Related papers: Rationality of hypersurfaces

We survey some results on real rational surfaces focused on their topology and their birational geometry.

We study the family of rational curves on arbitrary smooth hypersurfaces of low degree using tools from analytic number theory.

We prove analogues of several well-known results concerning rational morphisms between quadrics for the class of so-called quasilinear $p$-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric…

This is a survey on unramified cohomology with a view towards its applications to rationality problems.

A general strategy is given for the classification of graphs of rational surface singularities. For each maximal rational double point configuration we investigate the possible multiplicities in the fundamental cycle. We classify completely…

Various methods have been used to construct rational points and rational curves on rationally connected algebraic varieties. We survey recent advances in two of them, the descent and the fibration method, in a number-theoretical context…

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…

We discuss and prove a number of results for calculating characteristic cycles, or graded, enriched characteristic cycles. We concentrate particularly on results related to hypersurfaces.

One of the most powerful ideas in the study and classification of algebraic varieties is the notion of a model: that is, to single out an object, in the appropriate isomorphism class, with nice properties. This survey aims to define and…

This paper is about the question whether a cycle in the l-adic cohomology of a smooth projective variety over the rational numbers, which is algebraic over almost all finite fields, is also algebraic over the rationals. We use ultraproducts…

In this article we prove a result comparing rationality of algebraic cycles over the function field of a projective homogeneous variety under a linear algebraic group of type $F_4$ or $E_8$ and over the base field, which can be of any…

We propose and compare various techiques available to produce smooth cubic hypersurfaces over a non-algebraically-closed field which have rational points but which are not stably rational over their ground field.

We give examples of smooth $\k$-unirational line-free quartic hypersurfaces over a non algebraically closed field $\k$. Unlike other methods of proving unirationality, our method does not rely on existence of linear spaces on quartics.

Many combinatorial problems can be formulated as a polynomial optimization problem that can be solved by state-of-the-art methods in real algebraic geometry. In this paper we explain many important methods from real algebraic geometry, we…

This is a survey of the geometry of complex cubic fourfolds with a view toward rationality questions. Topics include classical constructions of rational examples, Hodge structures and special cubic fourfolds, associated K3 surfaces and…

The purpose of this paper is to show how Rees algebras can be applied in the study of singularities embedded in smooth schemes over perfect fields. In particular, we will study situations in which the multiplicity of a hypersurface is a…

We introduce new obstructions to rationality for geometrically rational threefolds arising from the geometry of curves and their cycle maps.

The circle method has been successfully used over the last century to study rational points on hypersurfaces. More recently, a version of the method over function fields, combined with spreading out techniques, has led to a range of results…

We give an introduction to the study of algebraic hypersurfaces, focusing on the problem of when two hypersurfaces are isomorphic or close to being isomorphic. Working with hypersurfaces and emphasizing examples makes it possible to discuss…

We complete the study of rationality problem for hypersurfaces $X_t\subset \mathbb{P}^4$ of degree $4$ invariant under the action of the symmetric group $S_6$.