Related papers: Negative curves on algebraic surfaces

We study normal analytic compactifications of C^2 and describe their singularities and configuration of curves at infinity, in particular improving and generalizing results of (Brenton, Math. Ann. 206:303--310, 1973). As a by product we…

We classify 'primitive normal compactifications' of C^2 (i.e. normal analytic surfaces containing C^2 for which the curve at infinity is irreducible), compute the moduli space of these surfaces and their groups of auomorphisms. In…

The surfaces considered are real, rational and have a unique smooth real $(-2)$-curve. Their canonical class $K$ is strictly negative on any other irreducible curve in the surface and $K^2>0$. For surfaces satisfying these assumptions, we…

When $a\ge2$, we show that a general pointed curve never interpolates through the expected number of points in the Hirzebruch surface $\mathcal{H}_a$, with one exception. In the exceptional case, the number of such interpolating maps is…

Let F be a polarized irreducible holomorphic symplectic fourfold, deformation equivalent to the Hilbert scheme parametrizing length-two zero-dimensional subschemes of a K3 surface. The homology group H^2(F,Z) is equipped with an integral…

I consider the class of surfaces $X$ over algebraically closed fields with numerical invariants given in the title. In characteristic zero, this class contains fake projective planes which were introduced by David Mumford. I prove that in…

A complex algebraic surface $S$ is a $\mathbb{Q}$-homology plane if $H_{i}(S,\mathbb{Q})=0$ for $i>0$. The Negativity Conjecture of Palka asserts that $\kappa(K_{X}+\tfrac{1}{2}D)=-\infty$, where $(X,D)$ is a log smooth completion of $S$.…

We present an effective criterion to determine if a normal analytic compactification of C^2 with one irreducible curve at infinity is algebraic or not. As a by product we establish a correspondence between normal algebraic compactifications…

We construct a smooth and projective surface over an arbitrary number field that is a counterexample to the Hasse principle but has the infinite etale Brauer-Manin set. We also construct a surface with a unique rational point and the…

We give counterexamples to the Kawamata-Viehweg vanishing theorem on ruled surfaces in positive characteristic, and prove that if there is a counterexample to the Kawamata-Viehweg vanishing theorem on a geometrically ruled surface f:X-->C,…

We study the algebraic exceptional set of a three-component curve $B$ with normal crossings on a Hirzebruch surface $\mathbb{F}_e$. If $K_{\mathbb{F}_{e}}+B$ is big and no component of $B$ is a fiber or the rational curve with negative…

We consider the question: ``How bad can the deformation space of an object be?'' The answer seems to be: ``Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.'' We show this for a number of…

In this note we study the local negativity for certain configurations of smooth rational curves in smooth surfaces with numerically trivial canonical class. We show that for such rational curves there is a bound for the so-called local…

The study of algebraic curves $\cX$ with numerous automorphisms in relation to their genus $g(\cX)$ is a well-established area in Algebraic Geometry. In 1995, Irokawa and Sasaki \cite{Sasaki} gave a complete classification of curves over…

In characteristic $p>0$ and for $q$ a power of $p$, we compute the number of nonplanar rational curves of arbitrary degrees on a smooth Hermitian surface of degree $q+1$ under the assumption that the curves have a parametrization given by…

In this paper, we obtain an explicit arithmetic intersection formula on a Hilbert modular surface between the diagonal embedding of the modular curve and a CM cycle associated to a non-biquadratic CM quartic field. This confirms a special…

We study deformations of rational curves and their singularities in positive characteristic. We use this to prove that if a smooth and proper surface in positive characteristic $p$ is dominated by a family of rational curves such that one…

Let f: Y -> CP^2 be a birational morphism of non-singular (rational) surfaces. We give an effective (necessary and sufficient) criterion for algebraicity of the surfaces resulting from contraction of the union of the strict transform of a…

The present paper concerns the question of the violation of the r-th inequality for extremal curves in the projective r-space, posed by T. Kato and G. Martens. We show that the answer is negative in many cases. The result is obtained by a…

Using the Donaldson-Auroux theory, we construct complete intersections in complex projective manifolds, which are negatively curved in various ways. In particular, we prove the existence of compact simply connected Kahler manifolds with…