A complex manifold X is Brody hyperbolic if every holomorphic map f : ℂ → X is constant. For compact complex manifolds this is equivalent to the condition that the Kobayashi pseudometric κ1 (see (1.12)) is a positive definite Finsler metric. One may verify the hyperbolicity of a manifold by exhibiting a Finsler metric with negative holomorphic sectional curvature. The construction of such a metric motivates the use of parametrized jet bundles, as defined by Green–Griffiths. (The theory of these bundles goes back to [Ehresmann 1952].) We examine the algebraic-geometric properties (ample, big, nef, spanned and the dimension of the base locus) of these bundles that are relevant toward the metric's existence. To do this, we start by determining (and computing) basic invariants of jet bundles. Then we apply Nevanlinna theory, via the construction of an appropriate singular Finsler metric of logarithmic type, to obtain precise extensions of the classical Schwarz Lemma on differential forms toward jets. Particularly, this allows direct control over the analysis of the jets j kf of a holomorphic map f : ℂ →X; namely, the image of j kf must be contained in the base locus of the jet differentials.For an algebraically nondegenerate holomorphic map we show by means of reparametrization that the algebraic closure of j kf is quite large while, under appropriate conditions, the base locus is relatively small. This contradiction shows that the map f must be algebraically degenerate. We apply this method to verify that a generic smooth hypersurface of ℙ 3, of degree d ≥ 5, is hyperbolic (see Corollary 7.21).