Involutions on surfaces with p g  = q = 0 and K 2 = 3

Rito, Carlos

doi:10.1007/s10711-011-9612-1

Cited by 7 publications

(6 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We have that S/i 1 is birational to an Enriques surface, S/i 3 is a rational surface and the bicanonical map of S is not composed with i 1 , i 2 and is composed with i 3 . The surface S/i 2 is birational to an Enriques surface in the We omit the proof for these facts: it is similar to the one given in [19] for an example with K 2 S = 3.…”

Section: Involutions On Smentioning

confidence: 87%

Some bidouble planes with p_g= q = 0 and 4 ≤ K²≤ 7

Rito

2015

Int. J. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Involutions On Smentioning

confidence: 87%

Some bidouble planes with p_g= q = 0 and 4 ≤ K²≤ 7

Rito

2015

Int. J. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Example 6.1 (K 2 S = 3). Rito gives an example in [Ri1,5.2] in which S is the minimal model of a bidouble cover of P 2 and K 2 S = 3. Write σ 1 , σ 2 , σ 3 for the 3 involutions of S corresponding to the bidouble cover.…”

Section: Numerical Campedelli Surfaces (K 2 = 2)mentioning

confidence: 99%

Some surfaces of general type for which Bloch's conjecture holds

Pedrini

Weibel

2016

Recent Advances in Hodge Theory

View full text Add to dashboard Cite

Abstract. We give many examples of surfaces of general type with p g = 0 for which Bloch's conjecture holds, for all values of K 2 = 9. Our surfaces are equipped with an involution.Let S be a smooth complex projective surface with p g (S) = 0. Bloch's conjecture states that the Albanese map A 0 (S) 0 → Alb(S) is an isomorphism, where A 0 (S) 0 is the Chow group of 0-cycles of degree 0 on S. It is known for all surfaces except those of general type (see [BKL]). For a surface S of general type with p g (S) = 0 we also have q(S) = 0, i.e., Alb(S) = 0 and the canonical divisor satisfies 1 ≤ K 2 ≤ 9. In the decades since this conjecture was formulated, surfaces of general type have become somewhat better understood. Two key developments have been (i) the results of S. Kimura on finite dimensional motives in [Ki] and (ii) the notion of the transcendental motive t 2 (S) which was introduced in [KMP]. This includes the theorem that if S is a surface with p g (S) = q(S) = 0 then Bloch's conjecture holds for S iff t 2 (S) = 0; see Lemma 1.5.In this paper we give motivic proofs of Bloch's conjecture for several examples of surfaces of general type for each value of K 2 between 1 and 8. This includes some numerical Godeaux surfaces, classical Campedelli surfaces, Keum-Naie surfaces, Burniat surfaces and Inoue's surfaces. All these surfaces carry an involution. We can say nothing about the remaining case K 2 = 9, because a surface of general type with p g = 0 and K 2 = 9 has no involution ([DMP, 2.3]). Bloch's conjecture is satisfied by all surfaces whose minimal models arise as quotients C 1 × C 2 /G of the product of two curves of genera ≥ 2 by the action of a finite group G. A complete classification of these surfaces has been given in [BCGP] and [BCG, 0.1]; the special case where G acts freely only occurs when K 2 S = 8. We also show in Corollary 7.8 that Bloch's conjecture holds for surfaces with an involution σ for which K 2 = 8 and S/σ is rational. The only known examples withDate: April 24, 2013.

show abstract

“…Remark 5.7. There has been a growing interest for surfaces of general type with p g = 0 having an involution, see [CCML07], [CMLP08], [Rit12] and [LS10]. The "intermediate" surface Y = (C × C)/G 0 has an involution given by σ : Y → X; it has q = 0 and K 2 Y = 2K 2 S , while p g = 0 in the cases A.1.1, A.3.1 and A.3.2, and p g = 1 in the others.…”

Section: The Classification Of the Surfacesmentioning

confidence: 99%

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

Frapporti

2013

Collect. Math.

View full text Add to dashboard Cite

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.

show abstract

Involutions on surfaces with p g = q = 0 and K 2 = 3

Cited by 7 publications

References 24 publications

Some bidouble planes with p_g= q = 0 and 4 ≤ K²≤ 7

Some bidouble planes with p_g= q = 0 and 4 ≤ K²≤ 7

Some surfaces of general type for which Bloch's conjecture holds

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

Contact Info

Product

Resources

About

Involutions on surfaces with p g = q = 0 and K 2 = 3

Cited by 7 publications

References 24 publications

Some bidouble planes with pg= q = 0 and 4 ≤ K2≤ 7

Some bidouble planes with pg= q = 0 and 4 ≤ K2≤ 7

Some surfaces of general type for which Bloch's conjecture holds

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

Contact Info

Product

Resources

About

Some bidouble planes with p_g= q = 0 and 4 ≤ K²≤ 7

Some bidouble planes with p_g= q = 0 and 4 ≤ K²≤ 7