A complex surface of general type with 𝑝_{𝑔}=0, 𝐾²=2 and 𝐻₁=ℤ/4ℤ

Abstract. We construct many new surfaces of general type with q = p g = 0 whose canonical model is the quotient of the product of two curves by the action of a finite group G, constructing in this way many new interesting fundamental groups which distinguish connected components of the moduli space of surfaces of general type.We indeed classify all such surfaces whose canonical model is singular (the smooth case was classified in an earlier work).As an important tool we prove a structure theorem giving a precise description of the fundamental group of quotients of products of curves by the action of a finite group G.

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“…Park, Park and Shin ( [PPS07]) showed the existence of simply connected surfaces, and of surfaces with torsion Z 2 ( [PPS08a]). …”

Section: Introductionmentioning

confidence: 99%

Quotients of products of curves, new surfaces with p g = 0 and their fundamental groups.

Bauer¹,

Catanese²,

Grunewald³

et al. 2012

ajm

105

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“…The answer to the question for the algebraic fundamental group is affirmative. Indeed, the last open case, Z 4 , is realized by our example and by a completely different construction found independently by [PPS10]. We note that the topological fundamental group of [PPS10] is not known.…”

Section: Introductionmentioning

confidence: 67%

Mixed quasi-étale surfaces, new surfaces of general type with $$p_g=0$$ and their fundamental group

Frapporti

2013

Collect. Math.

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We call a projective surface X mixed quasi-étale quotient if there exists a curve C of genus g(C) ≥ 2 and a finite group G that acts on C × C exchanging the factors such that X = (C × C)/G and the map C × C → X has finite branch locus. The minimal resolution of its singularities is called mixed quasi-étale surface. We study the mixed quasi-étale surfaces under the assumption that (C × C)/G 0 has only nodes as singularities, where G 0 ⊳ G is the index two subgroup of the elements that do not exchange the factors.We classify the minimal regular surfaces with pg = 0 whose canonical model is a mixed quasi-étale quotient as above. All these surfaces are of general type and as an important byproduct, we provide an example of a numerical Campedelli surface with topological fundamental group Z4, and we realize 2 new topological types of surfaces of general type. Three of the families we construct are Q-homology projective planes.

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“…As remarked in [BCP11] (see also [PPS13]), if H 1 (S, Z) is finite, it is isomorphic to the abelianization of π alg 1 (S) and so it has order ≤ 9, whence X cannot be minimal.…”

Section: Proof Beingmentioning

confidence: 82%

On semi‐isogenous mixed surfaces

Cancian

Frapporti

2017

Mathematische Nachrichten

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Let be a smooth projective curve and be a finite subgroup of Aut( ) 2 ⋊ ℤ 2 whose action is mixed, i.e. there are elements in exchanging the two isotrivial fibrations of × . Let 0 ⊲ be the index two subgroup ∩ Aut( ) 2 . If 0 acts freely, then ∶= ( × )∕ is smooth and we call it semi-isogenous mixed surface. In this paper we give an algorithm to determine semi-isogenous mixed surfaces with given geometric genus, irregularity and self-intersection of the canonical class. As an application we classify irregular semi-isogenous mixed surfaces with 2 > 0 and geometric genus equal to the irregularity; the regular case is subjected to some computational restrictions. In this way we construct new examples of surfaces of general type with = 1. We provide an example of a minimal surface of general type with 2 = 7 and = = 2. K E Y W O R D SSurfaces of general type, finite group actions M S C ( 2 0 1 0 ) 14J29, 14L30, 14Q10

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A complex surface of general type with 𝑝_{𝑔}=0, 𝐾²=2 and 𝐻₁=ℤ/4ℤ

Cited by 6 publications

References 22 publications

Quotients of products of curves, new surfaces with p g = 0 and their fundamental groups.

Quotients of products of curves, new surfaces with p g = 0 and their fundamental groups.

Mixed quasi-étale surfaces, new surfaces of general type with $$p_g=0$$ and their fundamental group

On semi‐isogenous mixed surfaces

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