New Examples of Hyperbolic Octic Surfaces in P

We show that a general small deformation of the union of two general cones in P of degree ≥ 4 is Kobayashi hyperbolic. Hence we obtain new examples of hyperbolic surfaces in P of any given degree d ≥ 8. It was shown by Clemens [2] that a very general surface Xd of degree d ≥ 5 in P 3 has no rational curves; G. Xu [11] showed that Xd also has no elliptic curves (and in fact has no curves of genus ≤ 2), i.e. Xd is algebraically hyperbolic. According to the Kobayashi Conjecture, Xd must even be Kobayashi hyperbolic, and hence does not possess non-constant entire curves C → Xd. The latter property is known to be open in the Hausdorff topology on the projective space of degree d surfaces [12], and it does hold for a very general surface of degree at least 15 [3, 5, 7]. Examples of hyperbolic surfaces in P have been given by many authors; see the references in our previous papers [9, 10], where more examples are given. So far, the minimal degree of known examples is 8; the first family of examples of degree 8 hyperbolic surfaces in P was found by Fujimoto [6] and independently by Duval [4]. In [10], we introduced a deformation method, which we used to construct a new degree 8 hyperbolic surface. In this note, we use a simple form of our deformation method to construct another degree 8 example, which is a deformation of the union of two quartic cones. Actually, our construction provides examples in any degree d ≥ 8. It follows from an observation by Mumford and Bogomolov, proved in [8], that every surface in P of degree at most 4 contains rational or elliptic curves. However, it remains unknown whether there exist hyperbolic surfaces in P in the remaining degrees d = 5, 6, 7. To describe our examples, we consider an algebraic curve C in a plane H ⊂ P. We let 〈C, p〉 denote the cone formed by the union of lines through a fixed point p ∈ P \ H and points of C. By a cone in P, we mean a cone of the form X = 〈C, p〉. If C ′ = X ∩H , where H ′ is an arbitrary plane not passing through p, then we also have X = 〈C , p〉. We observe that deg X = deg C. Theorem. For m, n ≥ 4, a general small deformation of the union X = X ′ ∪ X ′′ of two general cones in P of degrees m and n, respectively, is a hyperbolic surface of degree m+n. Proof. Let X = X ′ ∪ X ′′ ⊂ P be the union of two general cones of respective degrees m, n ≥ 4. We choose coordinates (z1 : z2 : z3 : z4) ∈ P 3 so that X , X ′′ are cones through 2000 Mathematics Subject Classification. 32Q45,32H25,14J70.