Related papers: On fibrations whose geometric fibers are nonreduce…

We prove that, under mild restrictions, the space of codimension-one foliations of degree one on a smooth projective complete intersection has two irreducible components of logarithmic type. We also prove that the same conclusion holds for…

Given a fibration over the circle, we relate the eigenspace decomposition of the algebraic monodromy, the homological finiteness properties of the fiber, and the formality properties of the total space. In the process, we prove a more…

In this paper we study flatness of the restriction on some special subgerms (e.g. the reduction and the unmixed part) of the total space of a flat morphism over a smooth base space. We give a relationship between reducedness of the total…

This work explores the space of foliations on projective spaces over algebraically closed fields of positive characteristic, with a particular focus on the codimension one case. It describes how the irreducible components of these spaces…

We investigate the existence, and lack of unicity, of a holomorphic fibration by discs transversal to a rational curve in a complex surface.

We study surfaces with one constant principal curvature in Riemannian and Lorentzian three-dimensional space forms. Away from umbilic points they are characterized as one-parameter foliations by curves of constant curvature, each of these…

Rational algebraic curves have been intensively studied in the last decades, both from the theoretical and applied point of view. In applications (e.g. level curves, linear homotopy deformation, geometric constructions in computer aided…

Here are studied pairs of transversal foliations with singularities, defined on the Elliptic region (where the Gaussian curvature $\mathcal K$ is positive) of an oriented surface immersed in $\mathbb R^3$. The leaves of the foliations are…

We study algebraic and geometric properties of metric spaces endowed with dilatation structures, which are emergent during the passage through smaller and smaller scales. In the limit we obtain a generalization of metric affine geometry,…

It is known that there is at least an invariant analytic curve passing through each of the components in the complement of nodal singularities, after the reduction of singularities of a germ of singular foliation in ${\mathbb C}^2,0$}.…

We show that there exists an admissible nonorientable genus $g$ Lefschetz fibration with only one singular fiber over a closed orientable surface of genus $h$ if and only if $g \geq 4$ and $h \geq 1$.

We study structures of embedded projective manifolds swept out by cubic varieties. We show if an embedded projective manifold is swept out by high-dimensional smooth cubic hypersurfaces, then it admits an extremal contraction which is a…

The aim of this paper is to classify reduction types of algebraic curves. Reduction types capture the discrete invariants of fibres in one-dimensional families of curves, and they have been described in genus 1, 2 and 3. For fixed genus…

A general model for geometric structures on differentiable manifolds is obtained by deforming infinitesimal symmetries. Specifically, this model consists of a Lie algebroid, equipped with an affine connection compatible with the Lie…

We prove some conditions for the existence of higher dimensional algebraic fibering of group extensions. This leads to various corollaries on incoherence of groups and some geometric examples of algebraic fibers of type $F_n$ but not…

For a general birational projection of a smooth nondegenerate projective $n$-fold from $\mathbb P^{n+c}$ to $\mathbb P^m$, $n<m\leq(n+c)/2$, all fibres have total length asymptotically bounded by $2^{\sqrt{n}+1} $ and the fibres are locally…

In this work, we investigate the behaviour of the covering gonality of a very general hypersurface in a product of projective spaces. Inspired by the work of Bastianelli, Ciliberto, Flamini and Suppino in [BCFS19] which addresses the case…

We prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed Riemannian manifolds in terms of their index and area, restricting to the case where the hypersurface has dimension less than seven. In…

We consider threefold del Pezzo fibrations over a curve germ whose central fiber is non-rational. Under the additional assumption that the singularities of the total space are at worst ordinary double points, we apply a suitable base change…

We prove that any compact surface with constant positive curvature and conical singularities can be decomposed into irreducible components of standard shape, glued along geodesic arcs connecting conical singularities. This is a spherical…