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We prove the following generalization of an example of Hartshorne:
Bounds for the maximal degree of certain Gröbner bases of simplicial toric ideals are given. These bounds are close to the bound stated in Eisenbud-Goto's Conjecture on the Castelnuovo-Mumford regularity.
We compare, for smooth monomial projective curves, the Castel-nuovo-Mumford regularity and the reduction number; we present an example where these two numbers differ. However, we show they coincide for a certain class of monomial curves. Furthermore, for smooth monomial curves we prove an inequality which is stronger than the one from the Eisenbud-Goto… (More)