Smoothings of Fano Varieties With Normal Crossing Singularities

Tziolas, Nikolaos

doi:10.1017/s0013091515000024

Cited by 7 publications

(9 citation statements)

References 12 publications

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“…See [Hu06,4.3] for details of this construction. The fact that semistable degeneration of Fano manifolds is unique follows from [Tzi15]. The theorem is proved.…”

Section: Maximal Degenerationmentioning

confidence: 87%

“…That is, X is unique up to isomorphism as an snc variety. From [Tzi15] it follows that this is equivalent to the claim that a maximal d-semistable snc Fano variety X = X i is unique.…”

Section: Maximal Degenerationmentioning

confidence: 99%

“…For k = n this construction coincides with the maximal degeneration described below. In [Tzi15] it is proved that any d-semistable snc Fano variety can be smoothed, that is, embedded as the special fiber in a semistable family. Hence, the classification of semistable degenerations of Fano manifolds is equivalent to the classification of d-semistable snc Fano varieties.…”

Section: Introductionmentioning

confidence: 99%

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Maximal log Fano manifolds are generalized Bott towers

Loginov¹,

Moraga²

2020

Preprint

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We prove that maximal log Fano manifolds are generalized Bott towers. As an application, we prove that in each dimension, there is a unique maximal snc Fano variety satisfying Friedman's d-semistability condition. Contents 1. Introduction 1 2. Sketch of the proof 3 3. Preliminaries 4 4. Adjunction theory 6 5. Geometry of maximal log Fano manifolds 8 6. Looking for a line 11 7. The case ρ(X) = ρ(D 1 ) + 1 14 8. The case ρ(X) = ρ(D 1 ) 17 9. Proof of Theorem 5.1 19 10. Maximal degeneration 20 Appendix A. Generalized Bott towers 25 References 26

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“…See [Hu06,4.3] for details of this construction. The fact that semistable degeneration of Fano manifolds is unique follows from [Tzi15]. The theorem is proved.…”

Section: Maximal Degenerationmentioning

confidence: 87%

“…That is, X is unique up to isomorphism as an snc variety. From [Tzi15] it follows that this is equivalent to the claim that a maximal d-semistable snc Fano variety X = X i is unique.…”

Section: Maximal Degenerationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Maximal log Fano manifolds are generalized Bott towers

Loginov¹,

Moraga²

2020

Preprint

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“…1 where e 3 is the "central" (−1) curve on D 1 ≃ D 2 which is a del Pezzo surface of degree 7. Thus, by [Tz15] the simple normal crossing Fano variety F 1 ∪ F 2 can be smoothed in a semistable family.…”

Section: One Hasmentioning

confidence: 99%

“…In [Tz15] Tziolas showed that any d-semistable simple normal crossing Fano variety can be smoothed, that is, included as the special fiber in a semistable family. Hence, the classification of semistable degenerations of Fano varieties is equivalent to the classification of d-semistable simple normal crossing Fano varieties.…”

Section: Introductionmentioning

confidence: 99%

On semistable degenerations of Fano varieties

Loginov¹

2019

Preprint

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Consider a family of Fano varieties π : X −→ B ∋ o over a curve germ with a smooth total space X. Assume that the generic fiber is smooth and the special fiber F = π −1 (o) has simple normal crossings. Then F is called a semistable degeneration of Fano varieties. We show that the dual complex of F is a simplex of dimension ≤ dim F . Simplices of any admissible dimension can be realized for any dimension of the fiber. Using this result and the Minimal Model Program in dimension 3 we reproduce the classification of the semistable degenerations of del Pezzo surfaces obtained by Fujita. We also show that the maximal degeneration is unique and has trivial monodromy in dimension ≤ 3.

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Deformations and moduli of irregular canonical covers with $$K^2=4p_g-8$$

et al. 2023

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In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying $$K^2 = 4p_g-8$$ K 2 = 4 p g - 8 , for any even integer $$p_g\ge 4$$ p g ≥ 4 . These surfaces also have unbounded irregularity q. We carry out our study by investigating the deformations of the canonical morphism $$\varphi :X\rightarrow {\mathbb {P}}^N$$ φ : X → P N , where $$\varphi $$ φ is a quadruple Galois cover of a smooth surface of minimal degree. These canonical covers are classified in Gallego and Purnaprajna (Trans Am Math Soc 360(10):5489-5507, 2008) into four distinct families, one of which is the easy case of a product of curves. The main objective of this article is to study the deformations of the other three, non trivial, unbounded families. We show that any deformation of $$\varphi $$ φ factors through a double cover of a ruled surface and, hence, is never birational. More interestingly, we prove that, with two exceptions, a general deformation of $$\varphi $$ φ is two-to-one onto its image, whose normalization is a ruled surface of appropriate irregularity. We also show that, with the exception of one family, the deformations of X are unobstructed even though $$H^2(T_X)$$ H 2 ( T X ) does not vanish. Consequently, X belongs to a unique irreducible component of the Gieseker moduli space. These irreducible components are uniruled. As a result of all this, we show the existence of infinitely many moduli spaces, satisfying the strict Beauville inequality $$p_g > 2q-4$$ p g > 2 q - 4 , with an irreducible component that has a proper "quadruple" sublocus where the degree of the canonical morphism jumps up. These components are above the Castelnuovo line, but nonetheless parametrize surfaces with non birational canonical morphisms. The existence of jumping subloci is a contrast with the moduli of surfaces with $$K^2 = 2p_g- 4$$ K 2 = 2 p g - 4 , studied by Horikawa. Irreducible moduli components with a jumping sublocus also present a similarity and a difference to the moduli of curves of genus $$g\ge 3$$ g ≥ 3 , for, like in the case of curves, the degree of the canonical morphism goes down outside a closed sublocus but, unlike in the case of curves, it is never birational. Finally, our study shows that there are infinitely many moduli spaces with an irreducible component whose general elements have non birational canonical morphism and another irreducible component whose general elements have birational canonical map.

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Smoothings of Fano Varieties With Normal Crossing Singularities

Cited by 7 publications

References 12 publications

Maximal log Fano manifolds are generalized Bott towers

Maximal log Fano manifolds are generalized Bott towers

On semistable degenerations of Fano varieties

Deformations and moduli of irregular canonical covers with $$K^2=4p_g-8$$

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