Arrangements of rational sections over curves and the varieties they define

Abstract

We introduce arrangements of rational sections over curves. They generalize line arrangements on ${\mathbb{P}}^2$. Each arrangement of $d$ sections defines a single curve in ${\mathbb{P}}^{d-2}$ through the Kapranov's construction of ${\overline{M}}_{0,d+1}$. We show a one-to-one correspondence between arrangements of $d$ sections and irreducible curves in $M_{0,d+1}$, giving also correspondences for two distinguished subclasses: transversal and simple crossing. Then, we associate to each arrangement $\mathcal{A}$ (and so to each irreducible curve in $M_{0,d+1}$) several families of nonsingular projective surfaces $X$ of general type with Chern numbers asymptotically proportional to various log Chern numbers defined by $\mathcal{A}$. For example, for the main families and over $\mathbb{C}$, any such $X$ is of positive index and ${\pi }_1(X)\simeq {\pi }_1(\overline{A})$, where $\overline{A}$ is the normalization of $A$. In this way, any rational curve in $M_{0,d+1}$ produces simply connected surfaces with Chern numbers ratio bigger than $2$. 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.