Related papers: Four Points and a Quadric
Admettant l'hypoth\`ese de Schinzel et la finitude des groupes de Tate-Shafarevich des courbes elliptiques sur les corps de nombres, toute intersection lisse de deux quadriques dans l'espace projectif de dimension n satisfait au principe de…
The purpose of this paper is to present projective geometry in a synthetic, visual and intuitive style through the central notion of harmonicity which leads to harmonic curves. This presentation includes new results, unpublished proofs of…
We show that the theorem of the three perpendiculars holds in any n-dimensional space form.
This paper proves a generalization of the Butterfly Theorem, a classical Euclidean result, which is valid in the complex projective plane.
We reprove the strong Hanani-Tutte theorem on the projective plane. In contrast to the previous proof by Pelsmajer, Schaefer and Stasi, our method is constructive and does not rely on the characterization of forbidden minors, which gives…
Arguably the simplest variation of this style of proof as we avoid reducing to the cubic case entirely.
It is well-known that Lagrange's four-square theorem, stating that every natural number may be written as the sum of four squares, may be proved using methods from the classical theory of modular forms and theta functions. We revisit this…
We prove that every polytope described by algebraic coordinates is the face of a projectively unique polytope. This provides a universality property for projectively unique polytopes. Using a closely related result of Below, we construct a…
Let X be a non-singular projective hypersurface of degree 4, which is defined over the rational numbers. Assume that X has dimension 39 or more, and that X contains a real point and p-adic points for every prime p. Then X is shown to…
We prove the Hasse principle for a smooth proper model of a geometrically integral non-conical intersection of two quadrics in the projective space of dimension 7 over a number field. This result generalizes the result of Heath-Brown who…
The fundamental theorem of affine geometry is a classical and useful result. For finite-dimensional real vector spaces, the theorem roughly states that a bijective self-mapping which maps lines to lines is affine. In this note we prove…
We give a geometric proof that Hasse principle holds for the following varieties defined over global function fields: smooth quadric hypersurfaces in odd characteristic, smooth cubic hypersurfaces of dimension at least $4$ in characteristic…
In this paper a generalized topological central point theorem is proved for maps of a simplex to finite-dimensional metric spaces. Similar generalizations of the Tverberg theorem are considered.
In this note we give a p-adic proof of Hodge symmetry for smooth, projective threefolds over complex numbers.
We give a new proof of Tietze Theorem on the convergence of infinite semi-regular continued fractions.
We present projective versions of the center point theorem and Tverberg's theorem, interpolating between the original and the so-called "dual" center point and Tverberg theorems. Furthermore we give a common generalization of these and many…
In this paper we formulate and prove a combinatorial version of the section conjecture for finite groups acting on finite graphs. We apply this result to the study of rational points and show that finite descent is the only obstruction to…
We prove that the Hasse principle holds for cubic threefolds with 9 singular points over a number field.
We give a short, elementary and explicit proof of the existence of Hilbert schemes of points on affine schemes. As a direct consequence we obtain the existence of the Hilbert scheme of points on any projective scheme, not necessarily of…
We prove the Tate conjecture for integral degree 4 classes on a smooth cubic hypersurface X of dimension 4 over an algebraic closure of a field finitely generated over its prime subfield.