[Paper Review] New examples of hyperbolic octic surfaces in $\PP^3$

Bernard Shiffman, Mikhail Zaidenberg|arXiv (Cornell University)|Jun 25, 2003

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper constructs new examples of Kobayashi hyperbolic surfaces in $$\mathbb{P}^3$$ by deforming the union of two general cones of degree $\geq 4$; the deformation method ensures the resulting surface of degree $m+n \geq 8$ is hyperbolic, providing the first explicit construction of hyperbolic surfaces in $\mathbb{P}^3$ for all degrees $d \geq 8$. The key result is that a general small deformation of such a union is hyperbolic, extending known examples beyond degree 8.

ABSTRACT

We show that a general small deformation of the union of two general cones in P3 of degree >= 4 is Kobayashi hyperbolic. Hence we obtain new examples of hyperbolic surfaces in P3 of any given degree d>= 8.

Motivation & Objective

• To construct new examples of Kobayashi hyperbolic surfaces in $\mathbb{P}^3$ for all degrees $d \geq 8$.
• To extend the known family of hyperbolic surfaces in $\mathbb{P}^3$ beyond the previously known degree 8 examples.
• To demonstrate that a general small deformation of the union of two cones of degree $\geq 4$ yields a hyperbolic surface.
• To provide a systematic deformation method that preserves hyperbolicity under small perturbations of singular surfaces.

Proposed method

• The method uses a deformation of the union $X = X' \cup X''$ of two general cones $X'$ and $X''$ of degrees $m,n \geq 4$ in $\mathbb{P}^3$, with vertices at distinct points not lying on the other cone.
• The deformation is defined by $X_t = \{f_1(z_1,z_2,z_3)f_2(z_1,z_2,z_4) + t f_\infty = 0\}$ for small $t \neq 0$, where $f_\infty$ is a general homogeneous polynomial of degree $m+n$.
• The proof relies on Brody’s Theorem, assuming the existence of non-constant entire curves and deriving a contradiction via Picard’s Theorem and transversality arguments.
• It analyzes the double curve $\Gamma = X' \cap X''$ and uses the projection $\pi_0: (z_1:z_2:z_3:z_4) \mapsto (z_1:z_2)$ to study fiber intersections and ensure injectivity of $\pi_0|\Gamma$ for general $X_\infty$.
• The argument shows that for a general $X_\infty$, the intersection $\Gamma \cap X_\infty$ has maximal cardinality under $\pi_0$, and no fiber contains two points from $\Gamma \cap X_\infty$, which prevents non-constant entire curves from existing.
• A contradiction is derived if such curves exist, using the fact that entire curves would be forced to lie in lines through the cone vertex, but such lines intersect $\Gamma \setminus X_\infty$ in at least three points, violating Picard’s Theorem.

Experimental results

Research questions

• RQ1 Can a general small deformation of the union of two cones of degree $\geq 4$ in $\mathbb{P}^3$ yield a hyperbolic surface?
• RQ2 Does the deformation method used in this paper produce hyperbolic surfaces of degree $d \geq 8$?
• RQ3 Is it possible to construct hyperbolic surfaces in $\mathbb{P}^3$ for all degrees $d \geq 8$ using cone unions and deformations?
• RQ4 What conditions on the base curves and projections ensure that the deformed surface remains hyperbolic?
• RQ5 Can the hyperbolicity of the deformed surface be established via contradiction using entire curves and Picard’s Theorem?

Key findings

• A general small deformation of the union of two cones of degree $m,n \geq 4$ in $\mathbb{P}^3$ results in a Kobayashi hyperbolic surface of degree $m+n \geq 8$.
• The construction provides explicit examples of hyperbolic surfaces in $\mathbb{P}^3$ for all degrees $d \geq 8$, extending previous degree 8 examples.
• The method ensures that no non-constant entire curves exist in the deformed surface by showing that any such curve would be forced to lie in a line through a cone vertex intersecting the double curve in at least three points.
• For a general $X_\infty$, the projection $\pi_0$ restricted to the double curve $\Gamma$ is injective on $\Gamma \cap X_\infty$, which is essential for the contradiction argument.
• The proof relies on transversality and continuity of zero sets under small perturbations, ensuring that the cardinality of $\pi_0(\Gamma \cap X_\infty)$ is maximal for general $X_\infty$.
• The construction is robust: even in the degenerate case where $f_1 = f_2$, the method recovers the original union of two cones, confirming consistency with prior constructions.