For a given a local ring (A, m), we study the fiber cone of ideals in A with analytic spread one. In this case, the fiber cone has a structure as a module over its Noether normalization which is a polynomial ring in one variable over the residue field. One may then apply the structure theorem for modules over a principal domain to get a complete description of the fiber cone as a module. We analyze this structure in order to study and characterize in terms of the ideal itself the arithmetical properties and other numerical invariants of the fiber cone as multiplicity, reduction number or Castelnuovo-Mumford regularity.

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In this paper we describe the structure of the tangent cone of a numerical semigroup ringwith multiplicity e (as a module over the Noether normalization determined by the fiber cone of the ideal generated by t e ) in terms of some classical invariants of the corresponding numerical semigroup. Explicit computations are also made by using the GAP system.

Abstract. We exhibit a set of minimal generators of the defining ideal of the Rees Algebra associated with the ideal of three bivariate homogeneous polynomials parametrizing a proper rational curve in projective plane, having a minimal syzygy of degree 2.

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