Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Fabrizio Catanese, Frederic Mangolte

TL;DR

The paper advances the understanding of real loci of geometrically rational threefolds fibred by rational curves by proving sharp bounds on the Seifert-fibered and lens-space components of real loci when the base is geometrically rational. The authors reduce the problem to real Du Val Del Pezzo surfaces of degree $1$ with small Picard number, classify possible configurations of $A^+_\mu$ singularities, and exclude most via intricate Euler-number and topology arguments, while realizing Seifert fibrations as projectivized tangent bundles to construct new geometric examples. They establish key inequalities, such as $k(N)\le 4$ and $\sum_l\left(1-\frac{1}{n_l}\right)\le 2$, and show that the base orbifold of a Seifert fibration over an orientable $|F|$ is nonnegative in Euler characteristic, hence spherical or Euclidean. The work links the minimal model program with real algebraic topology to constrain real threefolds, further generalizing Comessatti-type results and revealing new phenomena, such as Seifert fibrations over nonorientable hyperbolic base orbifolds on minimal real surfaces.

Abstract

Let W -> X be a real smooth projective 3-fold fibred by rational curves. J. Kollár 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 Kollár, 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.

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

TL;DR

The paper advances the understanding of real loci of geometrically rational threefolds fibred by rational curves by proving sharp bounds on the Seifert-fibered and lens-space components of real loci when the base is geometrically rational. The authors reduce the problem to real Du Val Del Pezzo surfaces of degree with small Picard number, classify possible configurations of singularities, and exclude most via intricate Euler-number and topology arguments, while realizing Seifert fibrations as projectivized tangent bundles to construct new geometric examples. They establish key inequalities, such as and , and show that the base orbifold of a Seifert fibration over an orientable is nonnegative in Euler characteristic, hence spherical or Euclidean. The work links the minimal model program with real algebraic topology to constrain real threefolds, further generalizing Comessatti-type results and revealing new phenomena, such as Seifert fibrations over nonorientable hyperbolic base orbifolds on minimal real surfaces.

Abstract

Let W -> X be a real smooth projective 3-fold fibred by rational curves. J. Kollár 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 Kollár, 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.

Paper Structure

This paper contains 15 sections, 21 theorems, 33 equations, 6 figures, 1 table.

Table of Contents

  1. Singular geometrically rational surfaces
  2. Reducing to seven configurations
  3. Three components
  4. Two components
  5. One component
  6. Smoothings of Du Val Del Pezzo surfaces
  7. Topology of connected components
  8. Exclusion of $A^+_3 + A_2^+ + 2A_1^+$ and $A^+_3 + 3A_1^+$
  9. The Euler number on an elliptic fibration
  10. Exclusion of $A^+_3 + 2A_2^+$ and $3A^+_2 + A_1^+$
  11. The Euler number of a real singular Del Pezzo surface
  12. Exclusion of $2A^+_3 + A_2^+$, $2A^+_2 + 2A_1^+$ and $A^+_2 + 3A_1^+$
  13. Real rationally connected Threefolds
  14. Two-dimensional orbifolds
  15. A Seifert fibration with base orbifold of hyperbolic type.

Key Result

Theorem 1

Let $W \to X$ be a real smooth projective 3-fold fibred by rational curves over a geometrically rational By ghs these assumptions are equivalent to: $W$ rationally connected and fibred by rational curves. surface $X$. Suppose that $W(\mathbb{R})$ is orientable. Then, for each connected component $N

Figures (6)

  • Figure 1: Three parabolae with two tacnodes.
  • Figure 2: The cut and the cylinder smoothings of the node $A^+_1$.
  • Figure 3: The +sphere smoothing of the cusp $A^+_2$.
  • Figure 4: The cut+sphere and the cylinder smoothings of the tacnode $A^+_3$.
  • Figure 5:
  • ...and 1 more figures

Theorems & Definitions (40)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 30 more