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We compare, for smooth monomial projective curves, the Castel- nuovo-Mumfordregularity and the reduction number; we present an example where these twonumbers differ. However, we show they coin- cide for a certain class ofmonomial curves. Furthermore, for smooth monomial curves we prove an inequalitywhich is stronger than the one from the Eisenbud-Goto conjecture.

Castelnuovo-Mumford regularity and reduction number of smooth monomial curves