# WEEKLY ACTIVITIES

Depth functions of symbolic powers of homogeneous ideals
 Speaker: Ngo Viet TrungTime: 9h00, Wednesday, January 24, 2018Location: 611-612, Building A6, Institute of Mathematics, 18 Hoang Quoc Viet, Cau Giay, HanoiAbstract:  We investigate the behavior of the function \$depth R/I^{(t)}\$, \$t ge 1\$, where \$I^{(t)}\$ is the \$t\$-th symbolic power of a homogeneous ideal \$I\$ in a polynomial ring \$R\$. Our first result shows that if \$I\$ is an arbitrary squarefree monomial ideal, the function \$depth R/I^{(t)}\$ is almost non-increasing in the sense that \$depth R/I^{(s)} ge depth R/I^{(t)}\$ for all \$s ge 1\$ and \$t in E(s)\$, where\$\$E(s) = {t in NN|  t = c(s-1)+r text{for} c ge 1, 1 le r le s-1, r le c}\$\$(e.g. \$depth R/I ge depth R/I^{(t)}\$ for \$t ge 2\$, \$depth R/I^{(2)} ge depth R/I^{(t)}\$ for \$t ge 3\$, \$depth R/I^{(3)} ge depth R/I^{(t)}\$ for \$t ge 5\$, \$depth R/I^{(4)} ge depth R/I^{(t)}\$ for \$t = 7,8\$ or \$t ge 10\$, etc.) We also show that there exist squarefree monomial ideals \$I\$ such that \$depth R/I^{(s)} < depth R/I^{(t)}\$ if \$t otin E(s)\$.If \$I\$ is an arbitrary monomial ideal, we know that the function \$depth R/I^{(t)}\$ is asymptotically periodic (i.e. periodic for \$t gg 0\$). However, it was not clear whether this function is always convergent. We show that if \$I^{(t)}\$ is integrally closed for \$t gg 0\$ (e.g. if \$I\$ is squarefree), then the function \$depth R/I^{(t)}\$ is convergent with  \$lim_{t to infty}depth R/I^{(t)} = liminf_{t to infty}depth R/I^{(t)}.\$ On the other hand, we are able to construct examples of monomial ideals for which the function \$depth R/I^{(t)}\$ is not  convergent. Our last result shows that for any asymptotically periodic positive numerical function \$phi(t)\$, there exist a polynomial ring \$R\$ and a homogeneous ideal \$I\$ such that \$depth R/I^{(t)} = phi(t)\$ for all \$t ge 1\$. It is not known whether there exists any function \$depth R/I^{(t)}\$ which is not asymptotically periodic at all.