2010/05/03 by Joël Merker, Merker, Joel
Mathematics · #13A05 #13A50 #13P10 #32Q45 #68W30 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.1005.0405
openalex publication_date 2010/05/03 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/30
Let X be a geometrically smooth n-dimensional projective algebraic complex\nhypersurface in Pn+1(C). Using Green-Griffiths jets, we establish the\nexistence of nonzero global algebraic differential equations that must be\nsatisfied by every nonconstant entire holomorphic curve C -> X if X is of\ngeneral type, namely if its degree d satisfies the optimal possible lower\nbound: d >= n + 3. The case n = 2 dates back to Green-Griffiths 1979, while\naccording to very recent advances (Invent. Math. 180, pp. 161-223, February\n2010), the best (and only) lower degree bound known previously in arbitrary\ndimension n was, using instead Demailly-Semple jets, something like d >=\n2n4 . n5n3, which, visibly, was far from the conjectured n + 3.\n