Real Enriques Surfaces

Degtyarev, Alex; Itenberg, Ilia; Kharlamov, Viatcheslav

doi:10.1007/bfb0103960

Cited by 88 publications

(131 citation statements)

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“…Then we get a real nonsingular curve A = s + A 1 in | − 2K F 4 | where A 1 is a real nonsingular curve in |12c + 3s| on RF 4 . Suppose that the fixed point set X ϕ (R) is homeomorphic to Σ g ∪ kS 2 . Then the region π(X ϕ (R)) (⊂ RF 4 ) is the disjoint union of an annulus with (g − 1) holes and k disks.…”

Section: 2mentioning

confidence: 99%

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On real anti-bicanonical curves with one double point on the 4-th real Hirzebruch surface

Saito¹

2015

jsing

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Abstract. We list up all the candidates for the real isotopy types of real anti-bicanonical curves with one real nondegenerate double point on the 4-th real Hirzebruch surface RF 4 by enumerating the connected components of the moduli space of real 2-elementary K3 surfaces of type (S, θ) ∼ = ((3, 1, 1), −id). We also list up all the candidates for the non-increasing simplest degenerations of real nonsingular anti-bicanonical curves on RF 4 . We find an interesting correspondence between the real isotopy types of real anti-bicanonical curves with one real nondegenerate double point on RF 4 and the non-increasing simplest degenerations of real nonsingular anti-bicanonical curves on RF 4 . This correspondence is very similar to the one provided by the rigid isotopic classification of real sextic curves on RP 2 with one real nondegenerate double point by I. Itenberg.

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Section: 2mentioning

confidence: 99%

“…The real part of F 4 is homeomorphic to the 2-dimensional torus or the empty set. On the other hand, F 3 has a unique real structure and its real part is homeomorphic to the Klein's bottle (see [2]). …”

Section: Introductionmentioning

confidence: 99%

On real anti-bicanonical curves with one double point on the 4-th real Hirzebruch surface

Saito¹

2015

jsing

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“…This formula is due to Rokhlin (at least in the case when ι * = 0, see [8], n • 4). It may be obtained from the generalised Rokhlin-Guillou-Marin congruence (see [2], Theorem 2.6.1, or [1], Théorème 3) by reducing it modulo 8 and plugging in an elementary algebraic property of the Brown invariant (see [2], 3.4.4(2)). Applying (1.1) to the characteristic submanifolds Σ ⊔ S and Σ, respectively, and using the orientability of Σ, we obtain that (The second congruence follows also from van der Blij's lemma, see [13], §II.5.)…”

Section: Viro Index (Cf [19] §4mentioning

confidence: 99%

Homology class of a Lagrangian Klein bottle

Nemirovski¹

2009

Izv. Math.

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Theorem 0.2 in the author's paper [14] asserts that a Lagrangian Klein bottle in a projective complex surface must have non-zero mod 2 homology class. A gap in the topological part of the proof of this result was pointed out by Leonid Polterovich. (It is erroneously claimed in §3.6 that a diffeomorphism of an oriented real surface acts in some natural way on the spinor bundle of the surface.)Recently, Vsevolod Shevchishin corrected both the statement and the proof of that theorem. On the one hand, he showed that the result is false as it stands by producing an example of a nullhomologous Lagrangian Klein bottle in a bi-elliptic surface. On the other hand, he proved that the conclusion holds true under an additional assumption which, in retrospect, appears to be rather natural. Shevchishin's proof of Theorem A uses the Lefschetz pencil approach proposed in [14], the combinatorial structure of mapping class groups, and the above description of symplectic manifolds with c 1 (X, ω)·[ω] > 0. The purpose of the present paper is to give an alternative proof in which the first two ingredients are replaced by somewhat more traditional four-manifold topology. It should be noted that though closer in spirit to the work of Polterovich [17] and , this argument has been found by interpreting the results obtained in [18] in other geometric terms.The contents of the paper should be clear from its section titles. For a more comprehensive discussion of Givental's Lagrangian embedding problem for the Klein bottle [6], see the introductions to [14] and [18].The author is grateful to Viatcheslav Kharlamov for several helpful remarks.

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“…This class consists of Abelian, hyperelliptic, K3-, and Enriques surfaces. They are all quasi-simple (see [2] and [1]; recall that, by definition, hyperelliptic and Enriques surfaces are respectively quotients of Abelian and K3-surfaces by free involutions). Furthermore, quasi-simplicity of hyperelliptic and Enriques surfaces extends to quasi-simplicity of the quotients of Abelian and K3-surfaces by certain finite group actions, see [3].…”

Section: Quasi-simplicitymentioning

confidence: 99%

“…This is what we call individual finiteness, which we understand as finiteness of the number of conjugacy classes of real structures on a given variety (note that individual finiteness understood in this way extends to hypersurfaces of degree 4 in projective spaces of dimension 3, see [2]). …”

Section: Finitenessmentioning

confidence: 99%