Real singular Del Pezzo surfaces and threefolds fibred by rational curves, I

TL;DR

This paper establishes bounds on the complexity of real threefolds fibred by rational curves, showing that for geometrically rational surfaces the number of certain singular fibers is at most 4, with specific cases having no such fibers.

Contribution

It proves that for real smooth projective threefolds fibred by rational curves over geometrically rational surfaces, the number of special fibers is at most 4, confirming a conjecture and refining Kollár's earlier bounds.

Findings

If the base surface is a torus, the real part of the threefold is connected and has no multiple fibers.

The bound k <= 4 is sharp, with examples achieving this maximum.

The results are derived from a detailed analysis of real singular Del Pezzo surfaces with Du Val singularities.

Abstract

Let W -> X be a real smooth projective threefold fibred by rational curves. Koll\'ar proved that if W(R) is orientable a connected component N of W(R) is essentially either a Seifert fibred manifold or a connected sum of lens spaces. Let k : = k(N) be the integer defined as follows: If g : N -> F is a Seifert fibration, one defines k : = k(N) as the number of multiple fibres of g, while, if N is a connected sum of lens spaces, k is defined as the number of lens spaces different from P^3(R). Our Main Theorem says: If X is a geometrically rational surface, then k <= 4. Moreover we show that if F is diffeomorphic to S^1xS^1, then W(R) is connected and k = 0. These results answer in the affirmative two questions of Koll\'ar who proved in 1999 that k <= 6 and suggested that 4 would be the sharp bound. We derive the Theorem from a careful study of real singular Del Pezzo surfaces with only Du Val singularities.