On the numerically and cohomologically trivial automorphisms of elliptic surfaces II: $\chi(S)>0$

TL;DR

This paper investigates the structure and bounds of numerically and cohomologically trivial automorphism groups of properly elliptic surfaces with positive Euler characteristic, revealing new finite group properties and explicit bounds.

Contribution

It demonstrates the existence of nontrivial automorphism groups for surfaces with positive Euler characteristic and provides explicit bounds and criteria for triviality, advancing understanding of automorphisms in elliptic surfaces.

Findings

Aut_Q(S) is always a 2-generated finite abelian group.

No absolute upper bound for |Aut_Q(S)|; bounds depend on invariants like χ(S), q(S), P_2(S).

Explicit upper bounds for |Aut_Z(S)| in special cases, including sharp bounds for isotrivial surfaces.

Abstract

In this second part we study first the group $Aut_{\mathbb Q}(S)$ of numerically trivial automorphisms of an algebraic properly elliptic surface $S$, that is, of a minimal algebraic surface with Kodaira dimension $\kappa(S)=1$, in the case $\chi(S) \geq 1$. Our first surprising result is that, against what has been believed for over 40 years, there exist nontrivial such groups for $p_g(S) >0$. Indeed, we show even that $Aut_{\mathbb Q}(S)$ is always a 2-generated finite abelian group, but there is no absolute upper bound for its cardinality. At any rate, we give explicit and essentially optimal upper bounds for $|Aut_{\mathbb Q}(S)|$ in terms of the numerical invariants of $S$, as $\chi(S)$, or the irregularity $q(S)$, or the bigenus $P_2(S)$. Moreover, we reach an almost complete description of the possible groups $Aut_{\mathbb Q}(S)$ and we give effective criteria for such surfaces to have trivial $Aut_{\mathbb Q}(S)$. Our second surprising results concern the quite elusive group $Aut_{\mathbb Z}(S)$ of cohomologically trivial automorphisms; we are able to give the explicit upper bounds for $|Aut_{\mathbb Z}(S)|$ in special cases: 9 when $p_g(S) =0$, and we achieve the sharp upper bound 3 when $S$ (i.e., the pluricanonical elliptic fibration) is isotrivial. Also in the non isotrivial case we produce subtle examples where $Aut_{\mathbb Z}(S)$ is a group of order 2 or 3.