Monomial ideals under ideal operations

In this paper, we show for a monomial ideal $I$ of $K[x_1,x_2,\ldots,x_n]$ that the integral closure \ol{I} is a monomial ideal of Borel type (Borel-fixed, strongly stable, lexsegment, or universal lexsegment respectively), if $I$ has the same property. We also show that the $k^{th}$ symbolic power $I^{(k)}$ of $I$ preserves the properties of Borel type, Borel-fixed and strongly stable, and $I^{(k)}$ is lexsegment if $I$ is stably lexsegment. For a monomial ideal $I$ and a monomial prime ideal $P$, a new ideal $J(I, P)$ is studied, which also gives a clear description of the primary decomposition of $I^{(k)}$. Then a new simplicial complex $_J\bigtriangleup$ of a monomial ideal $J$ is defined, and it is shown that $I_{_J\bigtriangleup^{\vee}} = \sqrt{J}$. Finally, we show under an additional weak assumption that a monomial ideal is universal lexsegment if and only if its polarization is a squarefree strongly stable ideal.Comment: 18 page

Stanley depth of monomial ideals with small number of generators

For a monomial ideal $I\subset S=K[x_1,...,x_n]$, we show that \sdepth(S/I)\geq n-g(I), where $g(I)$ is the number of the minimal monomial generators of $I$. If $I=vI'$, where $v\in S$ is a monomial, then we see that \sdepth(S/I)=\sdepth(S/I'). We prove that if $I$ is a monomial ideal $I\subset S$ minimally generated by three monomials, then $I$ and $S/I$ satisfy the Stanley conjecture. Given a saturated monomial ideal $I\subset K[x_1,x_2,x_3]$ we show that \sdepth(I)=2. As a consequence, \sdepth(I)\geq \sdepth(K[x_1,x_2,x_3]/I)+1 for any monomial ideal in $I\subset K[x_1,x_2,x_3]$.Comment: 7 pages. submitted to Central European Journal of Mathematic