Some Questions on bounds of Betti Numbers of Numerical Semigroup Rings

Abstract: J. Herzog proved in 1969 that the possible values of the first Betti number (minimal number of generators of the defining ideal) of numerical semigroup rings in embedding dimension 3 are 2 (complete intersection and Gorenstein) and 3 (the almost complete intersection).

In a conversation about this work, O. Zariski indicated a possible relation between Gorenstein rings and symmetric value semigroups. In response to that, E.Kunz proved (in 1970) that a one-dimensional, local, Noetherian, reduced, analytically irreducible ring is Gorenstein if and only if its value semigroup is symmetric. A question that remains open to date is whether the Betti numbers (or at least the first Betti number) of every numerical semigroup ring in embedding dimension e, are bounded above by a function of e.

In the years 1974 and 1975, two interesting classes of examples were given by T. Moh and H. Bresinsky. Moh’s example was that of a family of algebroid space curves and Bresinsky’s example was about a family of numerical semigroups in embedding dimension 4, with the common feature that there is no upper bound on the Betti numbers. Therefore, for embedding dimension 4 and above, the Betti numbers (or at least the first Betti number) are not bounded above by some “good” function of the embedding dimension e. A question that emerges is the following: Is there a natural way to generate such numerical semigroups in arbitrary embedding dimension? In this talk, we will discuss some recent observations in this direction, which is a joint work of the author with his collaborators Joydip Saha and Ranjana Mehta.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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