2012/03/14 by Jun Li, Yu-jong Tzeng · 21 citations
Mathematics · #Advanced Algebra and Geometry #Algebra over a field #Algebraic Geometry and Number Theory #Ample line bundle #Conjecture #Generalization #Geometry and complex manifolds #Gravitational singularity #Mathematical analysis #Mathematics #Polynomial #Projective line #Projective space #Projective test #Pure mathematics #Singularity #math.AG
paper · pdf · doi:10.1112/s0010437x13007756
published in Compositio Mathematica 150(7), 1169-1182 (Cambridge University Press) · 12 pages
arxiv created 2012/03/14 · openalex publication_date 2014/06/06 · arxiv updated 2019/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract Let \def \xmlpi #1\def \mathsfbi #1\boldsymbol \mathsf #1\let ≤ =\leqslant \let ≤ =\leqslant \let ≥ =\geqslant \let ≥ =\geqslant \def Pr \mathit Pr\def \Fr \mathit Fr\def \Rey \mathit ReS be a complex smooth projective surface and L be a line bundle on S . For any given collection of isolated topological or analytic singularity types, we show the number of curves in the linear system |L| with prescribed singularities is a universal polynomial of Chern numbers of L and S , assuming L is sufficiently ample. More generally, we show for vector bundles of any rank and smooth varieties of any dimension, similar universal polynomials also exist and equal the number of singular subvarieties cutting out by sections of the vector bundle. This work is a generalization of Göttsche’s conjecture.