Related papers: A counterexample to a conjecture on linear systems…

We proved a truncated second main theorem of level one with explicit exceptional sets for analytic maps into $\mathbb P^2$ intersecting the coordinate lines with sufficiently high multiplicities. As applications, we studied some cases of…

In this paper, besides a counterexample to Bloch's principle, normality criteria leading to counterexamples to the converse of Bloch's principle in several complex variables are proved. Some Picard-type theorems and their corresponding…

A famous conjecture attributed to Kodaira asks whether any compact Kaehler manifold can be approximated by projective manifolds. We confirm this conjecture on projectivized direct sums of three line bundles on three-dimensional complex tori…

We give an example of $C^k$-integrable almost complex structure that does not admit a corresponding $C^{k+1}$-complex coordinate system.

We consider the problem of bounding the dimension of the linear system of curves in ${\bf P}^2$ of degree $d$ with prescribed multiplicities $m_1,...,m_n$ at $n$ general points (\cite{Hir1},\cite{Hir2}). We propose a new method, based on…

In this paper we establish effective lower bounds on the degrees of the Debarre and Kobayashi conjectures. Then we study a more general conjecture proposed by Diverio-Trapani on the ampleness of jet bundles of general complete intersections…

In this paper we present a smallest possible counterexample to the Numerical Terao's Conjecture in the class of line arrangements in the complex projective plane. Our example consists of a pair of two arrangements with $13$ lines. Moreover,…

We introduce the notion of a combinatorial inverse system in non-commutative variables. We present two important examples, some conjectures and results. These conjectures and results were suggested and supported by computer investigations.

Let M be a Q-divisor on a smooth surface over C. In this paper we give criteria for very ampleness of the adjoint of the round-up of M. (Similar results for global generation were given by Ein and Lazarsfeld and used in their proof of…

We give a counterexample to the PIA (precise inversion of adjunction) conjecture for minimal log discrepancies. We also give a counterexample to the LSC conjecture for families.

We provide a broad class of counterexamples to a conjecture of L. de Branges concerning the superfluity of the continuity property in the axiomatic description of de Branges spaces.

In arXive:0705.3912 we studied triple-point defective very ample linear systems on regular surfaces, and we showed that they can only exist if the surface is ruled. In the present paper we show that we can drop the regularity assumption,…

We give an explicit counterexample to an entanglement inequality suggested in a recent paper [quant-ph/0005126] by Benatti and Narnhofer. The inequality would have had far-reaching consequences, including the additivity of the entanglement…

We give in this paper some squares of abelian surfaces that are counterexamples to a conjecture formulated by Zhang about the intersection of subvarieties and preperiodic points.

We settle the conjecture posed by Sziklai on the number of points of a plane curve over a finite field under the assumption that the curve is nonsingular.

It is conjectured that if a finite set of points in the plane contains many collinear triples then there is some structure in the set. We are going to show that under some combinatorial conditions such pointsets contain special…

We give a short proof of an inequality, conjectured by Tsfasman and proved by Serre, for the maximum number of points on hypersurfaces over finite fields. Further, we consider a conjectural extension, due to Tsfasman and Boguslavsky, of…

In this note we relate about the problem of evaluate the dimension of linear systems through fat points defined on generic $K3$ surfaces.

We give counterexamples to Okounkov's log-concavity conjecture for Littlewood-Richardson coefficients.

We describe examples showing the sharpness of Fujita's conjecture on adjoint bundles also in the general type case, and use these examples to formulate related bold conjectures on pluricanonical maps of varieties of general type.