Related papers: K-trivial structures on Fano complete intersection…

A topological hyperplane is a subspace of R^n (or a homeomorph of it) that is topologically equivalent to an ordinary straight hyperplane. An arrangement of topological hyperplanes in R^n is a finite set H such that k topological…

Let $X$ be a smooth projective surface such that linear and numerical equivalence of divisors on $X$ coincide and let $\sigma\subseteq |D|$ be a linear pencil on $X$ with integral general fibers. A fiber of $\sigma$ will be called special…

In this article we study how the birational geometry of a normal projective variety $X$ is influenced by a normal subvariety $A \subset X.$ One of the most basic examples in this context is provided by the following situation. Let $f:X\to…

Given a smooth projective variety $X$ of Kodaira dimension zero, we show that there exists a constant $m$ depending on two invariants of the general fiber of the Albanese map, such that $|mK_X|\neq\emptyset$ .

In this paper we extend to the singular setting the theory of Fano foliations developed in our previous paper. A Q-Fano foliation on a complex projective variety X is a foliation F whose anti-canonical class is an ample Q-Cartier divisor.…

A recent paper showed how to find sets of finite affine or projective planes constructed on a common set of points, so that lines of one plane meet lines of a different plane in at most two points. In this paper, those results are…

Affine structures on a Lie groupoid, including affine $k$-vector fields, $k$-forms and $(p,q)$-tensors are studied. We show that the space of affine structures is a 2-vector space over the space of multiplicative structures. Moreover, the…

We show that a line arrangement in the complex projective plane supports a nontrivial resonance variety if and only if it is the underlying arrangement of a "multinet," a multi-arrangement with a partition into three or more equinumerous…

A vector bundle whose projectivization becomes a weak Fano variety is called a weak Fano bundle. We present classification results for rank 2 weak Fano bundles on higher-dimensional quadrics $Q^n$ of dimension $\ge 5$.

The purpose of this paper is to compute the degree of irrationality of hypersurfaces of sufficiently high degree in various Fano varieties: quadrics, Grassmannians, products of projective space, cubic threefolds, cubic fourfolds, and…

Suppose X is a smooth projective 3-fold of general type and |mK_X| is composed of a pencil of surfaces with m>1. This pencil naturally induces a fibration f:X->C onto a smooth curve C after the Stein-factorization, which is the main objects…

Let $X$ be a normal, connected and projective variety over an algebraically closed field $k$. It is known that a vector bundle $V$ on $X$ is essentially finite if and only if it is trivialized by a proper surjective morphism $f:Y\to X$. In…

A line g is a transversal to a family F of convex polytopes in 3-dimensional space if it intersects every member of F. If, in addition, g is an isolated point of the space of line transversals to F, we say that F is a pinning of g. We show…

Let $X$ be a four-dimensional projective variety defined over the field of complex numbers with only terminal singularities. We prove that if the intersection number of the canonical divisor $K$ with every very general curve is positive…

We consider the multicanonical systems $\vert mK_{S}\vert$ of quasi-elliptic surfaces with Kodaira dimension $1$ in characteristic 2. We show that for any $m \geq 6$ $\vert mK_{S}\vert$ gives the structure of quasi-elliptic fiber space, and…

It is well known that all torsors under an affine algebraic group over an algebraically closed field are trivial. We note that under suitable conditions this also holds if the the group is not necessarily of finite type. This has an…

In the previous paper by Pereira and the author, it was proved that any pencil of plane curves of degree greater than one with irreducible generic fiber can have at most five completely reducible fibers although no examples with five such…

This article studies germs of holomorphic vector fields at the origin of C3 that are tangent to holomorphic foliations of codimension one. Two situations are considered. First, we assume hypotheses on the reduction of singularities of the…

Let $Z$ be a nondegenerate hypersurface in $d$-dimensional torus $(\mathbb{C}^*)^d$ defined by a Laurent polynomial $f$ with a $d$-dimensional Newton polytope $P$. The subset $F(P) \subset P$ consisting of all points in $P$ having integral…

We study conditions under which a finite simplicial complex $K$ can be mapped to $\mathbb R^d$ without higher-multiplicity intersections. An almost $r$-embedding is a map $f: K\to \mathbb R^d$ such that the images of any $r$ pairwise…