SINGULARITY OF MONOMIAL CURVES
Abstract
This work is centered around the question "How singular is a point on an algebraic or analytic variety?" A systematic way is introduced to solve this problem in the case of monomial curves, based on structure theorems for the derivation module and the canonical module of the curves in question. This approach leads to formulas for monomial space curves, which extend a result of J. Sylvester (1884) on the singularity degree of numerical semigroups generated by two elements. We then extend the computation of the Hilbert-Samuel polynomial, carried out by E. Matlis (1973) for 1-dimensional noetherian local Cohen-Macaulay rings of embedding dimension 2, from the case of monomial plane curves to some cases of monomial space curves. Difficulties turn up in embedding dimension 4, as it has been shown by H. Bresinsky (1975) that there exist monomial curves of any embedding dimension (GREATERTHEQ) 4 requiring arbitrarily large numbers of generators for their defining ideals (and having arbitrarily large types). We improve this and show that for all 4 (LESSTHEQ) s (ELEM) and for all 1 (LESSTHEQ) r (ELEM) there exists a monomial curve of embedding dimension s and type r. We are still able to obtain formulas of the kind mentioned above for Gorenstein monomial curves of embedding dimension 4. We then consider monomial projective curves; again, we first specialize to the case of space curves. The formulas derived for the affine curves enable us to write down explicitly the Hilbert polynomial of all monomial curves in ('3). Consideration of monomial curves in all ('n), n (ELEM) , leads us to a conjecture on a bound for their arithmetic genus depending on their degree, generalizing a bound obtained by G. Castelnuovo (1893) for smooth curves in ('3) over an algebraically closed field. Using the computation of the Hilbert polynomial and a construction due to H. Bresinsky and B. Renschuch (1980), we are able to prove the conjecture in a special case.
Degree
Ph.D.
Subject Area
Mathematics
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