28 citations · 136 across the 17 of their papers we have counts for
16 papers · 1 filter
Completions of affine spaces into Mori fiber spaces with non-rational fibers
Adrien Dubouloz, Takashi Kishimoto, Karol Palka
We describe a method to construct completions of affine spaces into total spaces of -factorial terminal Mori fiber spaces over the projective line. As an application we…
Complex planar curves homeomorphic to a line have at most four singular points
Mariusz Koras, Karol Palka
We show that a complex planar curve homeomorphic to the projective line has at most four singular points. If it has exactly four then it has degree five and is unique up to a proje…
Classification of planar rational cuspidal curves. II. Log del Pezzo models
Karol Palka, Tomasz Pełka
Let be a complex curve homeomorphic to the projective line. The Negativity Conjecture asserts that the Kodaira-Iitaka dimension of , whe…
The Jacobian Conjecture fails for pseudo-planes
Adrien Dubouloz, Karol Palka
A smooth complex variety satisfies the Generalized Jacobian Conjecture if all its étale endomorphisms are proper. We study the conjecture for -acyclic surfaces of negat…
Classification of planar rational cuspidal curves. I. C**-fibrations
Karol Palka, Tomasz Pełka
To classify complex rational cuspidal curves it remains to classify the ones with complement of log general type, i.e. the ones for which , wh…
The Coolidge-Nagata conjecture
Mariusz Koras, Karol Palka
Let be a complex rational cuspidal curve contained in the projective plane. The Coolidge-Nagata conjecture asserts that is Cremona equivalent to a lin…
The geometry of sporadic -embeddings into
Mariusz Koras, Karol Palka, Peter Russell
A closed algebraic embedding of into is 'sporadic' if for every curve isomorphic to an affine lin…
The Coolidge-Nagata conjecture, part I
Karol Palka
Let be a complex rational cuspidal curve contained in the projective plane and let be the minimal log resolution of singularit…
A new proof of the theorems of Lin-Zaidenberg and Abhyankar-Moh-Suzuki
Karol Palka
Using the theory of minimal models of quasi-projective surfaces we give a new proof of the theorem of Lin-Zaidenberg which says that every topologically contractible algebraic curv…
Cuspidal curves, minimal models and Zaidenberg's finiteness conjecture
Karol Palka
Let be a complex rational cuspidal curve and let be the minimal log resolution of singularities. We prove that has at…
The Coolidge-Nagata conjecture holds for curves with more than four cusps
Karol Palka
Let E be a plane rational curve defined over complex numbers which has only locally irreducible singularities. The Coolidge-Nagata conjecture states that E is rectifiable, i.e. it…
Classification of singular Q-homology planes. II. C^1- and C*-rulings
Karol Palka
A Q-homology plane is a normal complex algebraic surface having trivial rational homology. We classify singular Q-homology planes which are C^1- or C*-ruled. We analyze their compl…
Recent progress in the geometry of Q-acyclic surfaces
Karol Palka
We give a survey of results on the geometry of complex algebraic Q-acyclic surfaces, so-called 'Q-homology planes', including some recent results.
Singular Q-homology planes of negative Kodaira dimension have smooth locus of non-general type
Karol Palka, Mariusz Koras
We continue the program of classification of normal Q-acyclic surfaces defined over the field of complex numbers, so-called 'Q-homology planes'. Here we show that if a Q-homology p…
Exceptional singular Q-homology planes
Karol Palka
We consider singular Q-acyclic surfaces with smooth locus of non-general type. We prove that if the singularities are topologically rational then the smooth locus is C^1- or C*-rul…
Classification of singular Q-homology planes. I. Structure and singularities
Karol Palka
A Q-homology plane is a normal complex algebraic surface having trivial rational homology. We obtain a structure theorem for Q-homology planes with smooth locus of non-general type…