Related papers: On the Hasse principle for quartic hypersurfaces

If $X = V(f) \subset \mathbb P^N$ is a reduced complex hypersurface, the hessian of $f$ (or by abusing the terminology the hessian of $X$) is the determinant of the matrix of the second derivatives of the form $f$, that is the determinant…

We give a formula for the Hodge numbers of hypersurfaces in $\mathbb P^{4}$ with ordinary triple points.

We study an irreducible component H(X) of the Hilbert scheme Hilb^{2t+2}(X) of a smooth cubic hypersurface X containing two disjoint lines. For cubic threefolds, H(X) is always smooth, as shown in arXiv:2010.11622. We provide a second proof…

An affine hypersurface $M$ is said to admit a pointwise symmetry, if there exists a subgroup $G$ of ${\rm Aut}(T_p M)$ for all $p\in M$, which preserves (pointwise) the affine metric $h$, the difference tensor $K$ and the affine shape…

In this paper we determine the canonical arithmetic volume of hypersurfaces in smooth projective toric varieties. As a consequence, we prove a generalized Hodge index theorem on hypersurfaces in smooth projective toric varieties.

We study the integral Brauer--Manin obstruction for affine diagonal cubic surfaces, which we employ to construct the first counterexamples to the integral Hasse principle in this setting. We then count in three natural ways how such…

We construct an Enriques surface X over Q with empty \'etale-Brauer set (and hence no rational points) for which there is no algebraic Brauer-Manin obstruction to the Hasse principle. In addition, if there is a transcendental obstruction on…

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.

For a general cubic fourfold $X \subset \mathbb{P}^5$, we compute the Hodge numbers of the locus $S \subset F$ of lines of second type. We also give an upper bound of 6 for the degree of irrationality of the Fano scheme of lines of any…

The Hasse-Witt matrix of a hypersurface in ${\mathbb P}^n$ over a finite field of characteristic $p$ gives essentially complete mod $p$ information about the zeta function of the hypersurface. But if the degree $d$ of the hypersurface is…

We prove that the maximal number of conics, a priori irreducible of reducible, on a smooth spatial quartic surface is 800, realized by a unique quartic. We also classify quartics with many (at least 720) conics. The maximal number of real…

We classify all real hypersurfaces with three distinct constant principal curvatures in complex hyperbolic spaces of dimension greater than two.

Let k be a global field of characteristic not 2. The classical Hasse-Minkowski theorem states that if two quadratic forms become isomorphic over all the completions of k, then they are isomorphic over k as well. It is natural to ask whether…

Given a smooth projective variety of dimension $n-1\geq 1$ defined over a perfect field $k$ that admits a non-singular hypersurface modelin $\mathbb{P}^n_{\overline{k}}$ over $\overline{k}$, a fixed algebraic closure of $k$, it does not…

A very general hypersurface of dimension $n$ and degree $d$ in complex projective space is rational if $d \leq 2$, but is expected to be irrational for all $n, d \geq 3$. Hypersurfaces in weighted projective space with degree small relative…

We prove the sharp bound of at most 64 lines on complex projective quartic surfaces (resp. affine quartics) that are not ruled by lines. We study configurations of lines on certain non-K3 surfaces of degree four and give various examples of…

Let X/S be a quasi-projective morphism over an affine base. We develop in this article a technique for proving the existence of closed subschemes H/S of X/S with various favorable properties. We offer several applications of this technique,…

We count rational points of bounded height on the non-normal senary quartic hypersurface x 4 = (y 2 1 + $\times$ $\times$ $\times$ + y 2 4)z 2 in the spirit of Manin's conjecture.

Let F be a field of characteristic different from 2. The u-invariant and the Hasse number of a field F are classical and important field invariants pertaining to quadratic forms. These invariants measure the suprema of dimensions of…

We study Brauer-Manin obstructions to the Hasse principle and to weak approximation on algebraic surfaces over number fields.