Castelnuovo-Mumford regularity of simplicial semigroup rings with isolated singularity

Let $S = K[x_1, \ldots, x_n]$ be the polynomial ring in $n \geq 2$variables over a field $K$ and $\mathfrak{m}$ its graded maximal ideal. Let $f_1,\ldots, f_m \in S$ be homogeneous polynomials of degree $d-1\geq 2$ generating an $\mathfrak{m}$-primary ideal, and let $g_1,\ldots,g_r\in S$ be arbitrary homogeneous polynomials of degree $d$. In the present paper it will be proved that the Castelnuovo-Mumford regularity of the standard graded $K$-algebra $A=K[\{f_ix_j\}_{\substack{i=1,\ldots,m j=1,\ldots,n}}, g_1,\ldots, g_r]$is at most $(d-2)(n-1)$. By virtue of this result, it follows that the regularity of a simplicial semigroup ring $K[C]$ with isolated singularity is at most $e(K[C]) - \operatorname{codim}(K[C])$, where $e(K[C])$ is the multiplicity of $K[C]$and $\operatorname{codim}(K[C])$ is the codimension of $K[C]$.