Algebraic Surfaces and the Miyaoka-Yau Inequality

2017 ◽  
Author(s):  
Paula Tretkoff
Keyword(s):  
Differential Geometry ◽  
Algebraic Surface ◽  
Smooth Surface ◽  
General Type ◽  
Kodaira Dimension ◽  
Algebraic Surfaces ◽  
Surfaces Of General Type ◽  
Smooth Complex ◽  
Dimension 2

This chapter discusses complex algebraic surfaces, with particular emphasis on the Miyaoka-Yau inequality and the rough classification of surfaces. Every complex algebraic surface is birationally equivalent to a smooth surface containing no exceptional curves. The latter is known as a minimal surface. Two related birational invariants, the plurigenus and the Kodaira dimension, play an important role in distinguishing between complex surfaces. The chapter first provides an overview of the rough classification of (smooth complex connected compact algebraic) surfaces before presenting two approaches that, in dimension 2, give the Miyaoka-Yau inequality. The first, due to Miyaoka, uses algebraic geometry, whereas the second, due to Aubin and Yau, uses analysis and differential geometry. The chapter also explains why equality in the Miyaoka-Yau inequality characterizes surfaces of general type that are free quotients of the complex 2-ball.

scholarly journals Algebraic Surfaces of General Type with Small c21 in Positive Characteristic

2008 ◽  
Vol 191 ◽  
pp. 111-134 ◽  
Author(s):  
Christian Liedtke
Keyword(s):  
Characteristic Zero ◽  
General Type ◽  
Positive Characteristic ◽  
Algebraic Surfaces ◽  
Surfaces Of General Type ◽  
Arbitrary Characteristic ◽  
Numerical Invariants ◽  
Characteristic 2

AbstractWe 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.

Classification of surfaces of general type with Euler number 3

2013 ◽  
Vol 2013 (679) ◽  
pp. 1-22 ◽  
Author(s):  
Sai-Kee Yeung
Keyword(s):  
Projective Plane ◽  
Recent Work ◽  
Present Article ◽  
Smooth Surface ◽  
General Type ◽  
Euler Number ◽  
Projective Planes ◽  
Surfaces Of General Type ◽  
Surface Of General Type

Abstract The smallest topological Euler–Poincaré characteristic supported on a smooth surface of general type is 3. In this paper, we show that such a surface has to be a fake projective plane unless h1, 0(M) = 1. Together with the classification of fake projective planes given by Prasad and Yeung, the recent work of Cartwright and Steger, and a proof of the arithmeticity of the lattices involved in the present article, this gives a classification of such surfaces.

scholarly journals Projective normality and syzygies of algebraic surfaces

1999 ◽  
Vol 1999 (506) ◽  
pp. 145-180 ◽  
Author(s):  
F. J Gallego ◽  
B. P Purnaprajna
Keyword(s):  
General Type ◽  
Kodaira Dimension ◽  
Algebraic Surfaces ◽  
List Type ◽  
Singular Surfaces ◽  
Surfaces Of General Type ◽  
New Techniques ◽  
Uniform Bounds ◽  
Projective Normality ◽  
Koszul Cohomology

Abstract In this work we develop new techniques to compute Koszul cohomology groups for several classes of varieties. As applications we prove results on projective normality and syzygies for algebraic surfaces. From more general results we obtain in particular the following: Mukai's conjecture (and stronger variants of it) regarding projective normality and normal presentation for surfaces with Kodaira dimension 0, and uniform bounds for higher syzygies associated to adjoint linear series,effective bounds along the lines of Mukai's conjecture regarding projective normality and normal presentation for surfaces of positive Kodaira dimension, and,results on projective normality for pluricanonical models of surfaces of general type (recovering and strengthening results by Ciliberto) and generalizations of them to higher syzygies. In addition, we also extend the above results to singular surfaces.

ℤ3-actions on Horikawa surfaces

2021 ◽  
pp. 2150097
Author(s):  
Vicente Lorenzo
Keyword(s):  
Moduli Space ◽  
General Type ◽  
Admissible Pair ◽  
Algebraic Surfaces ◽  
The Other ◽  
Connected Components ◽  
Surfaces Of General Type ◽  
Canonical Models ◽  
Normal Surfaces ◽  
Stable Surfaces

Minimal algebraic surfaces of general type [Formula: see text] such that [Formula: see text] are called Horikawa surfaces. In this note, [Formula: see text]-actions on Horikawa surfaces are studied. The main result states that given an admissible pair [Formula: see text] such that [Formula: see text], all the connected components of Gieseker’s moduli space [Formula: see text] contain surfaces admitting a [Formula: see text]-action. On the other hand, the examples considered allow us to produce normal stable surfaces that do not admit a [Formula: see text]-Gorenstein smoothing. This is illustrated by constructing non-smoothable normal surfaces in the KSBA-compactification [Formula: see text] of Gieseker’s moduli space [Formula: see text] for every admissible pair [Formula: see text] such that [Formula: see text]. Furthermore, the surfaces constructed belong to connected components of [Formula: see text] without canonical models.

scholarly journals Complete algebraic vector fields on affine surfaces

2020 ◽  
Vol 31 (03) ◽  
pp. 2050018
Author(s):  
Shulim Kaliman ◽  
Frank Kutzschebauch ◽  
Matthias Leuenberger
Keyword(s):  
Algebraic Surface ◽  
Vector Fields ◽  
Algebraic Surfaces ◽  
Algebraic Vector ◽  
Dual Graphs ◽  
Holomorphic Automorphisms

Let [Formula: see text] be the subgroup of the group [Formula: see text] of holomorphic automorphisms of a normal affine algebraic surface [Formula: see text] generated by elements of flows associated with complete algebraic vector fields. Our main result is a classification of all normal affine algebraic surfaces [Formula: see text] quasi-homogeneous under [Formula: see text] in terms of the dual graphs of the boundaries [Formula: see text] of their SNC-completions [Formula: see text].

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