Very general surfaces of degree at least r plus 1 in general Fano threefolds with Picard number one are algebraically hyperbolic, except in three weighted cases where the proven threshold is r plus 2.
Demailly , Algebraic criteria for Kobayashi hyperbolic projective va rieties and jet differentials , Proc
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Algebraic hyperbolicity of surfaces in Fano threefolds with Picard number one
Very general surfaces of degree at least r plus 1 in general Fano threefolds with Picard number one are algebraically hyperbolic, except in three weighted cases where the proven threshold is r plus 2.