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Karol Palka

Publications and source records attributed to Karol Palka.

At least 19 recordsLinked to original sources
arXiv2025

Classification of del {P}ezzo surfaces of rank one. III. Height 3

This article is a part of a series aimed at classifying normal del Pezzo surfaces of Picard rank one over an algebraically closed field of arbitrary characteristic, up to an isomorphism. The key invariant guiding our classification is the height, defined as the minimal number $h$ such that the minimal resolution of singularities admits a $\mathbb{P}^1$-fibration whose fiber meets the exceptional divisor $h$ times. It is expected that every singular del Pezzo surface of rank one is of height $h\leq 4$, with minor exceptions in characteristics $2$ and $3$. Having settled the case $h\leq 2$ in our previous article arXiv:2412.21174, we now give a classification in case the height equals $3$.

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arXiv2024

Classification of del Pezzo surfaces of rank one. I. Height 1 and 2. II. Descendants with elliptic boundaries

This is the first article in a series aimed at classifying normal del Pezzo surfaces of Picard rank one over algebraically closed fields of arbitrary characteristic up to an isomorphism. Our guiding invariant is the height of a del Pezzo surface, defined as the minimal intersection number of the exceptional divisor of the minimal resolution and a fiber of some $\mathbb{P}^1$-fibration. The geometry of del Pezzo surfaces gets more constrained as the height grows; in characteristic $0$ no example of height bigger than $4$ is known. In this article, we classify del Pezzo surfaces of Picard rank one and height at most $2$; in particular we describe the non-log terminal ones. We also describe a natural class of del Pezzo surfaces which have descendants with elliptic boundary, i.e. whose minimal resolution has a birational morphism onto a canonical del Pezzo surface of rank one mapping the exceptional divisor to an anti-canonical curve.

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arXiv2024

On the structure of open del Pezzo surfaces

Let $(X,D)$ be an open log del Pezzo surface of rank one, that is, $X$ is a normal projective surface of Picard rank one, the boundary $D$ is a reduced nonzero divisor on $X$, and the anti-log canonical divisor $-(K_X+D)$ is ample. We show that, up to well described exceptions in characteristics 2, 3 and 5, the smooth part of $X\setminus D$ admits an $\mathbb{A}^1$- or an $\mathbb{A}^{1}_{*}$-fibration, which extends to a $\mathbb{P}^1$-fibration of the minimal log resolution of $(X,D)$. In characteristic 0 this improves a well-known structure theorem of Miyanishi-Tsunoda. Within the proof, we classify rational anti-canonical curves contained in smooth loci of canonical del Pezzo surfaces of rank one.

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arXiv2024

Almost minimal models of log surfaces

We generalize Miyanishi's theory of almost minimal models of log smooth surfaces with reduced boundary to the case of arbitrary log surfaces defined over an algebraically closed field. Given an MMP run of a log surface $(X,D)$ we define and construct its almost minimal model, whose underlying surface has singularities not worse than $X$ and which differs from a minimal model by a contraction of some curves supported in the boundary only. For boundaries of type $rD$, where $D$ is reduced and $r\in [0,1]\cap \mathbb{Q}$, we show that if $X$ is smooth or $r\in [0,\frac{1}{2}]$ then the construction respects $(1-r)$-divisorial log terminality and $(1-r)$-log canonicity. We show that the assumptions are optimal, too.

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arXiv2019

Classification of planar rational cuspidal curves. II. Log del Pezzo models

Let $E\subseteq \mathbb{P}^2$ be a complex curve homeomorphic to the projective line. The Negativity Conjecture asserts that the Kodaira-Iitaka dimension of $K_X+\frac{1}{2}D$, where $(X,D)\to (\mathbb{P}^{2},E)$ is a minimal log resolution, is negative. We prove structure theorems for curves satisfying this conjecture and we finish their classification up to a projective equivalence by describing the ones whose complement admits no $\mathbb{C}^{**}$-fibration. As a consequence, we show that they satisfy the Strong Rigidity Conjecture of Flenner-Zaidenberg. The proofs are based on the almost minimal model program. The obtained list contains one new series of bicuspidal curves.

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arXiv2018

The Jacobian Conjecture fails for pseudo-planes

A smooth complex variety satisfies the Generalized Jacobian Conjecture if all its étale endomorphisms are proper. We study the conjecture for $\mathbb{Q}$-acyclic surfaces of negative Kodaira dimension. We show that $G$-equivariant counterexamples for infinite group $G$ exist if and only if $G=\mathbb{C}^*$ and we classify them relating them to Belyi-Shabat polynomials. Taking universal covers we get rational simply connected $\mathbb{C}^*$-surfaces of negative Kodaira dimension which admit non-proper $\mathbb{C}^*$-equivariant étale endomorphisms. We prove also that for every integers $r\geq 1, k\geq 2$ the $\mathbb{Q}$-acyclic rational hyperplane $u(1+u^{r}v)=w^k$, which has fundamental group $\mathbb{Z}_k$ and negative Kodaira dimension, admits families of non-proper étale endomorphisms of arbitrarily high dimension and degree, whose members remain different after dividing by the action of the automorphism group by left and right composition.

