Article

# Genus of numerical semigroups generated by three elements

(Impact Factor: 0.6). 04/2011; 358. DOI: 10.1016/j.jalgebra.2012.02.010
Source: arXiv

ABSTRACT

Let H = < a, b, c > be a numerical semigroup generated by three elements and let R = k vertical bar H vertical bar be its semigroup ring over a field k. We assume that H is not symmetric and assume that the defining ideal of R is defined by maximal minors of the matrix ((X alpha)(Y beta)(Z gamma)(Y beta')(Z gamma')(X alpha')). Then we will show that the genus of H is determined by the Frobenius number F(H) and alpha beta gamma or alpha'beta'gamma' In particular, we show that H is pseudo-symmetric if and only if alpha beta gamma = 1 or alpha'beta'gamma' = 1. Also, we will give a simple algorithm to get all the pseudo-symmetric numerical semigroups H = < a. b, c > with given Frobenius number.

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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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