Surfaces with p<sub>g</sub> = q = 2, K<sup>2</sup> = 6, and Albanese Map of Degree 2

Penegini, Matteo; Polizzi, Francesco

doi:10.4153/cjm-2012-007-0

Cited by 15 publications

(31 citation statements)

References 36 publications

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“…Then the claim will follow if we show that rank(ε) = 3, and this can be done by using the same argument as in [24,Section 3].…”

Section: Lemma 33 We Havementioning

confidence: 95%

“…Proof Let us write down the cohomology exact sequence associated with the short exact sequence in the central row of , recalling first that S is a surface of general type and therefore

h^{0} (S, T_{S}) = 0

0 ⟶ H^{0} (S, α^{*} T_{A}) ≅ {double-struckC}^{2} ⟶ H^{0} (R + Z, N_{α}) ⟶ H^{1} (S, T_{S}) \overset{ε}{⟶} H^{1} (S, α^{*} T_{A}) .

Then the claim will follow if we show that

rank (ε) = 3

, and this can be done by using the same argument as in [, Section ]. More precisely, since

T_{A}

is trivial and the Albanese map induces an isomorphism

H^{1} (S, O_{S}) ≅ H^{1} (A, O_{A})

, then

H^{1} (S, α^{*} T_{A}) ≅ H^{1} (A, T_{A})

and we can see ε as the map

H^{1} (S, T_{S}) \to H^{1} (A, T_{A})

induced on first‐order deformations by the Albanese map.…”

Section: The Moduli Spacementioning

confidence: 97%

“…As in [, Section ] we have a commutative diagram

whose central column is the pullback via

β : S ⟶ B

of the sequence

0 ⟶ T_{B} \overset{d π}{⟶} π^{*} T_{A} ⟶ {scriptO}_{E} (- E) ⟶ 0,

see [, p. 73]. The normal sheaf

N_{α}

α : S \to A

is supported on the set of critical points of α, namely on the reducible divisor

R + Z

.…”

Section: The Moduli Spacementioning

confidence: 99%

“…and that, according to [20], is called the Tschirnhausen bundle of β. As in [24,Section 3] we have a commutative diagram . Furthermore, we observe that R 2 = −2 implies that the line bundle N β has negative degree on R, hence H 0 (R, N β ) = 0.…”

Section: The Moduli Spacementioning

confidence: 99%

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A family of surfaces with and Albanese map of degree 3

Pignatelli

Polizzi

2016

Mathematische Nachrichten

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“…Then the claim will follow if we show that rank(ε) = 3, and this can be done by using the same argument as in [24,Section 3].…”

Section: Lemma 33 We Havementioning

confidence: 95%

“…Proof Let us write down the cohomology exact sequence associated with the short exact sequence in the central row of , recalling first that S is a surface of general type and therefore

h^{0} (S, T_{S}) = 0

0 ⟶ H^{0} (S, α^{*} T_{A}) ≅ {double-struckC}^{2} ⟶ H^{0} (R + Z, N_{α}) ⟶ H^{1} (S, T_{S}) \overset{ε}{⟶} H^{1} (S, α^{*} T_{A}) .

Then the claim will follow if we show that

rank (ε) = 3

, and this can be done by using the same argument as in [, Section ]. More precisely, since

T_{A}

is trivial and the Albanese map induces an isomorphism

H^{1} (S, O_{S}) ≅ H^{1} (A, O_{A})

, then

H^{1} (S, α^{*} T_{A}) ≅ H^{1} (A, T_{A})

and we can see ε as the map

H^{1} (S, T_{S}) \to H^{1} (A, T_{A})

induced on first‐order deformations by the Albanese map.…”

Section: The Moduli Spacementioning

confidence: 97%

“…As in [, Section ] we have a commutative diagram

whose central column is the pullback via

β : S ⟶ B

of the sequence

0 ⟶ T_{B} \overset{d π}{⟶} π^{*} T_{A} ⟶ {scriptO}_{E} (- E) ⟶ 0,

see [, p. 73]. The normal sheaf

N_{α}

α : S \to A

is supported on the set of critical points of α, namely on the reducible divisor

R + Z

.…”

Section: The Moduli Spacementioning

confidence: 99%

Section: The Moduli Spacementioning

confidence: 99%

See 2 more Smart Citations

A family of surfaces with and Albanese map of degree 3

Pignatelli

Polizzi

2016

Mathematische Nachrichten

Self Cite

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“…• In [PP13a] we studied some surfaces S (originally constructed in [CH06]) with p g = q = 2 and K 2 = 5. Their Albanese map α: S −→ A is a generically finite triple cover of a (1, 2)-polarized abelian surface (A, ), branched over a divisor D ∈ |2 | with an ordinary quadruple point.…”

