Related papers: A Generalization of Desargues' Involution Theorem

Using toric geometry we prove a B\'ezout type theorem for weighted projective spaces.

We prove that small deformations of a projective variety of general type are also projective varieties of general type, with the same plurigenera. Version 2: small changes in first half. Improved version of the second half is now a separate…

The Linearization Theorem for proper Lie groupoids organizes and generalizes several results for classic geometries. Despite the various approaches and recent works on the subject, the problem of understanding invariant linearization…

$G$-deformability of maps into projective space is characterised by the existence of certain Lie algebra valued 1-forms. This characterisation gives a unified way to obtain well known results regarding deformability in different geometries.

We prove configuration theorems that generalize the Desargues, Pascal, and Pappus theorems. Our generalization of the Desargues theorem allows us to introduce the structure of an Abelian group on the (properly extended) set of triangles…

The goal of this paper is to give a simple proof of Deligne's conjecture (proven by Fujiwara) and to generalize it to the situation appearing in our joint project with David Kazhdan on the global Langlands correspondence over function…

A Galois theory of differential fields with parameters is developed in a manner that generalizes Kolchin's theory. It is shown that all connected differential algebraic groups are Galois groups of some appropriate differential field…

In this article we introduce the notion of projective isomonodromy, which is a special type of monodromy evolving deformation of linear differential equations, based on the example of the Darboux-Halphen equation. We give an algebraic…

What do Napoleon's and Varignon's theorems have in common? We claim that they both are examples of immediately regularizing natural polygon iterations in different planar geometries and present a new and analogous result in projective…

A point $P$ in projective space is said to be Galois with respect to a hypersurface if the function field extension induced by the projection from $P$ is Galois. We present a hyperplane section theorem for Galois points. Precisely, if $P$…

An algebraic deformation theory of coalgebra morphisms is constructed.

In this paper we prove that the branch curve of a general projection of a surface to the plane is irreducible, with only nodes and cusps.

Generalizations of the classical affine Lelieuvre formula to surfaces in projective three-dimensional space and to hypersurfaces in multi- dimensional projective space are given. A discrete version of the projective Lelieuvre formula is…

The classical theory of plane projective geometry is examined constructively, using both synthetic and analytic methods. The topics include Desargues's Theorem, harmonic conjugates, projectivities, involutions, conics, Pascal's Theorem,…

In this note we are going to consider a smooth projective surface equipped with an involution and study the action of the involution at the level of Chow group of zero cycles.

In this paper, we propose a geometrical proof of the generalized mirror transformation of genus 0 Gromov-Witten invariants of degree k hypersurface in CP^{N-1}.

We construct models of involution surface bundles over algebraic surfaces, degenerating over normal crossing divisors, and with controlled singularities of the total space.

Let X be a surface with quotient singularities which admits a smoothing to the plane. We prove that X is a deformation of a weighted projective plane P(a^2,b^2,c^2), where a,b,c is a solution of the Markov equation a^2+b^2+c^2=3abc. We also…

A multivariate Gauss-Lucas theorem is proved, sharpening and generalizing previous results on this topic. The theorem is stated in terms of a seemingly new notion of convexity. Applications to multivariate stable polynomials are given.

In this article we prove a theorem that will generalize the concurrence theorems that are leading to the Franke's point, Kariya's point, and to other remarkable points from the triangle geometry.