = ( Q + � Ν 1 Τ 1 Τ 1 Ν 1 � Ν 2 � 1 Τ 2 Τ 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 Theorem Let P be a simplicial affine convex polygonal semigroup such that F ∩ τ 1 and F ∩ τ 2 are not both segments. Then if int ( C ) = int ( P ) , the semigroup P is Cohen-Macaulay if and only 1 if P ∩ τ i = � n i � for i = 1 , 2 , if int ( C ) � = int ( P ) , the semigroup P is Cohen-Macaulay if and only 2 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: S is Gorenstein. 1 S is C-M and ∩ r i =1 Ap ( n i ) has a unique maximal element (with 2 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 ⊂ N 2 be the set � � ConvexHull ( { O , n 1 , n 2 , n 1 + n 2 } ) \ { On 1 , On 2 , n 1 + n 2 } ∪ { 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 ( n 1 ) ∩ Ap ( n 2 ) = 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 ∈ N 2 | P − n 1 , P − n 2 / ∈ P} , H ′′ = Υ ′ 2 ∩ { P ∈ N 2 | P − n 1 , P − n 2 / ∈ P} H ′′′ = ConvexHull ( { Q , Q + n 1 , Q + n 2 , Q + n 1 + n 2 } ) ∩ { P ∈ N 2 | P − n 1 , P − n 2 / ∈ 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 ( n 1 ) ∩ Ap ( n 2 ) = 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 + 2 k , 0) , (4 + k , k ) } , where k is an integer greater than or equal to 2. We determine explicitly the intersection Ap ( n 1 ) ∩ Ap ( n 2 ) . Ap ( n 1 ) ∩ Ap ( n 2 ) ∩ { y = 0 } = { (0 , 0) , (5 , 0) , (6 , 0) , (7 , 0) } Ap ( n 1 ) ∩ Ap ( n 2 ) ∩ { y = 1 } { (5 , 1) , (6 , 1) , (7 , 1) , (8 , 1) } = . . . . . . . . . = { (2 + k , k − 2) , (3 + k , k − 2) , Ap ( n 1 ) ∩ Ap ( n 2 ) ∩ { y = k − 2 } (4 + k , k − 2) , (5 + k , k − 2) } = { (3 + k , k − 1) , (4 + k , k − 1) , Ap ( n 1 ) ∩ Ap ( n 2 ) ∩ { y = k − 1 } (5 + k , k − 1) , (10 + k , k − 1) } Ap ( n 1 ) ∩ Ap ( n 2 ) ∩ { 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 ( n 1 ) ∩ Ap ( n 2 ) = { 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