Arrangements of rational sections over curves and the varieties they define

  • Giancarlo Urzúa

    Pontificia Universidad Católica de Chile, Santiago de Chile, Chile


We introduce arrangements of rational sections over curves. They generalize line arrangements on P2\mathbb P^2. Each arrangement of dd sections defines a single curve in Pd2\mathbb P^{d-2} through the Kapranov's construction of M0,d+1\overline{M}_{0,d+1}. We show a one-to-one correspondence between arrangements of dd sections and irreducible curves in M0,d+1M_{0,d+1}, giving also correspondences for two distinguished subclasses: transversal and simple crossing. Then, we associate to each arrangement A\mathcal A (and so to each irreducible curve in M0,d+1M_{0,d+1}) several families of nonsingular projective surfaces XX of general type with Chern numbers asymptotically proportional to various log Chern numbers defined by A\mathcal A. For example, for the main families and over C\mathbb C, any such XX is of positive index and π1(X)π1(A)\pi_1(X) \simeq \pi_1(\overline{A}), where A\overline{A} is the normalization of AA. In this way, any rational curve in M0,d+1M_{0,d+1} produces simply connected surfaces with Chern numbers ratio bigger than 22. 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.

Cite this article

Giancarlo Urzúa, Arrangements of rational sections over curves and the varieties they define. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 22 (2011), no. 4, pp. 453–486

DOI 10.4171/RLM/609