New examples of Kobayashi hyperbolic surfaces in \(\mathbb C\mathbb P^3\) (Q2571496)

The authors construct new examples of hyperbolic surfaces in \(\mathbb{P}_3(\mathbb{C})\) of any given degree \(d\) with \(d\geq 8\). Let \(C\) be an algebraic curve in a plane \(H\subset\mathbb{P}_3(\mathbb{C})\). Denote by \(X = \langle C,p\rangle\) the cone formed by the lines passing through a given point \(p\in\mathbb{P}_3(\mathbb{C})\setminus H\) and the points of \(C\). Notice that \(\deg X=\deg C\). Then the main result in this paper can be stated as follows: For arbitrary given integers \(m,n\geq 4\), a generic small deformation of the union \(X= X'\cup X''\) of two generic cones in \(\mathbb{P}_3(\mathbb{C})\) of degree \(m\) and \(n\), respectively, is a hyperbolic surface of degree \(m+ n\). It is noticed that a hyperbolic surface of degree 6 was constructed by \textit{J. Duval} [Math. Ann. 330, 473--476 (2004; Zbl 1071.14045)].