New bounds on Castelnuovo-Mumford regularity of monomial curves

Time: 9:30 -- 11: 00, Jan 21, 2026

Venue: Room 612, A6, Institute of Mathematics-VAST

Abstract: A monomial curve $C$ is defined by a sequence of coprime integers $0= a_0 < a_1 < \cdots < a_k =: d$. A gap of this sequence is $a_{i+1} - a_i - 1$. Gruson-Lazarsfeld-Peskine bound (1983) says that $\reg(C) \le d - k +2$, which is equal to the sum of all gaps plus 2. Lvovsky (1996) showed that it is enough to take the sum of two largest gaps plus 2.

In this talk,   under some specific conditions, we give several new bounds which are better than Lvovsky’s bound.  Our method relies on the study of Apery sets  and Frobenius numbers. From this we can give new criteria to check the (arithmetically) Cohen-Macaulay and Buchbaum property of $C$. Algorithms are provided to check these properties as well as to compute $\reg(C)$ and other invariants.  We also give an application to studying the structure of sumsets.

This is a joint work with D. Q. Tien.