In the paper under review, the authors study properties of vector bundles that are associated with nearly free curves and arrangements in the projective plane. Let \(C \subset \mathbb{P}^{2}_{\mathbb{C}}:=\mathbb{P}^{2}\) be a reduced curve of degree \(d\) defined as the zero locus of a homogeneous polynomial, \(f=0\). Let \(\mathcal{I}_{\nabla f}\) be the Jacobian ideal of \(f\) defined by \(\nabla f = [\partial_{x}\, f, \partial_{y} \, f, \partial_{z}\, f]\). Now we define \(\mathcal{T}_{C}\), a rank two reflexive sheaf, by the following short exact sequence \[ 0 \rightarrow \mathcal{T}_{C} \rightarrow \mathcal{O}^{3}_{\mathbb{P}^{2}} \rightarrow \mathcal{I}_{\nabla f}(d-1) \rightarrow 0. \]
We say that a reduced curve \(C \subset \mathbb{P}^{2}\) is nearly free with the exponents \(\mathrm{nexp}(C)= (a,b) \in \mathbb{N}^{2}\) with \(a\leq b\) if the associated vector bundle \(\mathcal{T}_{C}\) has a resolution of the form \[ 0 \rightarrow O_{\mathbb{P}^{2}}(-b-1) \rightarrow \mathcal{O}_{\mathbb{P}^{2}}(-a) \oplus \mathcal{O}_{\mathbb{P}^{2}}(-b)^{2} \rightarrow \mathcal{T}_{C} \rightarrow 0. \] The vector bundle \(\mathcal{T}_{C}\) associated with a nearly free curve \(C\) is called a nearly free vector bundle. The main aim of the paper under review is to understand the fundamental properties of nearly free vector bundles. It is worth emphasizing that this paper is strongly motivated by [T. Abe and A. Dimca, Int. J. Math. 29, No. 8, Article ID 1850055, 20 p. (2018; Zbl 1394.14020)]. The first result gives the following description of the nearly freeness.
Theorem. \(\mathcal{T}_{C}\) is nearly free with \(\mathrm{nexp}(C) = (a,b) \in \mathbb{N}^{2}\) if and only if there exists a point \(p \in \mathbb{P}^{2}\) such that \(\mathcal{T}_{C}\) splits in the following exact sequence \[ 0 \rightarrow \mathcal{O}_{\mathbb{P}^{2}}(-a) \rightarrow \mathcal{T}_{C} \rightarrow \mathcal{I}_{p}(-b+1) \rightarrow 0. \] The point \(p\) is called the jumping point for \(\mathcal{T}_{C}\). It has turned out recently that the properties of jumping points of nearly free reduced curves can be described by the so-called Bourbaki ideals of the syzygy modules \(\mathrm{AR}(f)\) as we can read in [A. Dimca and G. Sticlaru, Publ. Mat., Barc. 64, No. 2, 513–542 (2020; Zbl 1441.14140)].
Then the authors focus on the addition-deletion results applied to free arrangements that are strictly inspired by the aforementioned paper by Abe and Dimca. The first result in that context, Proposition 3.1, tells us how to obtain a nearly free arrangement of lines when we remove a special line from a free arrangement of lines. As we can read in the proof of that statement
“It shows that we can construct a nearly free arrangement with exponents \((a,b)\) by deleting a line in a free arrangement with the same exponents when this line is passing through exactly \(b\) triple points. That is why this process is known as deletion.”
Remark 3.4 shows that it is not always possible to delete a line from a given free arrangement to find a nearly free. As the authors explain, if we take the Hesse arrangement of \(12\) lines, then it is not possible to find a nearly free arrangement since there is no line such that it contains \(7\) triple points.
Let us consider the following example.
Let \(\mathcal{F}_{n}\) be the \(n\)-th Fermat arrangement of \(3n\) lines defined by the factors of \(Q_{n}(x,y,z) = (x^{n}-y^{n})(y^{n}-z^{n})(z^{n}-x^{n})\) with \(n\geq 4\). It is well known that this arrangement is free with the exponents \((n+1, 2n-2)\). Now we remove one line, and due to the perfectly symmetric picture, let us remove line \(\ell\) given by \(x-y\). Then we obtain an arrangement of \(3n-1\) lines which turns out to be nearly free with \(\mathrm{nexp}(\mathcal{F}_{n}) = (n+1, 2n-2)\). Observe that \(\ell\) is passing through exactly \(n\) triple points and, of course, \(n < b = 2n-2\).
Let us recall that the ancestor of Proposition 3.1 therein is the following result of Abe and Dimca.
Theorem. Let \(\mathcal{A}\) be an arrangement in \(\mathbb{P}^{2}\), \(H \in \mathcal{A}\) and \(\mathcal{B} :=\mathcal{A} \setminus \{H\}\). Also, let \(d_{1} \leq d_{2}\) be two non-negative integers. Then any two of the following conditions implies the third:

1)

\(\mathcal{A}\) is free with the exponents \((d_{1}, d_{2})\),

2)

\(\mathcal{B}\) is nearly free with \(\mathrm{nexp}(\mathcal{B}) = (d_{1}, d_{2})\),

3)

\(|\mathcal{A}^{H}|=d_{1}\).

Then the authors present an addition-type statement that involves adding one line passing through \(a-1\) triple points to a nearly free arrangement \(\mathcal{A}\) with \(\mathrm{nexp}(\mathcal{A})=(a+1,b+1)\) to obtain a free arrangement.
In the last part of the paper, the authors study the behavior of the jumping points for nearly free arrangement of lines.