Speaker:* Indranath Sengupta, IIT Gandhinagar, Gujarat, India*
Date/Time: *7 May 2021, 5:30pm IST/ 12:00pm GMT / 8:00am EDT* (joining
time 5:15pm IST).
Google meet link:
https://meet.google.com/kmy-ozko-jcq
Title: *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,
the reduced 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 examples 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.
For more information and links to previous seminars, visit the website of
VCAS:
https://sites.google.com/view/virtual-comm-algebra-seminar