Low pole order frames on vertical jets of the universal hypersurface

Details Low pole order frames on vertical jets of the universal hypersurface Low pole order frames on vertical jets of the universal hypersurface

2008-05-26

arxiv.org/abs/0805.3987v2

Mathematics

Algebraic Geometry

Major improvement yielding strong (instead of weak) Green-Griffiths algebraic degeneracy in 0811.2346

Scientific paper

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

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