Geometry

Tek terimli tam kesişim eğrileri ve azalmayan hilbert fonksiyonları

In this thesis, we first study the problem of determining set theoretic completeintersection (s.t.c.i.) projective monomial curves. We are also interested in findingthe equations of the hypersurfaces on which the monomial curve lie as set theoreticcomplete intersection. We find these equations for symmetric ArithmeticallyCohen-Macaulay monomial curves.We describe a method to produce infinitely many s.t.c.i. monomial curves inP^{n+1} starting from one single s.t.c.i. monomial curve in P^{n}. Our approach hasthe side novelty of describing explicitly the equations of hypersurfaces on whichthese new monomial curves lie as s.t.c.i.. On the other hand, semigroup gluingbeing one of the most popular techniques of recent research, we develop numericalcriteria to determine when these new curves can or cannot be obtained via gluing.Finally, by using the technique of gluing semigroups, we give infinitely manynew families of affine monomial curves in arbitrary dimensions with Cohen-Macaulay tangent cones. This gives rise to large families of 1-dimensional localrings with arbitrary embedding dimensions and having non-decreasing Hilbertfunctions. We also construct infinitely many affine monomial curves in A^{n+1}whose tangent cone is not Cohen Macaulay and whose Hilbert function is nondecreasingfrom a single monomial curve in A^{n} with the same property.