Tangent cones at curve and surface singularities
Let \((R,m)\) be a Gorenstein local ring, analytically unramified, \(I \subset R\) an \(m\)-primary ideal of \(R\), \(\overline I\) the integral closure of \(I\) and \(\delta (I)\) the reduction number of \(I\). There are some integers \(\{\overline e_d (I)\}\) such that \[ \text{length} (R/ \overline {I^{n + 1}}) = \overline e_0 (I) {n + d \choose d} - \overline e_1 (I) {n + d - 1 \choose d - 1} + \cdots + (- 1)^d \overline e_d (I) \] for sufficiently large \(n\). The paper gives sufficient conditions in terms of these invariants under which \(G(I) = R/I \oplus I/I^2 \oplus \ldots\) is Cohen-Macaulay or Gorenstein. -- Suppose \(\dim R = 1\), \(I\) is integrally closed and \(\overline e_1 (m) + \text{length} (R/ \overline m) = 1 + e(R)\). If \(e(R) = 2\) then \(G(I)\) is Gorenstein if and only if either \(I = m\), or \(I\) is the conductor \(C = (R : \overline R)\) of \(R\). If \(e(R) \leq 3\) then \(G(I)\) is Gorenstein if and only if either \(I = m\), or \(I = C\), or \(\delta (I) \leq 1\) and \(\text{length} (R/I) = e(R) - 1\). -- Now suppose \(\dim R = 2\). Then \(p_g (R) = \sup \{\overline e_2 (I) \mid I\) is an \(m\)-primary ideal\} is the geometric genus of \(R\). If \(R\) is pseudo-rational, that is \(p_g (R) = 0\), then the following conditions are equivalent: (i) \(G(I)\) is Gorenstein, (ii) \(\delta (I) \leq 1\) and \(e(I) = 2\cdot {length} (R/I)\). (iii) \(I\) is integrally closed and \(e(I) = 2\cdot {length} (R/I)\). If \(R\) is elliptic, that is \(p_g (R) = 1\) and \(I\) is integrally closed, then \(G(I)\) is Gorenstein if and only if either \(I = m\), or \(\delta (I) \leq 1\) and \(e(I) = 2\cdot {length} (R/I)\). -- If \(p_g (R) = 2\) then either \(e(R) = 2\), or \(\text{emb} (R) = e(R)\) and \(G(m)\) is Gorenstein. -- If \(p_g (R) = 3\) then \(G(m)\) is Gorenstein if and only if \(R\) is a quadratic hypersurface, a quartic hypersurface, or \(\text{emb} (R) = e(R)\).
scientific article; zbMATH DE number 681818
scientific article