On the intersection of two plane curves (Q5929466)

Let \(d\geq 3\), \(m\geq 1\) be integers, let \(|{\mathcal O}(d)|\) be the linear system of the curves of degree \(d\) in \(\mathbb{P}^2\) and let \(W_k: =\{(C,D): C\in|{\mathcal O}(m)|\) is irreducible, \(D\in|{\mathcal O}(d) |\) is smooth and \(\text{card} (C\cap D)=k\}\). Let \(p:W_k\to |{\mathcal O}(m) |\), \(q:W_k\to |{\mathcal O}(d) |\) be the canonical projection and let \(i(d,m): =\min\{k:q\) is dominant\}. -- Using a deformation-theoretical argument, \textit{G. Xu} [Am. J. Math. 118, No. 3, 611-620 (1996; Zbl 0872.14023)] proved that \(i(d,m) \geq d-2\). This result is related to a conjecture of S. Kobayashi and M. Zaidenberg concerning the hyperbolicity of the complement of a hypersurface in a projective space. Moreover, \(i(d,1)= d-2\) (take a bitangent line for \(d\geq 4)\) and \(i(d,m)= d-2\) for \(m\geq d\).NEWLINENEWLINENEWLINEIn the paper under review, the author shows that, for \(d>m\), NEWLINE\[NEWLINEi(d,m) \geq\min (dm-m(m+3)/2,2dm-2m^2-2)NEWLINE\]NEWLINE and conjectures that \(i(d,m)= dm-m(m+3)/2\). His proof is based on bounding the dimension of the tangent space to the fibre of \(p\) over an irreducible \(C\in|{\mathcal O}(m)|\) at a point \((C,D)\). In order to obtain this bound, he first resolves the singularities of \(C\) situated on \(C\cap D\) by successively blowing-up \(\mathbb{P}^2\) and then he works on the resulting surface. NEWLINENEWLINENEWLINEThe author's second main result is an analogue of G. Xu's result on the rational ruled surfaces \(\mathbb{F}_n\). His proof is of a quite different nature: It is based upon degeneration and induction using the theory of limit linear series of \textit{D. Eisenbud} and \textit{J. Harris} [Invent. Math. 85, 337-371 (1986; Zbl 0598.14003)]. In particular, he obtains a new proof of G. Xu's result on \(\mathbb{P}^2\).