Linearity Defects of Face Rings

Let $S = K[x_1, ..., x_n ]$ be a polynomial ring over a field $K$, and $E = K$ an exterior algebra. The "linearity defect" $ld_E(N)$ of a finitely generated graded $E$-module $N$ measures how far $N$ departs from "componentwise linear". It is known that $ld_E(N) < \infty$ for all $N$. But the value can be arbitrary large, while the similar invariant $ld_S(M)$ for an $S$-module $M$ is alway at most $n$. We show that if $I_\Delta$ (resp. $J_\Delta$) is the squarefree monomial ideal of $S$ (resp. $E$) corresponding to a simplicial complex $\Delta$ on ${1, >..., n}$, then $ld_E(E/J_\Delta) = ld_S(S/I_\Delta)$. Moreover, except some extremal cases, $ld$ is a topological invariant of the Alexander dual $\Delta^\vee$ of $\Delta$. We also show that, when $n > 3$, $ld_E(E/J_\Delta) = n-2$ (this is the largest possible value) if and only if $\Delta$ is an $n$-gon.Comment: 19pages. Section 5 is largely revised; particularly, the proof of Theorem 5.1 is simplified. To appear in J. Algebr

Alexander duality and Stanley depth of multigraded modules

We apply Miller's theory on multigraded modules over a polynomial ring to the study of the Stanley depth of these modules. Several tools for Stanley's conjecture are developed, and a few partial answers are given. For example, we show that taking the Alexander duality twice (but with different "centers") is useful for this subject. Generalizing a result of Apel, we prove that Stanley's conjecture holds for the quotient by a cogeneric monomial ideal.Comment: 18 pages. We have removed Lemma 2.3 of the previous version, since the proof contained a gap. This deletion does not affect the main results, while we have revised argument a little (especially in Sections in 2 and 3

Normal cyclic polytopes and cyclic polytopes that are not very ample

Let $d$ and $n$ be positive integers with $n \geq d + 1$ and $\tau_{1}, ..., \tau_{n}$ integers with $\tau_{1} < ... < \tau_{n}$. Let C_{d}(\tau_{1}, ..., \tau_{n}) \subset \RR^{d} denote the cyclic polytope of dimension $d$ with $n$ vertices $(\tau_{1},\tau_{1}^{2},...,\tau_{1}^{d}), ..., (\tau_{n},\tau_{n}^{2},...,\tau_{n}^{d})$. We are interested in finding the smallest integer $\gamma_{d}$ such that if $\tau_{i+1} - \tau_{i} \geq \gamma_{d}$ for $1 \leq i < n$, then $C_{d}(\tau_{1}, ..., \tau_{n})$ is normal. One of the known results is $\gamma_{d} \leq d (d + 1)$. In the present paper a new inequality $\gamma_{d} \leq d^{2} - 1$ is proved. Moreover, it is shown that if $d \geq 4$ with $\tau_{3} - \tau_{2} = 1$, then $C_{d}(\tau_{1}, ..., \tau_{n})$ is not very ample.Comment: 17 page