On Surfaces of General Type withpg = q = 1 Isogenous to a Product of Curves

Polizzi, Francesco

doi:10.1080/00927870801948676

Cited by 21 publications

(35 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The group is abelian, so (arguing as in the proof of [Pol1], theorem 6.3) α * ω n S in a sum of line bundles for each n ∈ N. By proposition 3.1 their smooth minimal models give a subfamily of M 2,3 ∪ M 4,2 ∪ M 3,1 ∪ M 6,1 . All Polizzi's surfaces have 4 nodes.…”

Section: Modulimentioning

confidence: 99%

See 1 more Smart Citation

Some (big) irreducible components of the moduli space of minimal surfaces of general type with $p_g= q = 1$ and $K^2 = 4$

Pignatelli¹

2009

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

View full text Add to dashboard Cite

Abstract. This paper is devoted to the irregular surfaces of general type having the smallest invariants, p g = q = 1. We consider the still unexplored case K 2 = 4, classifying those whose Albanese morphism has general fibre of genus 2 and such that the direct image of the bicanonical sheaf under the Albanese morphism is a direct sum of line bundles.We find 8 unirational families, and we prove that all are irreducible components of the moduli space of minimal surfaces of general type. This is unexpected because the assumption on the direct image bicanonical sheaf is a priori only a closed condition. One more unexpected property is that all these components have dimension strictly bigger than the expected one.

show abstract

Section: Modulimentioning

confidence: 99%

“…Bombieri's theorem on pluricanonical maps ensures that there is only a finite number of families of such surfaces, but recent results show that the number of these families is huge (see for instance [PK], [BCG], [BCGP] for the case p g = q = 0, [Pol1], [Pol2] for the case p g = q = 1, [Zuc] and [Pen] for the case p g = q = 2).…”

mentioning

confidence: 99%

Some (big) irreducible components of the moduli space of minimal surfaces of general type with $p_g= q = 1$ and $K^2 = 4$

Pignatelli¹

2009

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

View full text Add to dashboard Cite

show abstract

“…The calculation of N is due to Penegini and Rollenske, see [Pe08], except for the cases marked with ( * ), which were already studied in [Pol07]. The cases marked with ( * * ) also appeared in [Pol07], but the computation of N was missing.…”

Section: Introductionmentioning

confidence: 99%

“…Then α : S −→ E is the Albanese morphism of S and the genus g alb of the general Albanese fibre equals g(F ). It is proven in [Pol07,Proposition 2.3] that 3 ≤ g(F ) ≤ 5; in particular this allows us to control |G|. The covers f and h are determined by two suitable systems of generators for G, that we call V and W, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…The goal of the present paper is to give the full list of surfaces with p g = q = 1 that are isogenous to a product. Our work has to be seen as the sequel to the article [Pol07], which describe all unmixed cases with G abelian and some unmixed examples with G nonabelian. Apart from the complete list of the genera and groups occurring, our paper contains the first examples of surfaces of mixed type with q = 1.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The classification of surfaces with p_g = q = 1 isogenous to a product of curves

Carnovale

Polizzi

2009

Advg

Self Cite

View full text Add to dashboard Cite

Abstract. A smooth, projective surface S is said to be isogenous to a product if there exist two smooth curves C, F and a finite group G acting freely on C × F so that S = (C × F )/G. In this paper we classify all surfaces with pg = q = 1 which are isogenous to a product. IntroductionThe classification of smooth, complex surfaces S of general type with small birational invariants is quite a natural problem in the framework of algebraic geometry. For instance, one may want to understand the case where the Euler characteristic χ(O S ) is 1, that is, when the geometric genus p g (S) is equal to the irregularity q(S). All surfaces of general type with these invariants satisfy p g ≤ 4. In addition, if p g = q = 4 then the self-intersection K 2 S of the canonical class of S is equal to 8 and S is the product of two genus 2 curves, whereas if p g = q = 3 then K 2 S = 6 or 8 and both cases are completely described. On the other hand, surfaces of general type with p g = q = 0, 1, 2 are still far from being classified. We refer the reader to the survey paper [BaCaPi06] for a recent account on this topic and a comprehensive list of references. A natural way of producing interesting examples of algebraic surfaces is to construct them as quotients of known ones by the action of a finite group. For instance Godeaux constructed in [Go31] the first example of surface of general type with vanishing geometric genus taking the quotient of a general quintic surface of P 3 by a free action of Z 5 . In line with this Beauville proposed in [Be96, p. 118] the construction of a surface of general type with p g = q = 0, K 2 S = 8 as the quotient of a product of two curves C and F by the free action of a finite group G whose order is related to the genera g(C) and g(F ) by the equality |G| = (g(C) − 1)(g(F ) − 1). Generalizing Beauville's example we say that a surface S is isogenous to a product if S = (C × F )/G, for C and F smooth curves and G a finite group acting freely on C × F . A systematic study of these surfaces has been carried out in [Ca00]. They are of general type if and only if both g(C) and g(F ) are greater than or equal to 2 and in this case S admits a unique minimal realization where they are as small as possible. From now on, we tacitly assume that such a realization is chosen, so that the genera of the curves and the group G are invariants of S. The action of G can be seen to respect the product structure on C × F . This means that such actions fall in two cases: the mixed one, where there exists some element in G exchanging the two factors (in this situation C and F must be isomorphic) and the unmixed one, where G acts faithfully on both C and F and diagonally on their product. After [Be96], examples of surfaces isogenous to a product with p g = q = 0 appeared in [Par03] and [BaCa03], and their complete classification was obtained in [BaCaGr06]. The next natural step is therefore the analysis of the case p g = q = 1. Surfaces of general type with these invariants are the irregular ones with the lowest geometric genus ...

show abstract

On semi‐isogenous mixed surfaces

Cancian

Frapporti

2017

Mathematische Nachrichten

View full text Add to dashboard Cite

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

show abstract

On Surfaces of General Type withp_g = q = 1 Isogenous to a Product of Curves

Cited by 21 publications

References 20 publications

Some (big) irreducible components of the moduli space of minimal surfaces of general type with $p_g= q = 1$ and $K^2 = 4$

Some (big) irreducible components of the moduli space of minimal surfaces of general type with $p_g= q = 1$ and $K^2 = 4$

The classification of surfaces with p_g = q = 1 isogenous to a product of curves

On semi‐isogenous mixed surfaces

Contact Info

Product

Resources

About

On Surfaces of General Type withpg = q = 1 Isogenous to a Product of Curves

Cited by 21 publications

References 20 publications

Some (big) irreducible components of the moduli space of minimal surfaces of general type with $p_g= q = 1$ and $K^2 = 4$

Some (big) irreducible components of the moduli space of minimal surfaces of general type with $p_g= q = 1$ and $K^2 = 4$

The classification of surfaces with pg = q = 1 isogenous to a product of curves

On semi‐isogenous mixed surfaces

Contact Info

Product

Resources

About

On Surfaces of General Type withp_g = q = 1 Isogenous to a Product of Curves

The classification of surfaces with p_g = q = 1 isogenous to a product of curves