Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II

TL;DR

This paper provides sharp estimates on the topology of real 3-folds fibred by rational curves, extending classical results and constructing new examples using Seifert fibrations and tangent bundle techniques.

Contribution

It answers Kollár's questions by giving precise bounds on Seifert fibers and lens spaces, and generalizes Comessatti's theorem to non-orientable hyperbolic cases.

Findings

Connected components are Seifert fibred or sums of lens spaces.

Sharp bounds on the number and multiplicities of fibers.

Existence of non-orientable hyperbolic base orbifolds with minimal X.

Abstract

Let W -> X be a real smooth projective 3-fold fibred by rational curves. J. Koll\'ar proved that, if W(R) is orientable, then a connected component N of W(R) is essentially either a Seifert fibred manifold or a connected sum of lens spaces. Our Main Theorem, answering in the affirmative three questions of Koll\'ar, gives sharp estimates on the number and the multiplicities of the Seifert fibres and on the number and the torsions of the lens spaces when X is a geometrically rational surface. When N is Seifert fibred over a base orbifold F, our result generalizes Comessatti's theorem on smooth real rational surfaces: F cannot be simultaneously orientable and of hyperbolic type. We show as a surprise that, unlike in Comessatti's theorem, there are examples where F is non orientable, of hyperbolic type, and X is minimal. The technique we use is to construct Seifert fibrations as projectivized tangent bundles of Du Val surfaces.