2017/11/08 by Dinh Tuan Huynh, Huynh, Dinh Tuan, Duc‐Viet Vu +3
Mathematics · #30D35 #32A22 #32H30 #32Q45 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.1711.02996
openalex publication_date 2017/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For q≤ 3 smooth plane algebraic curves Ci having simple normal crossings, if the invariant logarithmic 2-jet differential bundle associated to (ℙ2(ℂ), ∑i=1q Ci) has a nonzero section vanishing on some ample divisor, then, for every algebraically nondegenerate entire holomorphic curve f\colonℂ→ℙ2(ℂ), we have a Second Main Theorem type estimate: Tf(r) ≤ c∑i=1q Nf[1](r,Ci) + o(Tf(r) )∥, where Tf(r) and Nf[1](r,Ci) stand for the order function and the 1--truncated counting functions in the Nevanlinna theory, and where the constant c=c(q,di)>0 can be computed explicitly. In particular, our result includes the case of 3 conics in ℙ2(ℂ). Moreover, we provide some new results concerning the algebraic degeneracy of certain complex surfaces, e.g., the complex hyperbolicity of a very generic surface of degree ≥ 15 in ℙ3(ℂ).