Gluing semigroups and strongly indispensable free resolutions

We study strong indispensability of minimal free resolutions of semigroup rings. We focus on two operations, gluing and extending, used in literature to produce more examples with a special property from the existing ones. We give a naive condition to determine whether gluing of two semigroup rings has a strongly indispensable minimal free resolution. As applications, we determine extensions of $3$-generated non-symmetric, $4$-generated symmetric and pseudo symmetric numerical semigroups as well as obtain infinitely many complete intersection semigroups of any embedding dimension, having strongly indispensable minimal free resolutions.Comment: Internat. J. Algebra Compu

Complete intersection monomial curves and non-decreasing Hilbert functions

Ankara : The Department of Mathematics and the Institute of Engineering and Sciences of Bilkent University, 2008.Thesis (Ph.D.) -- Bilkent University, 2008.Includes bibliographical references leaves 56-57.In this thesis, we first study the problem of determining set theoretic complete intersection (s.t.c.i.) projective monomial curves. We are also interested in finding the equations of the hypersurfaces on which the monomial curve lie as set theoretic complete intersection. We find these equations for symmetric Arithmetically Cohen-Macaulay monomial curves. We describe a method to produce infinitely many s.t.c.i. monomial curves in P n+1 starting from one single s.t.c.i. monomial curve in P n . Our approach has the side novelty of describing explicitly the equations of hypersurfaces on which these new monomial curves lie as s.t.c.i.. On the other hand, semigroup gluing being one of the most popular techniques of recent research, we develop numerical criteria to determine when these new curves can or cannot be obtained via gluing. Finally, by using the technique of gluing semigroups, we give infinitely many new families of affine monomial curves in arbitrary dimensions with CohenMacaulay tangent cones. This gives rise to large families of 1-dimensional local rings with arbitrary embedding dimensions and having non-decreasing Hilbert functions. We also construct infinitely many affine monomial curves in A n+1 whose tangent cone is not Cohen Macaulay and whose Hilbert function is nondecreasing from a single monomial curve in A n with the same property.Şahin, MesutPh.D

Codes on Subgroups of Weighted Projective Tori

We obtain certain algebraic invariants relevant to study codes on subgroups of weighted projective tori inside an $n$-dimensional weighted projective space. As application, we compute all the main parameters of generalized toric codes on these subgroups of tori lying inside a weighted projective plane of the form \Pp(1,1,a).Comment: supported by TUBITAK Project No:119F17

Multigraded Hilbert function and toric complete intersection codes

Let $X$ be a complete $n$-dimensional simplicial toric variety with homogeneous coordinate ring $S$. We study the multigraded Hilbert function $H_Y$ of reduced $0$-dimensional subschemes $Y$ in $X$. We provide explicit formulas and prove non-decreasing and stabilization properties of $H_Y$ when $Y$ is a $0$-dimensional complete intersection in $X$. We apply our results to computing the dimension of some evaluation codes on $0$-dimensional complete intersection in simplicial toric varieties.Comment: 22 pages, comments welcome, presented at ICM 2014, Seoul, KORE

On the minimal number of elements generating an algebraic set

Cataloged from PDF version of article.In this thesis we present studies on the general problem of finding the minimal number of elements generating an algebraic set in n-space both set and ideal theoretically.Şahin, MesutM.S