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.Comment: 24 page

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.Comment: 12 pages, 1 figur

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 $D$.Comment: 34 pages, 1 figur

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.Comment: 11 page

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.Comment: 50 page

Exceptional singular Q-homology planes

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*-ruled or the surface is up to isomorphism one of two exceptional surfaces of Kodaira dimension zero. For both exceptional surfaces the Kodaira dimension of the smooth locus is zero and the singular locus consists of a unique point of type A1 and A2 respectively. These surfaces can be constructed starting from two classical configurations of lines on the projective plane: the dual Hesse configuration and the complete quadrangle.Comment: 19 pages, 8 figure

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

Let E ⊆ P² be a complex curve homeomorphic to the projective line. The Negativity Conjecture asserts that the Kodaira–Iitaka dimension of K_X + 1/ 2 D , where (X, D) ⟶ (P², 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 complements admit no 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