On the conductor of the blowing-up of a one-dimensional Gorenstein local ring (Q1181438)

Let \(I\) be an \({\mathfrak m}\)-primary ideal of a one-dimensional Gorenstein local ring \((R,{\mathfrak m})\), and write \(S\), the first neighbourhood ring of \(I\), as \(I^ \delta:I^ \delta\) with \(\delta\) minimal \((\geq 1)\). Put \(g=\ell(S/R)\) and \(e=\) the multiplicity of \(I\). Extending work of \textit{F. Orecchia} and \textit{I. Ramella} [Manuscr. Math. 68, No. 1, 1-7 (1990; Zbl 0709.13010)] a main result is: \(R:S=I^ \delta\Leftrightarrow\delta=2g/e\Leftrightarrow\) the associated graded ring of \(I^ \delta\) is Gorenstein. Discussion involves \(g\), \(e\), \(\delta\) etc. particularly with \(I=\mathfrak{m}\), \(\delta=1,2\) or 3. With \(R\) analytically unramified, conditions for \(I\) to be quasi-normal are given.