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arXiv2016

Classification of planar rational cuspidal curves. I. C**-fibrations

To classify complex rational cuspidal curves $E\subseteq \mathbb{P}^2$ it remains to classify the ones with complement of log general type, i.e. the ones for which $κ(K_X+D)=2$, where $(X,D)$ is a log resolution of $(\mathbb{P}^2,E)$. It is conjectured that $κ(K_X+\frac{1}{2}D)=-\infty$ and hence $\mathbb{P}^2\setminus E$ is $\mathbb{C}^{**}$-fibered, where $\mathbb{C}^{**}=\mathbb{C}^1\setminus\{0,1\}$, or $-(K_X+\frac{1}{2}D)$ is ample on some minimal model of $(X,\frac{1}{2}D)$. Here we classify, up to a projective equivalence, those rational cuspidal curves for which the complement is $\mathbb{C}^{**}$-fibered. From the rich list of known examples only very few are not of this type. We also discover a new infinite family of bicuspidal curves with unusual properties.

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arXiv2015

Cuspidal curves, minimal models and Zaidenberg's finiteness conjecture

Let $E\subseteq \mathbb{P}^2$ be a complex rational cuspidal curve and let $(X,D)\to (\mathbb{P}^2,E)$ be the minimal log resolution of singularities. We prove that $\bar E$ has at most six cusps and we establish an effective version of the Zaidenberg Finiteness Conjecture (1994) concerning Eisenbud-Neumann diagrams of $E$. This is done by analysing the Minimal Model Program run for the pair $(X,\frac{1}{2}D)$. Namely, we show that $\mathbb{P}^2\setminus E$ is $\mathbb{C}^{**}$-fibred or for the log resolution of the minimal model the Picard rank, the number of boundary components and their self-intersections are bounded.

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arXiv2015

The Coolidge-Nagata conjecture

Let $E\subseteq \mathbb{P}^2$ be a complex rational cuspidal curve contained in the projective plane. The Coolidge-Nagata conjecture asserts that $E$ is Cremona equivalent to a line, i.e. it is mapped onto a line by some birational transformation of $\mathbb{P}^2$. In arXiv:1405.5917 the second author analyzed the log minimal model program run for the pair $(X,\frac{1}{2}D)$, where $(X,D)\to (\mathbb{P}^2,E)$ is a minimal resolution of singularities, and as a corollary he established the conjecture in case when more than one irreducible curve in $\mathbb{P}^2\setminus E$ is contracted by the process of minimalization. We prove the conjecture in the remaining cases.

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arXiv2015

A new proof of the theorems of Lin-Zaidenberg and Abhyankar-Moh-Suzuki

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 curve in the complex affine plane has equation $X^n=Y^m$ in some algebraic coordinates on the plane. This gives also a proof of the theorem of Abhyankar-Moh-Suzuki concerning embeddings of the complex line into the plane. Independently, we show how to deduce the latter theorem from basic properties of $\mathbb{Q}$-acyclic surfaces.

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arXiv2014

The Coolidge-Nagata conjecture, part I

Let $E\subseteq \mathbb{P}^2$ be a complex rational cuspidal curve contained in the projective plane and let $(X,D)\to (\mathbb{P}^2,E)$ be the minimal log resolution of singularities. Applying the log minimal model program to $(X,\frac{1}{2}D)$ we prove that if $E$ has more than two singular points or if $D$, which is a tree of rational curves, has more than six maximal twigs or if $\mathbb{P}^2\setminus E$ is not of log general type then $E$ is Cremona equivalent to a line, i.e. the Coolidge-Nagata conjecture for $E$ holds. We show also that if $E$ is not Cremona equivalent to a line then the morphism onto the minimal model contracts at most one irreducible curve not contained in $D$.

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arXiv2014

The geometry of sporadic $\mathbb{C}^*$-embeddings into $\mathbb{C}^2$

A closed algebraic embedding of $\mathbb{C}^*=\mathbb{C}^1\setminus\{0\}$ into $\mathbb{C}^2$ is 'sporadic' if for every curve $A\subseteq \mathbb{C}^2$ isomorphic to an affine line the intersection with $\mathbb{C}^*$ is at least $2$. Non-sporadic embeddings have been classified. There are very few known sporadic embeddings. We establish geometric and algebraic tools to classify them based on the analysis of the minimal log resolution $(X,D)\to (\mathbb{P}^2,U)$, where $U$ is the closure of $\mathbb{C}^*$ on $\mathbb{P}^2$. We show in particular that one can choose coordinates on $\mathbb{C}^2$ in which the type at infinity of the $\mathbb{C}^*$ and the self-intersection of its proper transform on $X$ are sharply limited.

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arXiv2012

The Coolidge-Nagata conjecture holds for curves with more than four cusps

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 can be transformed into a line by a birational automorphism of the plane. We show that if it is not rectifiable then the tree of the exceptional divisor for its minimal embedded resolution of singularities has at most nine maximal twigs. This settles the conjecture in case E has more than four singular points.

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arXiv2012

Classification of singular Q-homology planes. I. Structure and singularities

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. We show that if a Q-homology plane contains a non-quotient singularity then it is a quotient of an affine cone over a projective curve by an action of a finite group respecting the set of lines through the vertex. In particular, it is contractible, has negative Kodaira dimension and only one singular point. We describe minimal normal completions of such planes.

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arXiv2012

Classification of singular Q-homology planes. II. C^1- and C*-rulings

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 completions, the number of different rulings, the number of affine lines on it and we give constructions. Together with previously known results this completes the classification of Q-homology planes with smooth locus of non-general type. We show also that the dimension of a family of homeomorphic but non-isomorphic singular Q-homology planes having the same weighted boundary, singularities and Kodaira dimension can be arbitrarily big.

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