Arrangements of rational sections over curves and the varieties they define

TL;DR

This paper introduces arrangements of rational sections over curves, linking them to curves in moduli spaces, and constructs families of surfaces of general type with specific Chern number properties.

Contribution

It establishes a correspondence between rational section arrangements and curves in moduli spaces, and constructs new surfaces of general type with controlled Chern ratios.

Findings

Arrangements of rational sections correspond to irreducible curves in M_{0,d+1}.

Constructed surfaces have positive index and specified fundamental groups.

Produced simply connected surfaces with Chern ratios between 2 and 3.

Abstract

We introduce arrangements of rational sections over curves. They generalize line arrangements on P^2. Each arrangement of d sections defines a single curve in P^{d-2} through the Kapranov's construction of \bar{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 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 A. For example, for extended families over the complex numbers, one has that any such X is of positive index and \pi_1(X) = \pi_1(\bar{A}), where \bar{A} is the normalization of A. In this way, any rational curve in M_{0,d+1} produces simply connected surfaces with 2< c_1^2(X)/c_2(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 \'etale simply connected surfaces of general type in any characteristic violating any sort of Miyaoka-Yau inequality.