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Of the two techniques introduced by Bloch, Green-Griffiths and developed bySiu, Demailly to establish Kobayashi hyperbolicity of generic high degreecomplex algebraic hypersurfaces X in P^(n+1), the second one, initiated byClemens, Ein, Voisin and developed by Siu, Paun, Rousseau consists inconstructing meromorphic frames on the space of the so-called vertical k-jetsJ_vert^k (X_univ) of the universal hypersurface X_univ parametrizing all X inP^(n+1) of degree d. In 2004, Siu announced that there exists a constant c_nsuch that the twisting of the tangent bundle to J_vert^n (X_univ) by O (c_n) isglobally generated (frame property). The present article provides c_n = (n^2 +5n) / 2, recovering c_2 = 7 (Paun), c_3 = 12 (Rousseau). Applications toeffective degree estimates for algebraic degeneracy or hyperbolicity areexpected, especially in dimension n = 4, granted that the Demailly-Semplealgebra of jet polynomials invariant under reparametrization and under acertain unipotent action is, for n = k = 4, generated by 16 fundamentalbi-invariants enjoying 41 groebnerized syzygies.

Low pole order frames on vertical jets of the universal hypersurface