Related papers: Rational lines on diagonal hypersurfaces and subco…

We consider a smooth system of two homogeneous quadratic equations over the rationals in at least 13 variables. In this case, the Hasse principle is known to hold, thanks to the work of Mordell in 1959. The only local obstruction is over…

We prove the Hasse principle for a smooth projective variety $X\subset \PP^{n-1}_\Q$ defined by a system of two cubic forms $F,G$ as long as $n\geq 39$. The main tool here is the development of a version of Kloosterman refinement for a…

We prove new cases of the Hasse principle for Kummer surfaces constructed from 2-coverings of Jacobians of genus 2 curves, assuming finiteness of relevant Tate--Shafarevich groups. Under the same assumption, we deduce the Hasse principle…

Let C be an algebraic curve defined over a number field K, of positive genus and without K-rational points. We conjecture that there exists some extension field L over which C violates the Hasse principle, i.e., has points everywhere…

It is known that the Brauer--Manin obstruction to the Hasse principle is vacuous for smooth Fano hypersurfaces of dimension at least $3$ over any number field. Moreover, for such varieties it follows from a general conjecture of…

We show that if over some number field there exists a certain diagonal plane cubic curve that is locally solvable everywhere, but that does not have points over any cubic galois extension of the number field, then the algebraic part of the…

Conditionally on the $abc$ conjecture, we generalize previous work of Clark and the author to show that a superelliptic curve $C: y^n = f(x)$ of sufficiently high genus has infinitely many twists violating the Hasse Principle if and only if…

Let $C$ be a smooth projective curve defined over the finite field $\mathbb{F}_q$ ($q$ is odd) and let $K=\mathbb{F}_q(C)$ be its function field. Removing one closed point $C^\text{af} = C-\{\infty\}$ results in an integral domain…

For any pencil of conics or higher-dimensional quadrics over the rationals, with all degenerate fibres defined over the rationals, we show that the Brauer-Manin obstruction controls weak approximation. The proof is based on the Hasse…

We establish new upper bounds on the number of failures of the integral Hasse principle within the family of Markoff type cubic surfaces $x^2+ y^2+ z^2- xyz= a$ with $|a|\leq A$ as $A\to \infty$. Our bound improves upon existing work of…

We show that a closed piecewise-linear hypersurface immersed in $R^n$ ($n\ge 3$) is the boundary of a convex body if and only if every point in the interior of each $(n-3)$-face has a neighborhood that lies on the boundary of some convex…

A cubic hypersurface in $\mathbb{P}^n$ defined over $\mathbb{Q}$ is given by the vanishing locus of a cubic form $f$ in $n+1$ variables. It is conjectured that when $n \geq 4$, such cubic hypersurfaces satisfy the Hasse principle. This is…

The Hasse principle in number theory states that information about integral solutions to Diophantine equations can be pieced together from real solutions and solutions modulo prime powers. We show that the Hasse principle holds for…

In this paper, we prove three related results; (1) Extension of our result in [10] to all generic hypersurfaces. More precisely, the normal sheaf of a generic rational map $c_0$ to a generic hypersurface $X_0$ of $\mathbf P^n, n\geq 4$ has…

We develop a technique that allows us to prove results about subvarieties of general type hypersurfaces. As an application, we use a result of Clemens and Ran to prove that a very general hypersurface of degree (3n+1)/2 \leq d \leq 2n-3…

A central question in Arithmetic geometry is to determine for which polynomials $f \in \mathbb{Z}[t]$ and which number fields $K$ the Hasse principle holds for the affine equation $f(t) = N_{K/\mathbb{Q}}(\boldsymbol{x}) \neq 0$. Whilst…

For a given form $F\in \mathbb Z[x_1,\dots,x_s]$ we apply the circle method in order to give an asymptotic estimate of the number of $m$-tuples $\mathbf x_1, \dots, \mathbf x_m$ spanning a linear space on the hypersurface $F(\mathbf x) = 0$…

We establish the Hasse principle for $100\%$ of conic bundles over $\mathbb{P}^1_{\mathbb{Q}}$.

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…

In this paper we study the existence of rational points for the family of K3 surfaces over $\mathbb{Q}$ given by $$w^2 = A_1x_1^6 + A_2x_2^6 + A_3x_3^6.$$ When the coefficients are ordered by height, we show that the Brauer group is almost…