On the periodicity of the first Betti number of the semigroup ring under translations

Abstract

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT REQUEST OF AUTHOR.] Any curve C in any dimension can be described by a parameterization. In particular in the plane, that is dimension 2, the coordinates x and y are both given as a function of a third variable t, called parameter: x=x(t), y=y(t) Then each value of t determines a point (x,y) that lies on C. Given a curve, it is an interesting question in algebraic geometry to ask which equations do the coordinates of the points on this curve satisfy. Take a finite sequence of positive integers X. To every sequence X, we can associate a monomial curve C that is a curve whose parameterization is given by powers of t that are components of X. In this dissertation we are going to study the defining equations of a monomial curve C in dimension 4. It turns out that to study the defining equations of a monomial curve is equivalent to study the minimal generators of its defining P, called monomial prime ideal of X. A no degenerate monomial curve in dimension 4 needs at least three equations and in general many more. The minimal number of equations needed to define a curve is an invariant of the curve C and the associated prime ideal P and it is called the first Betti number of P. To any triplets X* equals to (a,b,c), we associate a family of monomial curves F* that corresponds to a family of primes ideals P*. Part of the Herzog-Srinivasan's conjecture states that under translations j, the first Betti number of P* are periodic with period given by a+b+c. Let p and t be natural numbers. Then what we proved is the following: 1) Let X*=(a,b,p(a+b)) or X*=(p(b+c),b,c). If (a+b+c) divides j+1, then the first Betti number of P* is 3, the smallest possible, and it occurs eventually with period a+b+c. 2) Let X*=(a,b,p(a+b)). If (a+b+c) divides [j+1-(a+b)t], then the first Betti number of P* is 4 and it occurs eventually with period a+b+c. 3) Let X*=(p(b+c),b,c). If (a+b+c) divides [j+1-(b+c)t], for any n natural number, then the first Betti number of P* is 4 and it occurs eventually with period a+b+c.4) Let X*=(a,b,p(a+b)) or X*=(p(b+c),b,c). If the first Betti number of P* is 3 then (a+b+c) divides j+1.--From public.pdf.