Low pole order frames on vertical jets of the universal hypersurface

By Joël Merker. 2008

51 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4ebaa8b2686283bf7346c2276eab2f5f68c589f8

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In Applied Mathematics

1. Damian Brotbek
2. Y. Siu
3. J. Demailly
4. Mihai Paun
5. Qi’an Guan
6. Xiangyu Zhou
7. Joël Merker
8. Junyan Cao

In Geometry and Topology

1. Damian Brotbek
2. Y. Siu
3. J. Demailly
4. Qi’an Guan
5. Mihai Paun
6. Xiangyu Zhou
7. Joël Merker
8. Junyan Cao

Citations in the corpus

• Hyperbolicity of generic high-degree hypersurfaces in complex projective space (2012, methodology): “Recently Diverio, Merker, and Rousseau in [8] used the theorem of Riemann–Roch to construct a holomorphic jet differential on X (α) vanishing on an ample divisor and then used Merker’s work [16] involving our method”
• On the hyperbolicity of general hypersurfaces (2016, result): “This motivated a lot of research over the last decade [52, 49, 22, 23, 41, 24, 26, 9, 10], which culminated with the work of Diverio, Merker and Rousseau [24], and the proof of the weak hyperbolicity of general hypersurfaces in P of degree d > 2 5 .”