Related papers: How often does a cubic hypersurface have a rationa…

Let X be a finite set of points in R^n. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of view of applications, such as polynomial optimization, we…

We extend non-emtpyness and irreducibility of Hassett divisors to the moduli spaces of $M$-polarizable cubic fourfolds for higher rank lattices $M$, which in turn provides a systematic approach for describing the irreducible components of…

We prove Manin's conjecture for a singular cubic surface S with a singularity of type E6. If U is the open subset of S obtained by deleting the unique line from S, then the number of rational points in U with anticanonical height bounded by…

Motivated by a recent question of Peyre, we apply the Hardy-Littlewood circle method to count "sufficiently free" rational points of bounded height on arbitrary smooth projective hypersurfaces of low degree that are defined over the…

We study the proportion of conics given by $(\mathcal{C}_{\mathbf{F}, \mathbf{y}}) : F_0(\mathbf{y})x_0^2 + F_1(\mathbf{y})x_1^2 = F_2( \mathbf{y})x_2^2 $ which have a rational point $\mathbf{x} = (x_0 :x_1:x_2) \in…

Let $\mathscr{E}\rightarrow\mathbb{P}^1_\mathbb{Q}$ be a non-trivial rational elliptic surface over $\mathbb{Q}$ with base $\mathbb{P}^1_\mathbb{Q}$ (with a section). We conjecture that any non-trivial elliptic surface has a Zariski-dense…

Let $X$ be a cubic surface over a global field $k$. We prove that a Brauer-Manin obstruction to the existence of $k$-points on $X$ will persist over every extension $L/k$ with degree relatively prime to $3$. In other words, a cubic surface…

Given a smooth subscheme of a projective space over a finite field, we compute the probability that its intersection with a fixed number of hypersurface sections of large degree is smooth of the expected dimension. This generalizes the case…

Let F be a finite field of characteristic p. We consider smooth surfaces over F(t) defined by an equation f+tg=0, where f and g are forms of degree d in 4 variables with coefficients in F, with d prime to p. We prove : For such surfaces…

Let $X\subset \P^5$ be a smooth cubic fourfold. A well known conjecture asserts that $X$ is rational if and only if there an Hodge theoretically associated K3 surface $S$. The surface $S$ can be associated to $X$ in two other different…

We determine exactly which positive rational numbers occur as squared edge lengths of regular $d$-simplices with vertices in $\mathbb{Q}^n$. The answer exhibits a sharp stabilization phenomenon: once $n-d\geq 3$, every positive rational…

A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. It is described by a system of four equations with respect to six variables.…

Cubic fourfolds of discriminant 24 contain special codimension-two algebraic cycles of degree 6 and self-intersection 20. Such cycles may be represented by singular scrolls or del Pezzo surfaces. A discriminant 24 cubic fourfold gives rise…

We study the product structure on the Chow ring (with rational coefficients) of a cubic hypersurface in projective space and prove that the image of the product map is as small as possible.

As was pointed out by Nikulin [8] and Vinberg [10], a right-angled polyhedron of finite volume in hyperbolic n-space $\mathbb{H}^n$ has at least one cusp for $n\geq 5$. We obtain non-trivial lower bounds on the number of cusps of such…

Generalizing a classical lemma of Castelnuovo, we characterize rational normal curves (resp. linearly normal elliptic curves) as curves $C\subset \PP^n$ such that the number of linearly independent hypersurfaces $Z\supset C$ of given…

We establish the Hasse Principle for systems of r simultaneous diagonal cubic equations whenever the number of variables exceeds 6r and the associated coefficient matrix contains no singular r x r submatrix, thereby achieving the…

Let $F_1,\dotsc,F_R$ be quadratic forms with integer coefficients in $n$ variables. When $n\geq 9R$ and the variety $V(F_1,\dotsc,F_R)$ is a smooth complete intersection, we prove an asymptotic formula for the number of integer points in an…

Toute hypersurface cubique lisse complexe de dimension au moins 2 dont l'\'equation est donn\'ee par l'annulation d'une somme de formes cubiques \`a variables s\'epar\'ees, chaque forme impliquant au plus trois variables, est…

In this paper we try to further explore the linear model of the moduli of rational maps. Our attempt yields following results. Let $X\subset \mathbf P^n$ be a generic hypersurface of degree $h$. Let $R_d(X, h)$ denote the open set of the…