In the paper under review, the authors study splitting types of vector bundles associated with curves in the complex projective plane from a point of view of the freeness and the near freeness.
Let \(C: f=0\) be a reduced curve in the complex projective plane and \(S = \mathbb{C}[x,y,z]\) the coordinate ring. We define the graded \(S\)-module \(\text{AR}(f) \subset S^{\oplus 3}\) of all relations by \[ \text{AR}(f)_{k} := \bigg\{(a,b,c) \in S_{k}^{\oplus 3} : a \cdot \frac{\partial f}{\partial x} + b \cdot \frac{\partial f}{\partial y} + c \cdot \frac{\partial f}{\partial z} = 0 \bigg\}. \] The module \(\text{AR}(f)\) is isomorphic to the logarithmic derivations killing \(f\) and its sheafification \(E_{C} := \widetilde{\text{AR}(f)}\) is a rank two vector bundle on \(\mathbb{P}^{2}\). Let us denote by \(\text{mdr}(f):= \text{min}\{k \, : \, \text{AR}(f)_{k} \neq 0\}\) and assume that \(\text{mdr}(f) \geq 1\). For a line \(L\) in \(\mathbb{P}^{2}\) the pair of integers \((d_{1}^{L},d_{2}^{L})\) with \(d_{1}^{L} \leq d_{2}^{L}\) such that \(E_{C}|_{L} \simeq \mathcal{O}_{L}(-d_{1}^{L}) \oplus \mathcal{O}_{L}(-d_{2}^{L})\) is called the splitting type of \(E_{C}\) along \(L\). For any line \(L\) in \(\mathbb{P}^{2}\) we set \[ I(C,L) = (d-1)^{2}-d_{1}^{L}d_{2}^{L}. \] Let \(N(f) = \hat{J_{f}}/J_{f}\) with \(J_{f}\) being the Jacobian ideal of \(f\) in \(S\) spanned by the partial derivatives \(f_{x},f_{y},f_{z}\) of \(f\), and \(\hat{J_{f}}\) is the saturation of the ideal \(J_{f}\) with respect to the maximal ideal \(\mathfrak{m} = (x,y,z)\) in \(S\). Let \(\nu(f) = \text{dim} \, N(f)_{[T/2]}\) where \([\cdot]\) denotes the integral part and \(T=3(d-2)\). We know that the curve \(C : f=0\) is free (i.e., the module \(\text{AR}(f)\) is free) iff \(\nu(C) = 0\), and nearly free iff \(\nu(C)=1\). Finally, for a curve \(C\) we define its global Tjurina number by \[ \tau(C) = \sum_{p \in C} \tau(C,p). \] Now we are in a position to formulate the main results of the paper.
Theorem 1. For any line \(L\) in \(\mathbb{P}^{2}\) and any generic line \(L_{0}\) in \(\mathbb{P}^{2}\) the following holds:
1) \(\text{max}(\text{mdr}(f) - \nu(C),0) \leq d_{1}^{L} \leq d_{1}^{L_{0}} \leq \text{min}(\text{mdr}(f),[(d-1)/2])\).
2) \(I(C,L) \geq I(C,L_{0}) = \tau(C) + \nu(C)\).
In particular, the reduced curve \(C: f=0\) in \(\mathbb{P}^{2}\) is free (nearly free, respectively) iff \(I(C,L_{0}) = \tau(C)\,\,\)(\(I(C,L_{0}) = \tau(C)+1\), respectively).
Corollary 1. Let \(c_{E_{C}}(t) = 1+c_{1}(E_{C})t + c_{2}(E_{C})t^{2} \in \mathbb{Z}[t]\) be the Chern polynomial of the vector bundle \(E_{C}\). The curve \(C\) is free (nearly free, respectively) iff there is a line \(L\) in \(\mathbb{P}^{2}\) such that \(c_{2}(E_{C}) = d_{1}^{L}d_{2}^{L}\,\) (\(c_{2}(E_{C}) = d_{1}^{L}d_{2}^{L}+1\), respectively).
Corollary 2. For any reduced curve \(C: f=0\) in \(\mathbb{P}^{2}\) and any line \(L\) in \(\mathbb{P}^{2}\), we have \[ \tau(C) \leq (d-1)^{2} -d_{1}^{L}d_{2}^{L}. \] Theorem 2. Let \(C : f=0\) be a reduced curve of degree \(d\) in the complex projective plane. Then the following conditions are equivalent:
1) the Chern polynomial \(c_{E_{C}}(t) = 1+c_{1}(E_{C})t + c_{2}(E_{C})t^{2} \in \mathbb{Z}[t]\) of the vector bundle \(E_{C}\) has real roots;
2) \(\tau(C) \geq \frac{3}{4}(d-1)^{2}\);
3) \(d_{1}^{L_{0}} \leq \frac{d-1}{2} - \sqrt{\nu(C)}\) for a generic line \(L_{0}\).
Moreover, these properties imply that \(\text{mdr}(f)\) coincides with \(d_{1}^{L_{0}}\) for a generic line \(L_{0}\) and they are implied by either \(\text{mdr}(f) < \frac{d}{4}\) or \(\text{mdr}(f) \leq \frac{d-1}{2} - \sqrt{\nu(C)}\).