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A hyperplane arrangement $\cA$ is said to be free if the corresponding Jacobian ideal $J_\cA$ is Cohen-Macaulay. If $\cA$ is free then $J_\cA$ is unmixed (i.e. equidimensional). Freeness is an important property, yet its presence is not…

Freeness is an important property of a hypersurface arrangement, although its presence is not well understood. A hypersurface arrangement in $\PP^n$ is free if $S/J$ is Cohen-Macaulay (CM), where $S = K[x_0,\ldots,x_n]$ and $J$ is the…

In the theory of hyperplane arrangements, the most important and difficult problem is the combinatorial dependency of several properties. In this atricle, we prove that Terao's celebrated addition-deletion theorem for free arrangements is…

Let ${\mathcal C}= \bigcup_{i=1}^n C_i \subseteq \mathbb{P}^2$ be a collection of smooth rational plane curves. We prove that the addition-deletion operation used in the study of hyperplane arrangements has an extension which works for a…

We consider the triple $(\mathcal{A},\mathcal{A}',\mathcal{A}^H)$ of hyperplane arrangements and the division of their characteristic polynomials. We show that the freeness of $\mathcal{A}^H$ and the division of $\chi(\mathcal{A};t)$ by…

Hyperplane Arrangements of rank $3$ admitting an unbalanced Ziegler restriction are known to fulfill Terao's conjecture. This long-standing conjecture asks whether the freeness of an arrangement is determined by its combinatorics. In this…

We study the classes of free and plus-one generated hyperplane arrangements. Specifically, we describe how to compute the associated prime ideals of the Jacobian ideal of such an arrangement from its lattice of intersection. Moreover, we…

We study the free path problem, i.e., if we are given two free arrangements of hyperplanes, then we can connect them by free arrangements or not. We prove that if an arrangement $\mathcal{A}$ and $\mathcal{A} \setminus \{H,L\}$ are free,…

In this paper, we show how to compute using Fitting ideals the nonfree locus of the moduli space of arrangements of a rank $3$ simple matroid, i.e., the subset of all points of the moduli space which parametrize nonfree arrangements. Our…

We show that the deletion theorem of a free arrangement is combinatorial, i.e., whether we can delete a hyperplane from a free arrangement keeping freeness depends only on the intersection lattice. In fact, we give an explicit sufficient…

We prove Terao conjecture saying that the freeness is determined by the combinatorics for arrangements of 13 lines in the complex projective plane and that the property of being nearly free is combinatorial for line arrangements of up to 12…

Let $I\subset \mathbb C[x,y,z]$ be an ideal of height 2 and minimally generated by three homogeneous polynomials of the same degree. If $I$ is a locally complete intersection we give a criterion for $\mathbb C[x,y,z]/I$ to be arithmetically…

We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong…

In this article we describe two new characterizations of freeness for hyperplane arrangements via the study of the generic initial ideal and of the sectional matrix of the Jacobian ideal of arrangements.

In this paper, we introduce the notion of a complete hypertetrahedral arrangement $\mathcal{A}$ in $\mathbb{P}^{n}$. We address two basic problems. First, we describe the local freeness of $\mathcal{A}$ in terms of smaller complete…

Let A = (A,V) be a complex hyperplane arrangement and let L(A) denote its intersection lattice. The arrangement A is called supersolvable, provided its lattice L(A) is supersolvable, a notion due to Stanley. Jambu and Terao showed that…

Athanasiadis studied arrangements obtained by adding shifted hyperplanes to the braid arrangement. Similarly, Bailey studied arrangements obtained by adding tilted hyperplanes to the braid arrangement. These two kinds of arrangements are…

The discriminantal arrangement is the space of configurations of $n$ hyperplanes in generic position in a $k$ dimensional space (see \cite{MS}). Differently from the case $k=1$ in which it corresponds to the well known braid arrangement,…

A central question in arrangement theory is to determine whether the characteristic polynomial $\Delta_q$ of the algebraic monodromy acting on the homology group $H_q(F(\mathcal{A}),\mathbb{C})$ of the Milnor fiber of a complex hyperplane…

Let $\mathcal{H}$ be a hyperplane arrangement in $\mathbb{CP}^n$. We define a quadratic form $Q$ on $\mathbb{R}^{\mathcal{H}}$ that is entirely determined by the intersection poset of $\mathcal{H}$. Using the Bogomolov-Gieseker inequality…