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

Fabrizio Catanese, Frédéric Mangolte

TL;DR

The paper investigates how the topology of real loci of real projective threefolds fibred by rational curves is constrained by the geometry of the base. By a detailed study of real Du Val Del Pezzo surfaces, especially degree 1, via a double-cover anticanonical model and a plane-model analysis, it derives sharp bounds on the number of real singularities contributing to the real topology. Through Werther and Seifert-fibration techniques, it proves a tight bound $k(N) \le 4$ for orientable components and shows $k=0$ with connected $W(\mathbb{R})$ when the base is topologically a torus $S^1\times S^1$, strengthening Kollár’s earlier estimates. These results answer two of Kollár’s questions from 1999 and highlight the role of real singular Del Pezzo surfaces in constraining 3-manifold topology of real loci.

Abstract

Let W -> X be a real smooth projective threefold fibred by rational curves. Kollár 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 Kollár 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.

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

TL;DR

The paper investigates how the topology of real loci of real projective threefolds fibred by rational curves is constrained by the geometry of the base. By a detailed study of real Du Val Del Pezzo surfaces, especially degree 1, via a double-cover anticanonical model and a plane-model analysis, it derives sharp bounds on the number of real singularities contributing to the real topology. Through Werther and Seifert-fibration techniques, it proves a tight bound for orientable components and shows with connected when the base is topologically a torus , strengthening Kollár’s earlier estimates. These results answer two of Kollár’s questions from 1999 and highlight the role of real singular Del Pezzo surfaces in constraining 3-manifold topology of real loci.

Abstract

Let W -> X be a real smooth projective threefold fibred by rational curves. Kollár 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 Kollár 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.

Paper Structure

This paper contains 4 sections, 16 theorems, 14 equations, 5 figures.

Table of Contents

  1. Real Du Val surfaces
  2. Singular Del Pezzo surfaces of degree 1
  3. Real rationally connected Threefolds
  4. Seifert fibrations over a torus

Key Result

Theorem 1

Let $X$ be a projective surface defined over $\mathbb{R}$. Suppose that $X$ is geometrically rational with Du Val singularities. Then a connected component $M$ of the topological normalization $\overline {X(\mathbb{R})}$ contains at most 4 Du Val singular points which are either not of type $A^-$ or

Figures (5)

  • Figure 4: Two irreducible components.
  • Figure 5: Two irreducible components.
  • Figure 6: The point $O$ is a globally nonseparating $A_1$ singular point.
  • Figure 7: The cusp gives rise to a singular point of type $A^-_2$.
  • Figure :

Theorems & Definitions (34)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Definition 1.1
  • Theorem 1.2
  • Example 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Lemma 1.7
  • ...and 24 more