Classification theory of non-complete algebraic surfaces

Classification of Complex Algebraic Surfaces

In this note we present the classical Enriques’ classification theorem for complex algebraic surfaces. We’ll recall basic facts about the theory of complex surfaces (structure theorems for birational maps), and discuss (using a modern (=Mori) approach) some important results like the Castelnuovo’s rationality criterion and the classification of minimal ruled surfaces. Finally, after the descrip...

An extended complete Chebyshev system of 3 Abelian integrals related to a non-algebraic Hamiltonian system

In this paper, we study the Chebyshev property of the 3-dimentional vector space $E =langle I_0, I_1, I_2rangle$, where $I_k(h)=int_{H=h}x^ky,dx$ and $H(x,y)=frac{1}{2}y^2+frac{1}{2}(e^{-2x}+1)-e^{-x}$ is a non-algebraic Hamiltonian function. Our main result asserts that $E$ is an extended complete Chebyshev space for $hin(0,frac{1}{2})$. To this end, we use the criterion and tools developed by...

Direct decompositions of non-algebraic complete lattices

For a given complete lattice L, we investigate whether L can be decomposed as a direct product of directly indecomposable lattices. We prove that this is the case if every element of L is a join of join-irreducible elements and dually, thus extending to non-algebraic lattices a result of L. Libkin. We illustrate this by various examples and counterexamples.

CLASSIFICATION OF COMPLEX ALGEBRAIC SURFACES from the point of view of MORI THEORY

On the Complete Classification of Extremal Log Enriques Surfaces

We show that there are exactly, up to isomorphisms, seven rational extremal log Enriques surfaces Z and construct all of them; among them types D19 and A19 have been shown of certain uniqueness by M. Reid. We also prove that the (degree 3 or 2) canonical covering of each of these seven Z has either X3 or X4 as its minimal resolution. Here X3 (resp. X4) is the unique K3 surface with Picard numbe...