Topology of holomorphic Lefschetz pencils on the four-torus

Hamada, Noriyuki; Hayano, Kenta

doi:10.2140/agt.2018.18.1515

Cited by 8 publications

(12 citation statements)

References 22 publications

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“…(1) The genus of f is equal to Proof. (1) is merely a consequence of the adjunction formula, and we can prove (2) in the same way as that for [11,Lemma 3.1]. In what follows we will prove (3).…”

Section: Preliminariesmentioning

confidence: 53%

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Classification of genus-$1$ holomorphic Lefschetz pencils

Hamada¹,

Hayano²

2020

Preprint

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In this paper, we classify relatively minimal genus-1 holomorphic Lefschetz pencils up to smooth isomorphism. We first show that such a pencil is isomorphic to either the pencil on P 1 × P 1 of bi-degree (2, 2) or a blow-up of the pencil on P 2 of degree 3, provided that no fiber of a pencil contains an embedded sphere. (Note that one can easily classify genus-1 Lefschetz pencils with an embedded sphere in a fiber.) We further determine the monodromy factorizations of these pencils and show that the isomorphism class of a blowup of the pencil on P 2 of degree 3 does not depend on the choice of blown-up base points. We also show that the genus-1 Lefschetz pencils constructed by Korkmaz-Ozbagci (with nine base points) and Tanaka (with eight base points) are respectively isomorphic to the pencils on P 2 and P 1 ×P 1 above, in particular these are both holomorphic.

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Section: Preliminariesmentioning

confidence: 53%

“…It is known, however, that the composition is a Lefschetz pencil provided that L is very ample and the projection is generic. Moreover, the smooth isomorphism class of the Lefschetz pencil does not depend on the choice of this projection (see [32,11]).…”

Section: Preliminariesmentioning

confidence: 99%

Classification of genus-$1$ holomorphic Lefschetz pencils

Hamada¹,

Hayano²

2020

Preprint

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show abstract

“…Proof (1) is merely a consequence of the adjunction formula, and we can prove (2) in the same way as that for [11,Lemma 3.1]. In what follows we will prove (3).…”

Section: The Genus Of F Is Equal Tomentioning

confidence: 60%

Section: The Genus Of F Is Equal Tomentioning

confidence: 99%

Classification of genus-1holomorphic Lefschetz pencils

Hamada¹,

Hayano²

2021

Turk J Math

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In this paper, we classify relatively minimal genus-1 holomorphic Lefschetz pencils up to smooth isomorphism. We first show that such a pencil is isomorphic to either the pencil on P 1 × P 1 of bi-degree (2, 2) or a blow-up of the pencil on P 2 of degree 3 , provided that no fiber of a pencil contains an embedded sphere. (Note that one can easily classify genus-1 Lefschetz pencils with an embedded sphere in a fiber.) We further determine the monodromy factorizations of these pencils and show that the isomorphism class of a blow-up of the pencil on P 2 of degree 3 does not depend on the choice of blown-up base points. We also show that the genus-1 Lefschetz pencils constructed by Korkmaz-Ozbagci (with nine base points) and Tanaka (with eight base points) are respectively isomorphic to the pencils on P 2 and P 1 × P 1 above, in particular these are both holomorphic.

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“…In what follows we further construct other new relations in the same way as (19), so the procedures will be simplified. We will again use the same symbols as those in (19).…”

Section: Applying Those Lantern Relations We Altermentioning

confidence: 99%

Sections of the Matsumoto-Cadavid-Korkmaz Lefschetz fibration

Hamada¹

2016

Preprint

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We give a maximal set of disjoint (−1)-sections of the well-known Lefschetz fibration constructed by Matsumoto, Cadavid and Korkmaz. In fact, we obtain several such sets for a fixed genus, which implies that the Matsumoto-Cadavid-Korkmaz Lefschetz fibration has more than one supporting minimal Lefschetz pencils. We also determine the diffeomorphism types of the obtained supporting minimal Lefschetz pencils.

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Topology of holomorphic Lefschetz pencils on the four-torus

Cited by 8 publications

References 22 publications

Classification of genus-$1$ holomorphic Lefschetz pencils

Classification of genus-$1$ holomorphic Lefschetz pencils

Classification of genus-1holomorphic Lefschetz pencils

Sections of the Matsumoto-Cadavid-Korkmaz Lefschetz fibration

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