Section: Proof If No Reducible Curve Inmentioning

confidence: 99%

A note on surfaces with p_g = q = 2 and an irrational fibration

Penegini

Polizzi

2017

Advances in Geometry

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We study several examples of surfaces with p g = q = 2 and maximal Albanese dimension that are endowed with an irrational fibration. IntroductionThe classification of surfaces S of general type with χ( S ) = 1, i.e. p g (S) = q(S), is currently an active area of research, see for instance [BCP06]. In this case some wellknown results imply p g ≤ 4 . While surfaces with p g = q = 4 and p g = q = 3 have been completely described in One of the most useful techniques used to understand the geometry of an algebraic surface S is the study of its fibrations f : S −→ C, where C is a smooth curve. If g(C) ≥ 1 we say that the f is irrational, and if all its smooth fibres are isomorphic we say that f is isotrivial. The study of irrational fibrations on surfaces with p g = q = 2 was started by Zucconi in [Z03a]. Later on, however, it was found that Zucconi's results were incomplete: see [Pe11], where the first author deals with the isotrivial case.If the image of the Albanese map of S is a curve then everything is known: S is a socalled generalized hyperelliptic surface, i.e. a quotient S = (C 1 × C 2 )/G by the diagonal action of a finite group G on C 1 × C 2 such that the Galois morphism C 1 −→ C 1 /G is unramified and C 2 /G ∼ = 1 , see [Ca00, Z03b, Pe11]. Therefore it only remains to investigate the case where S has maximal Albanese dimension, i.e. its Albanese map α: S −→ Alb(S) is generically finite; in this situation the base of any irrational fibration on S is an elliptic curve E (see Proposition 1.3). An important invariant of the fibration f is the push-forward of the relative canonical bundle f * ω S/E (see for instance [BHPV03, Chapter III]), and the aim of this paper is to explicitly calculate this invariant in many different examples related to our previous work on the subject, see [PP13a, PP13b, PP14].This article is organized as follows. In Section 1 we fix the notation and the terminology and we state some technical results needed in the sequel of the paper. Moreover, by using methods borrowed from [Fu78a,Fu78b,Ba00,CD13], we deduce the following structure result, see Propositions 1.6 and 1.7. This theorem corrects and extends Zucconi's results quoted above. For instance, in [Z03a] only the case where r = 1 is considered, and the existence of non-isotrivial irrational fibrations is overlooked. See Remark 1.9 for more details.In Section 2 we provide several examples with r = 1 and r ≥ 2, both in the isotrivial case (Examples 2.2, 2.3, 2.4, 2.5) and in the non-isotrivial one (Examples 2.6, 2.10, 2.11, 2.12). In particular, we show the existence of surfaces S with p g

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Quotients of the square of a curve by a mixed action, further quotients and Albanese morphisms

Pignatelli

2019

Rev Mat Complut

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We study mixed surfaces, the minimal resolution S of the singularities of a quotient (C × C)/G of the square of a curve by a finite group of automorphisms that contains elements not preserving the factors. We study them through the further quotientsAs first application we prove that if the irregularity is at least 3, then S is also minimal. The result is sharp.The main result is a complete description of the Albanese morphism of S through a determined further quotient (C ×C)/G ′ that is anétale cover of the symmetric square of a curve. In particular, if the irregularity of S is at least 2, then S has maximal Albanese dimension.We apply our result to all the semi-isogenous mixed surfaces of maximal Albanese dimension constructed by Cancian and Frapporti, relating them with the other constructions of surfaces of general type having the same invariants in the literature.

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Surfaces with p_g = q = 2, K² = 6, and Albanese Map of Degree 2

Cited by 15 publications

References 36 publications

A family of surfaces with and Albanese map of degree 3

A family of surfaces with and Albanese map of degree 3

A note on surfaces with p_g = q = 2 and an irrational fibration

Quotients of the square of a curve by a mixed action, further quotients and Albanese morphisms

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Surfaces with pg = q = 2, K2 = 6, and Albanese Map of Degree 2

Cited by 15 publications

References 36 publications

A family of surfaces with and Albanese map of degree 3

A family of surfaces with and Albanese map of degree 3

A note on surfaces with pg = q = 2 and an irrational fibration

Quotients of the square of a curve by a mixed action, further quotients and Albanese morphisms

Contact Info

Product

Resources

About

Surfaces with p_g = q = 2, K² = 6, and Albanese Map of Degree 2

A note on surfaces with p_g = q = 2 and an irrational fibration