On the cohomologically trivial automorphisms of elliptic surfaces I: $χ(S)=0$

Research output: Working paper/PreprintPreprint

Authors

Research Organisations

View graph of relations

Details

Original languageEnglish
Publication statusE-pub ahead of print - 29 Aug 2024

Abstract

In this first part we describe the group $Aut_{\mathbb{Z}}(S)$ of cohomologically trivial automorphisms of a properly elliptic surface (a minimal surface $S$ with Kodaira dimension $\kappa(S)=1$), in the initial case $ \chi(\mathcal{O}_S) =0$. In particular, in the case where $Aut_{\mathbb{Z}}(S)$ is finite, we give the upper bound 4 for its cardinality, showing more precisely that if $Aut_{\mathbb{Z}}(S)$ is nontrivial, it is one of the following groups: $\mathbb{Z}/2, \mathbb{Z}/3, (\mathbb{Z}/2)^2$. We also show with easy examples that the groups $\mathbb{Z}/2, \mathbb{Z}/3$ do effectively occur. Respectively, in the case where $Aut_{\mathbb{Z}}(S)$ is infinite, we give the sharp upper bound 2 for the number of its connected components.

Keywords

    math.AG, math.CV, 14J50, 14J80, 14J27, 14H30, 14F99, 32L05, 32M99, 32Q15, 32Q55

Cite this

On the cohomologically trivial automorphisms of elliptic surfaces I: $χ(S)=0$. / Catanese, Fabrizio; Frapporti, Davide; Gleissner, Christian et al.
2024.

Research output: Working paper/PreprintPreprint

Catanese, F., Frapporti, D., Gleissner, C., Liu, W., & Schütt, M. (2024). On the cohomologically trivial automorphisms of elliptic surfaces I: $χ(S)=0$. Advance online publication.
Catanese F, Frapporti D, Gleissner C, Liu W, Schütt M. On the cohomologically trivial automorphisms of elliptic surfaces I: $χ(S)=0$. 2024 Aug 29. Epub 2024 Aug 29.
Catanese, Fabrizio ; Frapporti, Davide ; Gleissner, Christian et al. / On the cohomologically trivial automorphisms of elliptic surfaces I: $χ(S)=0$. 2024.
Download
@techreport{a2345c7eebdb4c5fb192457650b01805,
title = "On the cohomologically trivial automorphisms of elliptic surfaces I: $χ(S)=0$",
abstract = " In this first part we describe the group $Aut_{\mathbb{Z}}(S)$ of cohomologically trivial automorphisms of a properly elliptic surface (a minimal surface $S$ with Kodaira dimension $\kappa(S)=1$), in the initial case $ \chi(\mathcal{O}_S) =0$. In particular, in the case where $Aut_{\mathbb{Z}}(S)$ is finite, we give the upper bound 4 for its cardinality, showing more precisely that if $Aut_{\mathbb{Z}}(S)$ is nontrivial, it is one of the following groups: $\mathbb{Z}/2, \mathbb{Z}/3, (\mathbb{Z}/2)^2$. We also show with easy examples that the groups $\mathbb{Z}/2, \mathbb{Z}/3$ do effectively occur. Respectively, in the case where $Aut_{\mathbb{Z}}(S)$ is infinite, we give the sharp upper bound 2 for the number of its connected components. ",
keywords = "math.AG, math.CV, 14J50, 14J80, 14J27, 14H30, 14F99, 32L05, 32M99, 32Q15, 32Q55",
author = "Fabrizio Catanese and Davide Frapporti and Christian Gleissner and Wenfei Liu and Matthias Sch{\"u}tt",
note = "49 pages, to appear in a volume of the Taiwanese Journal of Mathematics dedicated to Yurii (Gennadievich) Prokhorov on the occasion of his 60th birthday",
year = "2024",
month = aug,
day = "29",
language = "English",
type = "WorkingPaper",

}

Download

TY - UNPB

T1 - On the cohomologically trivial automorphisms of elliptic surfaces I: $χ(S)=0$

AU - Catanese, Fabrizio

AU - Frapporti, Davide

AU - Gleissner, Christian

AU - Liu, Wenfei

AU - Schütt, Matthias

N1 - 49 pages, to appear in a volume of the Taiwanese Journal of Mathematics dedicated to Yurii (Gennadievich) Prokhorov on the occasion of his 60th birthday

PY - 2024/8/29

Y1 - 2024/8/29

N2 - In this first part we describe the group $Aut_{\mathbb{Z}}(S)$ of cohomologically trivial automorphisms of a properly elliptic surface (a minimal surface $S$ with Kodaira dimension $\kappa(S)=1$), in the initial case $ \chi(\mathcal{O}_S) =0$. In particular, in the case where $Aut_{\mathbb{Z}}(S)$ is finite, we give the upper bound 4 for its cardinality, showing more precisely that if $Aut_{\mathbb{Z}}(S)$ is nontrivial, it is one of the following groups: $\mathbb{Z}/2, \mathbb{Z}/3, (\mathbb{Z}/2)^2$. We also show with easy examples that the groups $\mathbb{Z}/2, \mathbb{Z}/3$ do effectively occur. Respectively, in the case where $Aut_{\mathbb{Z}}(S)$ is infinite, we give the sharp upper bound 2 for the number of its connected components.

AB - In this first part we describe the group $Aut_{\mathbb{Z}}(S)$ of cohomologically trivial automorphisms of a properly elliptic surface (a minimal surface $S$ with Kodaira dimension $\kappa(S)=1$), in the initial case $ \chi(\mathcal{O}_S) =0$. In particular, in the case where $Aut_{\mathbb{Z}}(S)$ is finite, we give the upper bound 4 for its cardinality, showing more precisely that if $Aut_{\mathbb{Z}}(S)$ is nontrivial, it is one of the following groups: $\mathbb{Z}/2, \mathbb{Z}/3, (\mathbb{Z}/2)^2$. We also show with easy examples that the groups $\mathbb{Z}/2, \mathbb{Z}/3$ do effectively occur. Respectively, in the case where $Aut_{\mathbb{Z}}(S)$ is infinite, we give the sharp upper bound 2 for the number of its connected components.

KW - math.AG

KW - math.CV

KW - 14J50, 14J80, 14J27, 14H30, 14F99, 32L05, 32M99, 32Q15, 32Q55

M3 - Preprint

BT - On the cohomologically trivial automorphisms of elliptic surfaces I: $χ(S)=0$

ER -

By the same author(s)