Rank Selection and Depth Conditions for Balanced Simplicial Complexes

We prove some new rank selection theorems for balanced simplicial complexes. Specifically, we prove that if a balanced simplicial complex satisfies Serre's condition $(S_{\ell})$ then so do all of its rank selected subcomplexes. We also provide a formula for the depth of a balanced simplicial complex in terms of reduced homologies of its rank selected subcomplexes. By passing to a barycentric subdivision, our results give information about Serre's condition and the depth of any simplicial complex. Our results extend rank selection theorems for depth proved by Stanley, Munkres, and Hibi.

F-THRESHOLDS OF GRADED RINGS

The $a$-invariant, the $F$-pure threshold, and the diagonal $F$-threshold are three important invariants of a graded $K$-algebra. Hirose, Watanabe, and Yoshida have conjectured relations among these invariants for strongly $F$-regular rings. In this article, we prove that these relations hold only assuming that the algebra is $F$-pure. In addition, we present an interpretation of the $a$-invariant for $F$-pure Gorenstein graded $K$-algebras in terms of regular sequences that preserve $F$-purity. This result is in the spirit of Bertini theorems for projective varieties. Moreover, we show connections with projective dimension, Castelnuovo–Mumford regularity, and Serre’s condition $S_{k}$. We also present analogous results and questions in characteristic zero.

BLOWUPS AND FIBERS OF MORPHISMS

Our object of study is a rational map defined by homogeneous forms $g_{1},\ldots ,g_{n}$, of the same degree $d$, in the homogeneous coordinate ring $R=k[x_{1},\ldots ,x_{s}]$ of $\mathbb{P}_{k}^{s-1}$. Our goal is to relate properties of $\unicode[STIX]{x1D6F9}$, of the homogeneous coordinate ring $A=k[g_{1},\ldots ,g_{n}]$ of the variety parameterized by $\unicode[STIX]{x1D6F9}$, and of the Rees algebra ${\mathcal{R}}(I)$, the bihomogeneous coordinate ring of the graph of $\unicode[STIX]{x1D6F9}$. For a regular map $\unicode[STIX]{x1D6F9}$, for instance, we prove that ${\mathcal{R}}(I)$ satisfies Serre’s condition $R_{i}$, for some $i>0$, if and only if $A$ satisfies $R_{i-1}$ and $\unicode[STIX]{x1D6F9}$ is birational onto its image. Thus, in particular, $\unicode[STIX]{x1D6F9}$ is birational onto its image if and only if ${\mathcal{R}}(I)$ satisfies $R_{1}$. Either condition has implications for the shape of the core, namely, $\text{core}(I)$ is the multiplier ideal of $I^{s}$ and $\text{core}(I)=(x_{1},\ldots ,x_{s})^{sd-s+1}.$ Conversely, for $s=2$, either equality for the core implies birationality. In addition, by means of the generalized rows of the syzygy matrix of $g_{1},\ldots ,g_{n}$, we give an explicit method to reduce the nonbirational case to the birational one when $s=2$.