On the Normality of a Class of Monomial Ideals via the Newton Polyhedron

Abstract

Let $$I=\left\langle x_{1}^{a_{1}},\ldots ,x_{n}^{a_{n}}\right\rangle \subset R=K[x_{1},\ldots ,x_{n}]$$ with $$a_{1},\ldots ,a_{n}$$ positive integers and K a field, and let J be the integral closure of I. A criterion for the normality of J is developed. This criterion is used to show that J is normal if and only if the integral closure of the ideal $$\langle x_{1}^{b_{1}},\ldots ,x_{n}^{b_{n}},\ldots ,x_{r}^{b_{r}}\rangle \subset R[x_{n+1},\ldots ,x_{r}]$$ is normal, where $$b_{i}\in \left\{ a_{1},\ldots ,a_{n}\right\} $$ for all i, this generalizes the work of Al-Ayyoub (Rocky Mt Math 39(1):1–9, 2009). If $$l=lcm (a_{1},\ldots ,a_{n})$$ and the integral closure of $$\left\langle x_{1}^{a_{1}},\ldots ,x_{n}^{a_{n}},x_{n+1}^{l}\right\rangle \subset R[x_{n+1}]$$ is not normal, then we show that the integral closure of $$\left\langle x_{1}^{a_{1}},\ldots ,x_{n}^{a_{n}},x_{n+1}^{s}\right\rangle $$ is not normal for any $$s>l$$ . Also, we give a shorter proof of a main result of Coughlin (Classes of Normal Monomial Ideals. Ph.D. thesis, 2004).