Optimal bounds for T-singularities in stable surfaces

Rana, Julie; Urzúa, Giancarlo

doi:10.1016/j.aim.2019.01.029

Cited by 15 publications

(40 citation statements)

References 21 publications

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“…Since the closure of the locus of smooth I-surfaces is irreducible of dimension 28, we conclude that the general surface obtained via this construction is not smoothable. This confirms the infinitesimal computations of [23], where it is shown that the obstruction space for Q-Gorenstein deformations is non-zero for these surfaces.…”

Section: Remark 317supporting

confidence: 87%

“…As explained in the introduction of [23], the log Bomolov-Miyaoka-Yau inequality gives d ≤ 34 for a stable I-surface with a T-singularity of type 1 4d (1, 2d − 1), a weaker bound than Proposition 3.4 (i).…”

Section: Remark 35mentioning

confidence: 96%

“…An example of a T-singular I-surface of type 1 25 (1,14) can be found in [23] right after the proof of Thm. 3.2.…”

Section: The Case Of Index N =mentioning

confidence: 99%

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I-surfaces with one T-singularity

Franciosi

Pardini

Rana

et al. 2021

Boll Unione Mat Ital

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We classify normal stable surfaces with $$K_X^2 = 1$$ K X 2 = 1 , $$p_g = 2$$ p g = 2 and $$q=0$$ q = 0 with a unique singular point which is a non-canonical T-singularity, thus exhibiting two divisors in the main component and a new irreducible component of the moduli space of stable surfaces $${{\overline{{{\mathfrak {M}}}}}}_{1,3}$$ M ¯ 1 , 3 .

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Section: Remark 317supporting

confidence: 87%

Section: Remark 35mentioning

confidence: 96%

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I-surfaces with one T-singularity

Franciosi

Pardini

Rana

et al. 2021

Boll Unione Mat Ital

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“…After the first version of this paper was prepared, in June 2017, we learned that a very similar (slightly stronger, and optimal) result had been proved by Rana and Urzúa[RU19]. They use algebro-geometric rather than symplectic methods, so do not recover our purely symplectic Theorem 1.4.…”

mentioning

confidence: 88%

“…The authors would like to thank Weiyi Zhang for extremely helpful discussions about how −1spheres can degenerate under Gromov compactness (see Remark 2.14) and how our paper might generalise beyond the p g > 0 setting, Paul Hacking for useful comments and for making us aware of the ongoing work of Rana and Urzúa, Julie Rana, and Giancarlo Urzúa for constructive discussions once we learned about their work [RU19], and the anonymous referee for their comments.…”

Section: Acknowledgementsmentioning

confidence: 99%

Bounds on Wahl singularities from symplectic topology

Smith¹

2020

Alg. Geom.

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Let X be a minimal surface of general type with p g > 0 (equivalently, b + > 1) and let K 2 be the square of its canonical class. Building on work of Khodorovskiy and Rana, we prove that if X develops a Wahl singularity of length in a Q-Gorenstein degeneration, then 4K 2 + 7. This improves on the current best-known upper bound due to Lee 400 K 2 4 . Our bound follows from a stronger theorem constraining symplectic embeddings of certain rational homology balls in surfaces of general type. In particular, we show that if the rational homology ball B p,1 embeds symplectically in a quintic surface, then p 12, partially answering the symplectic version of a question of Kronheimer.is a continued fraction expansion. The number is called the length of the singularity. The index of the singularity is p and is bounded above by 2 .Theorem 1.1. Let X be a minimal surface of general type with positive geometric genus p g > 0 which has a finite set of Du Val and Wahl singularities. Then the length of any of the Wahl singularities satisfies 4K 2 X + 7 .Remark 1.2. The best bound for Wahl singularities we could find in the algebraic geometry literature 1 is due to Lee [Lee99, Theorem 23], who showed that if X is a stable surface of general type having a Wahl singularity of length , and if the minimal model S of the minimal resolution X of X has general type, then 400 K 2 X 4 .

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Unimodal singularities and boundary divisors in the KSBA moduli of a class of Horikawa surfaces

Gallardo,

Pearlstein,

Schaffler

et al. 2023

Mathematische Nachrichten

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Smooth minimal surfaces of general type with , , and constitute a fundamental example in the geography of algebraic surfaces, and the 28‐dimensional moduli space of their canonical models admits a modular compactification via the minimal model program. We describe eight new irreducible boundary divisors in such compactification parameterizing reducible stable surfaces. Additionally, we study the relation with the GIT compactification of and the Hodge theory of the degenerate surfaces that the eight divisors parameterize.

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Optimal bounds for T-singularities in stable surfaces

Cited by 15 publications

References 21 publications

I-surfaces with one T-singularity

I-surfaces with one T-singularity

Bounds on Wahl singularities from symplectic topology

Unimodal singularities and boundary divisors in the KSBA moduli of a class of Horikawa surfaces

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