Complete intersection monomial curves and non-decreasing Hilbert functions

Abstract

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.