arXiv subjects

Antonio Macchia

Publications and source records attributed to Antonio Macchia.

21 records · Page 2Linked to original sources

arXiv2014

Simplicial complexes of whisker type

Let $I\subset K[x_1,\ldots,x_n]$ be a zero-dimensional monomial ideal, and $Δ(I)$ be the simplicial complex whose Stanley--Reisner ideal is the polarization of $I$. It follows from a result of Soleyman Jahan that $Δ(I)$ is shellable. We give a new short proof of this fact by providing an explicit shelling. Moreover, we show that $Δ(I)$ is even vertex decomposable. The ideal $L(I)$, which is defined to be the Stanley--Reisner ideal of the Alexander dual of $Δ(I)$, has a linear resolution which is cellular and supported on a regular CW-complex. All powers of $L(I)$ have a linear resolution. We compute $\mathrm{depth}\ L(I)^k$ and show that $\mathrm{depth}\ L(I)^k=n$ for all $k\geq n$.

math.AC↗

arXiv2014

On the ideal of orthogonal representations of a graph in $\mathbb{R}^2$

In this paper, we study orthogonal representations of simple graphs $G$ in $\mathbb{R}^d$ from an algebraic perspective in case $d = 2$. Orthogonal representations of graphs, introduced by Lovász, are maps from the vertex set to $\mathbb{R}^d$ where non-adjacent vertices are sent to orthogonal vectors. We exhibit algebraic properties of the ideal generated by the equations expressing this condition and deduce geometric properties of the variety of orthogonal embeddings for $d=2$ and $\mathbb{R}$ replaced by an arbitrary field. In particular, we classify when the ideal is radical and provide a reduced primary decomposition if $\sqrt{-1} \not\in K$. This leads to a description of the variety of orthogonal embeddings as a union of varieties defined by prime ideals. In particular, this applies to the motivating case $K = \mathbb{R}$.

math.AC↗

arXiv2013

The arithmetical rank of the edge ideals of graphs with whiskers

We consider the edge ideals of large classes of graphs with whiskers and for these ideals we prove that the arithmetical rank is equal to the big height. Then we extend these results to other classes of squarefree monomial ideals, generated in any degree, proving that the same equality holds.

math.AC↗