2016/09/13 by Karol Palka, Tomasz Pełka
Mathematics · #Algebraic Geometry and Number Theory #Complement (music) #Degree (music) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Planar #Resolution (logic) #math.AG #msc:14H50 #msc:14J17 #msc:14R25
paper · pdf · doi:10.1112/plms.12049
published as Proc. London Math. Soc. 115 (2017), No. 3, 638-692 · 50 pages
arxiv created 2016/09/13 · openalex created_date 2016/10/21 · openalex publication_date 2017/06/23 · arxiv updated 2019/04/30 · openalex updated_date 2026/08/05
To classify planar complex rational cuspidal curves E ⊆ P 2 it remains to classify the ones with complement of log general type, that is, the ones for which κ ( K X + D ) = 2 , where ( X , D ) is a log resolution of ( P 2 , E ) . It is conjectured that κ ( K X + 1 2 D ) = − ∞ and hence P 2 ∖ E is C ∗ ∗ -fibered, where C ∗ ∗ = C 1 ∖ 0 , 1 , or − ( K X + 1 2 D ) is ample on some minimal model of ( X , 1 2 D ) . Here we classify, up to a projective equivalence, those rational cuspidal curves for which the complement is C ∗ ∗ -fibered. From the rich list of known examples only very few are not of this type. We also discover a new infinite family of bicuspidal curves with unusual properties.