Classification of Enriques surfaces with finite automorphism group in characteristic 2

Katsura, Takeshi; Kondō, Shigeyuki; Martin, Gebhard

doi:10.48550/arxiv.1703.09609

Cited by 5 publications

(26 citation statements)

References 27 publications

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“…There exists an Enriques surface S over F with a quasi-elliptic fibration π : S → B := P 1 (cf. [KKM,Section 11]). Then π has multiple fibres (cf.…”

Section: Boundedness Of Regular Curvesmentioning

confidence: 99%

Invariants of algebraic varieties over imperfect fields

Tanaka¹

2019

Preprint

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We introduce four invariants of algebraic varieties over imperfect fields, each of which measures either geometric nonnormality or geometric non-reducedness. The first objective of this article is to establish fundamental properties of these invariants. We then apply our results to curves over imperfect fields. In particular, we establish a genus change formula and prove the boundedness of non-smooth regular curves of genus one. We also compute our invariants for some explicit examples.2010 Mathematics Subject Classification. 14G17, 14D06. Key words and phrases. imperfect fields, generic fibres, positive characteristic. 1 2 HIROMU TANAKA 5.1. Definition 36 5.2. Base changes 37 5.3. Behaviour under morphisms 38 5.4. Frobenius factorisation 39 5.5. Q-Gorenstein index 40 6. Frobenius length of geometric non-reducedness m F (X/k) 42 6.1. Definition 42 6.2. Base changes 43 6.3. Behaviour under morphisms 44 6.4. Frobenius factorisation 44 7. Thickening exponents ǫ(X/k) 45 7.1. Definition 45 7.2. Base changes 47 7.3. Behaviour under morphisms 49 8. Relation between invariants 50 8.1. Geometric generic embedding dimension 50 8.2. Geometric non-reducedness 51 8.3. Geometric non-normality 52 8.4. Summary of relations 53 9. Examples 53 9.1. q-Fermat complete intersections 54 9.2. Curves of genus zero 63 9.3. Curves of genus one 65 10. Genus change formula 67 11. Boundedness of regular curves 70 11.1. Very ampleness 70 11.2. Boundedness 72 11.3. Non-smooth geometrically reduced curves of genus one 75 References 79

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“…There exists an Enriques surface S over F with a quasi-elliptic fibration π : S → B := P 1 (cf. [KKM,Section 11]). Then π has multiple fibres (cf.…”

Section: Boundedness Of Regular Curvesmentioning

confidence: 99%

Invariants of algebraic varieties over imperfect fields

Tanaka¹

2019

Preprint

View full text Add to dashboard Cite

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“…We shall facilitate that X admits a 3-torsion section at Q = (0, 0), precisely it takes the general shape X : y 2 + a 1 xy + a 3 y = x 3 (10) (with reducible fibers at the zeroes of a 3 , presently of Kodaira types I 6 and IV ). It follows from divisibility considerations in K[s], or more general from the theory of Mordell-Weil lattices [25], that P.Q = 2.…”

Section: Explicit Equations For Root Typementioning

confidence: 99%

“…But then comparing the substitution into (10) with the relation obtained from ı * P = −P , we read off ν = 1. We write out g = g 2 s 2 +g 1 s+g 0 (with g 2 +g 1 = 0, for otherwise g = h) and solve for the substitution into (10),…”

Section: Explicit Equations For Root Typementioning

confidence: 99%

“…The subtleties and special properties which they already display over the complex numbers are further augmented in characteristic two where additional intriguing features arise. Despite great recent progress (see [10], [15]), there are still many open problems, both on the abstract and the explicit side. This paper contributes to both sides by considering Enriques surfaces which carry a rank nine configuration of smooth rational curves.…”

Section: Introductionmentioning

confidence: 99%

“…All the root types in question can be found in the first column of Table 1. Theorem 1.2 largely builds on deep work of Dolgachev-Liedtke (unpublished as of yet) and Katsura-Kondō-Martin [10] which provide a plentitude of constructions and examples of classical Enriques surfaces with presribed configurations of smooth rational curves. At present our methods do not yield enough information to fully classify the moduli of the classical Enriques surfaces from Theorem 1.2 (whose dimensions do in fact vary, largely due to the presence of quasi-elliptic fibrations); neither do they apply directly to the remaining class of supersingular Enriques surfaces (compare Remark 10.2).…”

Section: Introductionmentioning

confidence: 99%

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Q_l-cohomology projective planes and singular Enriques surfaces in characteristic two

Schütt

2017

Preprint

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We classify singular Enriques surfaces in characteristic two supporting a rank nine configuration of smooth rational curves. They come in one-dimensional families defined over the prime field, paralleling the situation in other characteristics, but featuring novel aspects. Contracting the given rational curves, one can derive algebraic surfaces with isolated ADEsingularities and trivial canonical bundle whose Q ℓ -cohomology equals that of a projective plane. Similar existence results are developed for classical Enriques surfaces. We also work out an application to integral models of Enriques surfaces (and K3 surfaces).

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Enriques surfaces with finite automorphism group in positive characteristic

Martin¹

2019

Alg. Geom.

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We classify Enriques surfaces with smooth K3 cover and finite automorphism group in arbitrary positive characteristic. The classification is the same as over the complex numbers except that some types are missing in small characteristics. Moreover, we give a complete description of the moduli of these surfaces. Finally, we realize all types of Enriques surfaces with finite automorphism group over the prime fields F p and Q whenever they exist.

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Classification of Enriques surfaces with finite automorphism group in characteristic 2

Cited by 5 publications

References 27 publications

Invariants of algebraic varieties over imperfect fields

Invariants of algebraic varieties over imperfect fields

Q_l-cohomology projective planes and singular Enriques surfaces in characteristic two

Enriques surfaces with finite automorphism group in positive characteristic

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