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Algebraic families of smooth hyperbolic surfaces of low degree in \(\mathbb{P}_ \mathbb{C}^ 3\). (English) Zbl 0887.32007
In [S. Kobayashi, Hyperbolic manifolds and holomorphic mappings, Marcel Dekker, New York (1970; Zbl 0207.37902)], Kobayashi conjectured that a generic hypersurface of \(\mathbb{P}_{\mathbb{C}}^n\) of sufficiently high degree \(d\) (expected bound being \(d \geq 2n-1\)) is hyperbolic. It is true for \(n=2\), but for \(n>2\) only some examples are known. For \(n=3\), Brody and Green obtained the first example of a smooth hyperbolic surface in \(\mathbb{P}_{\mathbb{C}}^3\) of any degree \(d=2p \geq 50\) [R. Brody and M. Green, Duke Math. J. 44, 873-874 (1977; Zbl 0383.32009)], and Nadel obtained such examples of any degree \(d=6p+3\geq 21\) [A. M. Nadel, Duke Math. J. 58, No. 3, 749-771 (1989; Zbl 0686.32015)].
In the present paper, the author improves Nadel’s technique and produces examples of smooth hyperbolic surfaces in \(\mathbb{P}_{\mathbb{C}}^3\) of arbitrary degree \(d \geq 14\), which is closer to the expected bound \(5\). Furthermore, if \(H_{3,d}\) is the subset of all hyperbolic surfaces in the projective space \(P_{3,d}\) of all surfaces of degree \(d\) in \(\mathbb{P}_{\mathbb{C}}^3\), and if \(d>9+ \sum_{i=0}^3 k_i\) where at least two of the integers \(k_i\) are \(\geq 2\), then the author proves that \(H_{3,d}\) contains a Zariski open subset of \(P_{3,d}\) of dimension \(\sum_{i=0}^3 \binom{k_i+3}{k_i} -1\).

32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
14H10 Families, moduli of curves (algebraic)
14H20 Singularities of curves, local rings
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
Full Text: DOI EuDML
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