ABHYANKAR 82nd BIRTHDAY CONFERENCE
SPEAKER: Hema Srinivasan
TITLE: Number of equations defining the ideals of Monomial Curves.
ABSTRACT: Given a monomial curve in $n+1$ dimensional affine space in it's
parametric form $x_i =3D t^{m_i},0\le i\le n$, it can be uniquely determined
by an $n-tuple$ of positive integers ${\bf a } = (a_1, \ldots a_n)$
and a non negative integer $k$ with $m_i =1+k+ \sum_{t=1}^{i} a_t$.
Although the number of generators for the ideal $I$ defining the monomial
curve is unbounded for $n\ge 4$, it is still possible that the number of
generators for the ideal $I({\bf a},k)$, when you fix $\bf a$
is bounded for all $k$. Juergen Herzog and I conjecture that it is eventually
periodic in $k$ with a possible period $\sum _ia_i$ and further all the
Betti numbers of the monomial curve are also eventually periodic in $k$.
It is true in dimension three or less and for all monomial curves given by
arithmetic sequences by a result of Gimenez, Sengupta and myself.
We will discuss the problem and more recent progress in it.