Article
Genus of numerical semigroups generated by three elements
Journal of Algebra (Impact Factor: 0.6). 04/2011; 358. DOI: 10.1016/j.jalgebra.2012.02.010
Source: arXiv
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 "By the result of Kleiman mentioned above, e(r * (JM (X)), r * (N D )) = e(JM (X), N D ). By results of Watanabe et al [17], δ(X l ) = l. Hence the Milnor number is 2δ = 2l, so e(JM (X l ), N D (X l )) = µ−2, and so e(JM (X l ), N D (X l )) plus the multiplicity over Y of the polar curve of N D (X l ) is µ − 2+ m + 1 = µ + m − 1, which is the multiplicity over Y of the relative polar curve of a smoothing of the family of space curves X l as predicted. "
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ABSTRACT: We continue the development of the study of the equisingularity of isolated singularities, in the determinantal case. This version of the paper includes a substantial amount of new material (76% larger). The new material introduces the idea of the landscape of singularity, which includes the allowable deformations of the singularity and associated structure useful for equisingularity questions. Fixing a presentation matrix M of a determinantal singularity means viewing the singularity as a section via M of the set of matrices of a given or smaller rank. Varying M gives the allowable deformations of X. This version also includes a description of the conormal varieties of the rank singularities, which is applied to our machinery. There is also an example of a determinantal singularity which is a member of two Whitney equisingular families, whose generic elements have topologically distinct smoothings. This example shows that it is impossible to find an invariant which depends only on an analytic space $X$ with an isolated singularity, whose value is independent of parameter for all Whitney equisingular deformations of $X$, and which is determined by the geometry of a smoothing of $X$. 
 "Corollary 4.2 has been reported by H. Nari [14] (see [15] also) at the 32nd Symposium on Commutative Algebra in Japan (Hayama, 2010). Our research is independent of [14] [15]. "
Dataset: Almost Gorenstein Rings
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ABSTRACT: The notion of almost Gorenstein ring given by Barucci and Fr{\"o}berg \cite{BF} in the case where the local rings are analytically unramified is generalized, so that it works well also in the case where the rings are analytically ramified. As a sequel, the problem of when the endomorphism algebra $\m : \m$ of $\m$ is a Gorenstein ring is solved in full generality, where $\m$ denotes the maximal ideal in a given CohenMacaulay local ring of dimension one. Characterizations of almost Gorenstein rings are given in connection with the principle of idealization. Examples are explored. 
 "Corollary 4.2 has been reported by H. Nari [14] (see [15] also) at the 32nd Symposium on Commutative Algebra in Japan (Hayama, 2010). Our research is independent of [14] [15]. "
Article: Almost Gorenstein rings
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ABSTRACT: The notion of almost Gorenstein ring given by Barucci and Fr{\"o}berg \cite{BF} in the case where the local rings are analytically unramified is generalized, so that it works well also in the case where the rings are analytically ramified. As a sequel, the problem of when the endomorphism algebra $\m : \m$ of $\m$ is a Gorenstein ring is solved in full generality, where $\m$ denotes the maximal ideal in a given CohenMacaulay local ring of dimension one. Characterizations of almost Gorenstein rings are given in connection with the principle of idealization. Examples are explored.