Normal reduction numbers for normal surface singularities with application to elliptic singularities of Brieskorn type

In this paper, we give a formula for normal reduction number of an integrally closed $\mathfrak{m}$-primary ideal of a $2$-dimensional normal local ring $(A,\mathfrak{m})$ in terms of the geometric genus $p_g(A)$ of $A$. Also we compute the normal reduction number of the maximal ideal of Brieskorn hypersurfaces. As an application, we give a short proof of a classification of Brieskorn hypersurfaces having elliptic singularities.