Related papers: Fibres of embedding dimension 2 in generic project…

We bound the complexity of the fibers of the generic linear projection of a smooth variety in terms of a new family of invariants. These invariants are closely related to ideas of John Mather, and we give a simple proof of his bound on the…

A generalised notion of connection on a fibre bundle E over a manifold M is presented. These connections are characterised by a smooth distribution on E which projects onto a (not necessarily integrable) distribution on M and which, in…

We prove asymptotics for the proportion of fibres with a rational point in a conic bundle fibration. The basis of the fibration is a general hypersurface of low degree.

We contribute a new algebraic method for computing the orthogonal projections of a point onto a rational algebraic surface embedded in the three dimensional projective space. This problem is first turned into the computation of the finite…

Let f:X->Y be an algebraic fiber space such that the general fiber has a good minimal model. We show that if f is the Iitaka fibration or if f is the Albanese map of relative dimension no more than three, then X has a good minimal model.

We assume that the existence and termination conjecture for flips holds. A complex projective manifold is said to be {\it of almost general type} if the intersection number of the canonical divisor with every very general curve is strictly…

An abelian fibration is a proper projective surjective map of complex varieties with general fiber an abelian variety. Consider a multiple fiber of an abelian fibration, and let $m_1, ..., m_k$ be the multiplicities of its irreducible…

The theory of frames normal for general connections on differentiable bundles is developed. Links with the existing theory of frames normal for covariant derivative operators (linear connections) in vector bundles are revealed. The…

We discuss the flatness property of some fiber type contractions of complex smooth projective varieties of arbitrary dimensions. We relate the flatness of some morphisms having one-dimensional fibers with their conic bundles structures,…

Let $\f: X \ra Z$ be a proper surjective map from a smooth complex manifold $X$ onto a normal variety $Z$. If $\f$ has connected fibers and $-K_X$ is $\f$-ample then $\f$ is called a good contraction. In the present paper we study good…

A finite morphism $f:X\to \mathbb P^2$ of a a smooth irreducible projective surface $X$ is called an almost generic cover if for each point $p\in \mathbb P^2$ the fibre $f^{-1}(p)$ is supported at least on $deg(f)-2$ distinct points and $f$…

A common assumption in representation learning is that globally well-distributed embeddings support robust and generalizable representations. This focus has shaped both training objectives and evaluation protocols, implicitly treating…

The concept of generalised (in the sense of Colombeau) connection on a principal fibre bundle is introduced. This definition is then used to extend results concerning the geometry of principal fibre bundles to those that only have a…

It is easy to imagine that a subvariety of a vector bundle, whose intersection with every fibre is a vector subspace of constant dimension, must necessarily be a sub-bundle. We give two examples to show that this is not true, and several…

A 2-dimensional direction-length framework is a collection of points in the plane which are linked by pairwise constraints that fix the direction or length of the line segments joining certain pairs of points. We represent it as a pair…

In this partly expository paper we discuss conditions for the global injectivity of $C^2$ semi-algebraic local diffeomorphisms $f:\mathbb{R}^n \to \mathbb{R}^n$. In case $n > 2$, we consider the foliations of $\mathbb{R}^n$ defined by the…

We consider fibrations of genus 2 over complex surfaces. The purpose of this paper is primarily to provide a geometric description of the possible structures of the fibration on a neighborhood of a singular fiber. In particular it is shown…

In this note we discuss a common misconception, namely that embeddings are always used to reduce the dimensionality of the item space. We show that when we measure dimensionality in terms of information entropy then the embedding of sparse…

The measurement of distance between two objects is generalized to the case where the objects are no longer points but are one-dimensional. Additional concepts such as non-extensibility, curvature constraints, and non-crossing become central…

We prove that the r-th pluricanonical maps of threefolds of general type have birationally bounded fibers if $r\geqslant 2$. Similarly, we prove that the r-th pluricanonical maps of fourfolds of general type have birationally bounded fibers…