Threefolds with negative Kodaira dimension and positive irregularity (original) (raw)

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

The Michigan Mathematical Journal, 2008

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

Annales scientifiques de l'École normale supérieure

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.

Characterization of the 4-canonical birationality of algebraic threefolds

Mathematische Zeitschrift, 2008

In this article we present a 3-dimensional analogue of a well-known theorem of E. Bombieri (in 1973) which characterizes the bi-canonical birationality of surfaces of general type. Let X be a projective minimal 3-fold of general type with Q-factorial terminal singularities and the geometric genus p g (X) ≥ 5. We show that the 4-canonical map ϕ 4 is not birational onto its image if and only if X is birationally fibred by a family C of irreducible curves of geometric genus 2 with K X • C 0 = 1 where C 0 is a general irreducible member in C .

Explicit birational geometry of threefolds of general type, I

Annales scientifiques de l'École normale supérieure, 2010

Let V be a complex nonsingular projective 3-fold of general type. We prove P 12 (V) := dim H 0 (V, 12K V) > 0 and P m0 (V) > 1 for some positive integer m 0 ≤ 24. A direct consequence is the birationality of the pluricanonical map ϕ m for all m ≥ 126. Besides, the canonical volume Vol(V) has a universal lower bound ν(3) ≥ 1 63•126 2 .

On the birational geometry of conic bundles over the projective space

2021

Let π : Z → P be a general minimal n-fold conic bundle with a hypersurface BZ ⊂ P n−1 of degree d as discriminant. We prove that if d ≥ 4n+1 then −KZ is not pseudo-effective, and that if d = 4n then none of the integral multiples of −KZ is effective; in the 3-fold case, thanks to a result of J. Kollár, this also proves the equivalent birational statements. Finally, we provide examples of smooth unirational n-fold conic bundles π : Z → P with discriminant of arbitrarily high degree.

3 New Examples of Calabi–Yau Threefolds and Genus Zero Surfaces

2016

We classify the subgroups of the automorphism group of the product of 4 projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi-Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is non-trivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K 2 = 3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.

New examples of Calabi-Yau threefolds and genus zero surfaces

2012

We classify the subgroups of the automorphism group of the product of 4 projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi-Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is non-trivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K^2=3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.