Related papers: Rational morphisms between quasilinear hypersurfac…

The main result is that a quasi-projective surface has negative log Kodaira dimension (i.e. no log pluricanonical sections) iff it is dominated by images of the affine line. This follows from our main intermediate result, that the smooth…

We show that a wide class of hypersurfaces in all dimensions are not stably rational. Namely, for all d at least about 2n/3, a very general complex hypersurface of degree d in P^{n+1} is not stably rational. The statement generalizes…

We construct the first examples of rationally convex surfaces in the complex plane with hyperbolic complex tangencies. In fact, we give two very different types of rationally convex surfaces: those that admit analytic fillings by…

We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the degeneration method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X of bidegree…

We give an exposition of graded and microformal geometry, and the language of $Q$-manifolds. $Q$-manifolds are supermanifolds endowed with an odd vector field of square zero. They can be seen as a non-linear analogue of Lie algebras (in…

In this note we study rational curves on degree $p^r+1$ Fermat hypersurface in $\PP^{p^r+1}_k$, where $k$ is an algebraically closed field of characteristic $p$. The key point is that the presence of Frobenius morphism makes the behavior of…

I use local differential geometric techniques to prove that the algebraic cycles in certain extremal homology classes in Hermitian symmetric spaces are either rigid (i.e., deformable only by ambient motions) or quasi-rigid (roughly…

We study the groups of automorphisms of rational algebraic surfaces that admit a relatively minimal pencil of curves of arithmetic genus one over an algebraically closed field of arbitrary characteristic. In particular, we classify such…

Linearly projecting smooth projective varieties provides a method of obtaining hypersurfaces birational to the original varieties. We show that in low dimension, the resulting hypersurfaces only have Du Bois singularities. Moreover, we…

Let k be a finite field with characteristic exceeding 3. We prove that the space of rational curves of fixed degree on any smooth cubic hypersurface over k with dimension at least 11 is irreducible and of the expected dimension.

For any field k of characteristic at most 5 we exhibit an explicit smooth quartic surface in projective threespace over k with trivial automorphism group over the algebraic closure of k. We also show how this can be extended to higher…

We propose a framework to give a precise meaning to the intuitive notion of "family of real forms of a variety parametrised by a variety" and study some fundamental properties of this notion. As an illustration, for any $n \geq 1$, we…

This paper develops the basic theory of formal schemes over fields in the supersymmetric setting. We introduce the notion of a formal superscheme and investigate some of its fundamental properties. Particular emphasis is placed on the study…

A strong consequence of quadratic forms becoming hyperbolic over the function field of a form is established. This result is invoked to obtain a new characterisation of hyperbolicity over function fields, and to recover a number of…

We study the representability of motivic spheres by smooth varieties. We show that certain explicit "split" quadric hypersurfaces have the $\mathbb A^1$-homotopy type of motivic spheres over the integers and that the $\mathbb A^1$-homotopy…

We study non-degenerate CR geometries of hypersurface type that are symmetric in the sense that, at each point, there is a CR transformation reversing the CR distribution at that point. We show that such geometries are either flat or…

In this work we study some problems related with algebraic hypersurfaces invariant by foliations on weighted projective spaces $\mathbb{P}_{\mathbb{C}}(\varpi_0,...,\varpi_n)$ generalizing some results known for $\p$, as for example: the…

We study rationality constructions for smooth complete intersections of two quadrics over nonclosed fields. Over the real numbers, we establish a criterion for rationality in dimension four.

This article deals with the existence of hypersurfaces minimizing general shape functionals under certain geometric constraints. We consider as admissible shapes orientable hypersurfaces satisfying a so-called reach condition, also known as…

Grothendieck-Chow motives of quadric hypersurfaces have provided many insights into the theory of quadratic forms. Subsequently, the landscape of motives of more general projective homogeneous varieties has begun to emerge. In particular,…