Related papers: Conic bundles and iterated root stacks

In 1982 V.G. Sarkisov proved the existense of standard models of conic fibrations over algebraically closed fields of $\operatorname{char}\neq 2$. In this paper we will prove the analogous result for three-dimensional conic fibrations over…

Yanchevski\u{i} had asked whether conic bundle surfaces over $\mathbf{P}^1_k$ are unirational when $k$ is a finite field. We give a partial answer to his question by showing that for quasi-finite fields $k$ (e.g. finite fields) a regular…

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…

We give a proof of generalizations of the classical Arakelov inequality valid for the degree $d$ of the relative canoincal bundle of a family of curves of genus $g$ over a complete curve of genus $p$ under the assumption that the monodromy…

Projective spaces for finite-dimensional vector spaces over general fields are considered. The geometry of these spaces and the theory of line bundles over these spaces is presented. Particularly, the space of global regular sections of…

A conic bundle is a contraction $X\to Z$ between normal varieties of relative dimension $1$ such that $-K_X$ is relatively ample. We prove a conjecture of Shokurov which predicts that, if $X\to Z$ is a conic bundle such that $X$ has…

We study the linearization of line bundles and the local structure of actions of connected linear algebraic groups, in the setting of seminormal varieties. We show that several classical results about normal varieties extend to that…

We study stable rationality of conic bundles $X$ over $\mathbb{P}^1$ defined over non-closed field $k$ via the cohomology of the Galois group of finite field extension $k'/k$ with action on the geometric Picard lattice of $X$.

We show that a standard conic bundle over a minimal rational surface is rational and its Jacobian splits as the direct sum of Jacobians of curves if and only if its derived category admits a semiorthogonal decomposition by exceptional…

We review the concept of a graded bundle as a natural generalisation of a vector bundle. Such geometries are particularly nice examples of more general graded manifolds. With hindsight there are many examples of graded bundles that appear…

We use the theory of logarithmic line bundles to construct compactifications of spaces of roots of a line bundle on a family of curves, generalising work of a number of authors. This runs via a study of the torsion in the tropical and…

We describe a polynomial complexity algorithm for reducing transition matrices, for vector bundles glued along a clutching-type cover of a real anisotropic conic, to canonical block diagonal forms. This is a generalization, to the real…

A stratified bundle is a vector bundle which is a D-module. We show that regular singularity of stratified bundles on smooth varieties in positive characteristic is preserved by pullback and that regular singularity can be checked on…

We investigate principal bundles over a root stack. In case of dimension one, we generalize the criterion of Weil and Atiyah for a principal bundle to have an algebraic connection.

We study projective manifolds with nonamenable and non-residually finite fundamental groups. We generalize the uniformization theorem of our earlier note. We generalize a classical theorem of Maltsev about finitely generated subgroups of…

We study stable rationality properties of conic bundles over rational surfaces.

Using a Zariski topology associated to a finite field extensions, we give new proofs and generalize the primitive and normal basis theorems.

Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard…

This expository paper is concerned with the rationality problems for three-dimensional algebraic varieties with a conic bundle structure. We discuss the main methods of this theory. We sketch the proofs of certain principal results, and…

Let $ \mathcal{D} = \{D_{1}, ..., D_{\ell}\} $ be a multi-degree arrangement with normal crossings on the complex projective space $ \mathbf{P}^{n} $, with degrees $ d_{1}, ..., d_{\ell} $; let $ \Omega_{\mathbf{P}^{n}}^{1}(\log…