On free and plus-one generated curves arising from free curves by addition-deletion of a line (Q6660260)

In this important paper the author studies the homological behaviour of reduced plane curves under addition/deletion technique. Let \(S = \mathbb{C}[x,y,z]\) and consider \(C : f=0\) a reduced plane curve of degree \(d\geq 3\) given by \(f \in S\). We denote by \(J_{f}\) the Jacobian ideal generated by the partial derivatives of \(f\) and we define \(M(f) = S/J_{f}\), the Milnor algebra of \(f\). Consider the graded \(S\)-module of Jacobian syzygies of \(f\), namely \N\[\ND_{0}(f) = \{ (a,b,c) \in S^{3} : a\cdot \partial_{x}f + b\cdot \partial_{y}f + c\cdot \partial_{z}f=0\}.\N\]\NThe graded Milnor algebra \(M(f)\) has a minimal resolution of the form \N\[\N0\rightarrow F_{3} \rightarrow F_{2} \rightarrow F_{1} \rightarrow F_{0},\N\]\Nwhere \(F_{0} = S\), \(F_{1} = S^{3}(1-d)\) and the morphism \(F_{1} \rightarrow F_{0}\) is given by \N\[\N(a,b,c) \mapsto a\cdot \partial_{x}f + b\cdot \partial_{y}f + c\cdot \partial_{z}f.\N\]\NWith the above notation, the graded \(S\)-module \(D_{0}(f)\) has the following minimal resolution \N\[\N0 \rightarrow F_{3}(d-1) \rightarrow F_{2}(d-1).\N\]\NWe say that \(C : f=0\) is an \(m\)-syzygy curve if the module \(F_{2}\) has rank \(m\) and then the module \(D_{0}(f)\) is generated by \(m\) homogeneous syzygies \(r_{1}, \ldots, r_{m}\) of degrees \(d_{j} = \deg \, (r_{j})\) ordered such that\N\[\N1 \leq d_{1} \leq d_{2} \leq \cdots \leq d_{m}.\N\]\NThe degrees \((d_{1}, \dots, d_{m})\) are called the exponents of \(C\). A reduced plane curve \(C\) is free when \(m=2\) since in that scenario the \(S\)-module \(D_{0}(f)\) is free of rank \(2\). Moreover, a reduced plane curve \(C\) is plus-one generated if \(C\) is \(3\)-syzygy with \(d_{1}+d_{2}=d\).\N\NThe first main contribution of the author is the following generalization of Abe's result on the addition/deletion for free line arrangements to the case of reduced plane curves admitting arbitrary singularities.\N\NTheorem 1. Let \(C : f=0\) be a reduced curve in \(\mathbb{P}^{2}_{\mathbb{C}}\) and let \(L\) be a line such that \(L\) is not an irreducible component of \(C\). Consider \(C' = C \cup L\) and assume that \(C'\) is free, then \(C\) is either free or plus-one generated.\N\NTheorem 2. Let \(C : f=0\) be a reduced curve in \(\mathbb{P}^{2}_{\mathbb{C}}\) and let \(L\) be a line such that \(L\) is not an irreducible component of \(C\). Consider \(C ' = C \cup L\) and assume that \(C\) is free, then \(C'\) is either free or plus-one generated.\N\NIn order to show the above results the author introduces new invariants of singularities. For an isolated hypersurface singularity \((X,0)\) we define \N\[\N\epsilon(X,0) = \mu(X,0) - \tau(X,0),\N\]\Nwhere \(\mu\) denotes the Milnor number and \(\tau\) denotes the Tjurina number of \((X,0)\). Now for curves \(C_{1}, C_{2}\) and \(C = C_{1} \cup C_{2}\) and a point \(q \in C_{1} \cap C_{2}\) one sets \N\[\N\epsilon(C_{1}, C_{2})_{q} = \epsilon(C_{1} \cup C_{2},q) - \epsilon(C_{1}, q),\N\] \Nand then we define the invariant of the pair \((C_{1}, C_{2})\) by \N\[\N\epsilon(C_{1},C_{2}) = \sum_{q \in C_{1} \cap C_{2}}\epsilon(C_{1},C_{2})_{q}.\N\]\NThe understanding of this invariant seems to be a very crucial thing, for instance we have the following fundamental conjecture.\N\NConjecture 1. For any two curves \(C_{1}\), \(C_{2}\) without common irreducible components and any intersection point \(q \in C_{1} \cap C_{2}\), one has \N\[\N\epsilon(C_{1}, C_{2})_{q} \geq 0.\N\]\NObviously this conjecture holds if singularities are quasi-homogeneous.\N\NA very important application of the results obtained by the author is devoted to cuspidal curves.\N\NConsider an affine plane curve \(X : g(x,y)=0\) given by a reduced polynomial \(g \in R = \mathbb{C}[x,y]\) of degree \(d\). Then the projective closure \(\overline{X}\) of \(X\) is the curve in \(\mathbb{P}^{2}_{\mathbb{C}}\) given by \[\Nf(x,y,z) = z^{d}g(x/z , y/z).\N\]\NRecall that a contractible irreducible affine plane curve \(X\) is given in a suitable coordinate system in \(\mathbb{C}^{2}\) by \(u^{p}-v^{q}=0\) for some relatively prime integers \(p,q \geq 1\). In particular, \(X\) has at most a unique singular point \(a\) which is a cusp of type \((p,q)\).\N\NTheorem 3. With the notation as above, assume that \(X\) is irreducible and contractible and has a cusp of type \((p,q)\) such that either \(p\) or \(q\) is relatively prime to \(d+1\), where \(d =\deg X = \deg \overline{X}\). Then the projective closure \(\overline{X}\) of the affine plane curve \(X\) is either free or plus-one generated.