2004/02/03 by Ai-Ko Liu, Liu, Ai-Ko · 2 citations
Mathematics · #14N99 #53D99 #Advanced Combinatorial Mathematics #Advanced Operator Algebra Research #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric and Algebraic Topology #Symplectic Geometry (math.SG) #math.AG #math.SG #msc:14N99 #msc:53D99
paper · pdf · doi:10.48550/arxiv.math/0402045
116 pages, 8 figures
arxiv created 2004/02/03 · openalex publication_date 2004/02/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the long paper "Family Blowup formula, Admissible Graphs and the Enumeration of Singular Curves (I)" (appearing in JDG), the author solved the enumeration problem of nodal (or general singular) curve counting on algebraic surfaces by using techniques from differential topology/symplectic geometry (including family Seiberg-Witten theory) and some ideas derived from Taubes' "SW=Gr". In the current paper, we offer an algebraic proof of the "universality theorem", showing that the counting of nodal curves for 5δ-1 very ample complete linear systems are controled by universal polynomials of the characteristic classes. The theorem (implicitly) was the backbone of the earlier long paper(cited above). In the current paper, we derive the result by using intersection theory and the concept of localized contributions of top Chern classes, and therefore relaxing the dependence on the symplectic techniques.