2009/10/26 by Giancarlo Urzúa, Giancarlo Urzua, Urzua, Giancarlo
Mathematics · Social Sciences · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Vietnamese History and Culture Studies #math.AG #math.CO
paper · pdf · doi:10.48550/arxiv.0910.4928
33 pages, accepted for publication on Rendiconti Lincei: Matematica e Applicazioni
openalex publication_date 2009/10/26 · arxiv created 2011/04/04 · arxiv updated 2011/04/05 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
We introduce arrangements of rational sections over curves. They generalize line arrangements on P2. Each arrangement of d sections defines a single curve in Pd-2 through the Kapranov's construction of M0,d+1. We show a one-to-one correspondence between arrangements of d sections and irreducible curves in M0,d+1, giving also correspondences for two distinguished subclasses: transversal and simple crossing. Then, we associate to each arrangement A (and so to each irreducible curve in M0,d+1) several families of nonsingular projective surfaces X of general type with Chern numbers asymptotically proportional to various log Chern numbers defined by A. For example, for extended families over the complex numbers, one has that any such X is of positive index and π1(X) = π1(A), where A is the normalization of A. In this way, any rational curve in M0,d+1 produces simply connected surfaces with 2< c12(X)/c2(X) <3. Inequalities like these come from log Chern inequalities, which are in general connected to geometric height inequalities (see Appendix). Along the way, we show examples of étale simply connected surfaces of general type in any characteristic violating any sort of Miyaoka-Yau inequality.