Zur Theorie des eindeutigen Entsprechens algebraischer Gebilde

Nöther, M.

doi:10.1007/bf02106598

Cited by 29 publications

(8 citation statements)

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“…We start with Noether's inequality [Noe,Abschnitt 11]. Probably it is known to the experts that it holds in arbitrary characteristic.…”

Section: Noether's Inequality and Horikawa Surfacesmentioning

confidence: 99%

Algebraic Surfaces of General Type with Small c²₁ in Positive Characteristic

Liedtke¹

2008

Nagoya Mathematical Journal

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Abstract. We establish Noether's inequality for surfaces of general type in positive characteristic. Then we extend Enriques' and Horikawa's classification of surfaces on the Noether line, the so-called Horikawa surfaces. We construct examples for all possible numerical invariants and in arbitrary characteristic, where we need foliations and deformation techniques to handle characteristic 2. Finally, we show that Horikawa surfaces lift to characteristic zero.

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“…We start with Noether's inequality [Noe,Abschnitt 11]. Probably it is known to the experts that it holds in arbitrary characteristic.…”

Section: Noether's Inequality and Horikawa Surfacesmentioning

confidence: 99%

Algebraic Surfaces of General Type with Small c²₁ in Positive Characteristic

Liedtke¹

2008

Nagoya Mathematical Journal

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“…510-511] and Note 7.5 below.) Max Noether [55] discovered this remarkable formula in 1870's in the framework of the theory of adjoints of algebraic surfaces. For comments on its early history and for a modern direct proof see Gray [29, §2], Hulek [42, §3] and Piene [58].…”

Section: Introductionmentioning

confidence: 99%

On the Twelve-Point Theorem for $\ell$-Reflexive Polygons

Dais

2019

Electron. J. Combin.

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It is known that, adding the number of lattice points lying on the boundary of a reflexive polygon and the number of lattice points lying on the boundary of its polar, always yields 12. Generalising appropriately the notion of reflexivity, one shows that this remains true for " -reflexive polygons". In particular, there exist (for this reason) infinitely many (lattice inequivalent) lattice polygons with the same property. The first proof of this fact is due to Kasprzyk and Nill [46]. The present paper contains a second proof (which uses tools only from toric geometry) as well as the description of complementary properties of these polygons and of the invariants of the corresponding toric log del Pezzo surfaces. 1 2 denote the euclidean norm of any x ∈ R 2 . Proposition 1.1. For any nonempty subset N of R 2 the following conditions are equivalent:(i) N is a discrete subgroup of the additive group R 2 (i.e., n − n ∈ N for all n, n ∈ N, and for every n ∈ N there is an ε ∈ R >0 , s.t. B ε (n) ∩ N = {n}, where B ε (n) := x ∈ R 2 x − n ≤ ε ), and N spans the entire R 2 as R-vector space.(ii) There exists a set {b 1 , b 2 } of two R-linear independent vectors b 1 , b 2 ∈ R 2 such that N = {k 1 b 1 + k 2 b 2 | k 1 , k 2 ∈ Z} .

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“…Algebraic surfaces with geometric genus p g = 4 have been a very natural object of study since Noether's seminal paper [Noe70] in the 19-th century. Because their canonical map has image Σ 1 which most of the times is a surface in the projective 3-dimensional space P 3 , defined therefore by a single polynomial equation.…”

Section: Introductionmentioning

confidence: 99%

The moduli space of even surfaces of general type with K2=8, pg=
Catanese
¹
,
Liu
²
,
Pignatelli
³

2014
Journal de Mathématiques Pures et Appliquées
5
0
7
0
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Even surfaces of general type with K 2 = 8, pg = 4 and q = 0 were found by Oliverio [Ol05] as complete intersections of bidegree (6, 6) in a weighted projective space P (1, 1, 2, 3, 3).In this article we prove that the moduli space of even surfaces of general type with K 2 = 8, pg = 4 and q = 0 consists of two 35-dimensional irreducible components intersecting in a codimension one subset (the first of these components is the closure of the open set considered by Oliverio). All the surfaces in the second component have a singular canonical model, hence we get a new example of a generically nonreduced moduli space.Our result gives a posteriori a complete description of the half-canonical rings of the above even surfaces. The method of proof is, we believe, the most interesting part of the paper. After describing the graded ring of a cone we are able, combining the explicit description of some subsets of the moduli space, some deformation theoretic arguments, and finally some local algebra arguments, to describe the whole moduli space. This is the first time that the classification of a class of surfaces can only be done using moduli theory: up to now first the surfaces were classified, on the basis of some numerical inequalities, or other arguments, and later on the moduli spaces were investigated. FABRIZIO CATANESE, WENFEI LIU, AND ROBERTO PIGNATELLIsurfaces which we mentioned above are surfaces with 4 ≤ K 2 ≤ 7; by the work of Ciliberto and Catanese, [C81] and [Cat99], existence is known for each 4 ≤ K 2 ≤ 28.Irregular surfaces with p g = 4 were later investigated in [CS02]: in this case K 2 ≥ 8 since, by [De82], one has K 2 ≥ 2p g for irregular surfaces 1 ; while K 2 ≥ 12 if the canonical map has degree 1.Surfaces with p g = 4 and K 2 = 4 were classified by Noether and Enriques, but it took the work of Horikawa and Bauer ( [Ho76a, Ho76b, Ho78, B01]) to finish the classification of the surfaces with p g = 4 and 4 ≤ K 2 ≤ 7 (necessarily regular). These are 'essentially' classified, in the sense that the moduli space is shown to be a union of certain (explicitly described) locally closed subsets: but there is missing complete knowledge of the incidence structure of these subsets of the moduli space. We refer to the survey [BCP06b] for a good account of the range 4 ≤ K 2 ≤ 7, and to [Cat97] for a previous more general survey (containing the construction of several new examples).Minimal surfaces with K 2 = 8, p g = 4, q = 0 have been the object of further work by several authors [C81, CFM97, Ol05]. The surfaces constructed by Ciliberto have a birational canonical map, are not even, and have a trivial torsion group H 1 (S, Z) (unlike the ones considered in [CFM97]); the ones constructed by Oliverio are simply connected (see [D82]), and they are even (meaning that the canonical divisor is divisible by two: i.e., the second Stiefel Whitney class w 2 (S) = 0, equivalently, the intersection form is even).Therefore, for K 2 = 8, p g = 4, q = 0 there are at least three different topological types [Ol05, Remark 5.4], contras...

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Zur Theorie des eindeutigen Entsprechens algebraischer Gebilde

Cited by 29 publications

References 1 publication

Algebraic Surfaces of General Type with Small c²₁ in Positive Characteristic

Algebraic Surfaces of General Type with Small c²₁ in Positive Characteristic

On the Twelve-Point Theorem for $\ell$-Reflexive Polygons

The moduli space of even surfaces of general type with K2=8, pg=
Catanese
¹
,
Liu
²
,
Pignatelli
³

2014
Journal de Mathématiques Pures et Appliquées
5
0
7
0
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Zur Theorie des eindeutigen Entsprechens algebraischer Gebilde

Cited by 29 publications

References 1 publication

Algebraic Surfaces of General Type with Small c21 in Positive Characteristic

Algebraic Surfaces of General Type with Small c21 in Positive Characteristic

On the Twelve-Point Theorem for $\ell$-Reflexive Polygons

The moduli space of even surfaces of general type with K2=8, pg=Catanese1, Liu2, Pignatelli3 2014Journal de Mathématiques Pures et Appliquées5070View full textAdd to dashboardCite

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About

Algebraic Surfaces of General Type with Small c²₁ in Positive Characteristic

Algebraic Surfaces of General Type with Small c²₁ in Positive Characteristic

The moduli space of even surfaces of general type with K2=8, pg=
Catanese
¹
,
Liu
²
,
Pignatelli
³

2014
Journal de Mathématiques Pures et Appliquées
5
0
7
0
View full text Add to dashboard Cite