Related papers: Unirationality of cubic hypersurfaces

We exhibit geometric conditions on a family of toric hypersurfaces under which the value of a canonical normal function at a point of maximal unipotent monodromy is irrational.

We sharpen to nearly optimal the known asymptotic and explicit bounds for the number of $\mathbb{F}_q$-rational points on a geometrically irreducible hypersurface over a (large) finite field. The proof involves a Bertini-type probabilistic…

We prove that the spaces of rational curves on del Pezzo surfaces are either irreducible or empty, with a unique exception.

We show that over any algebraically closed field of positive characteristic, there exists a smooth rational surface which violates Kawamata-Viehweg vanishing.

We prove that hypersurfaces defined by irreducible square-free polynomials have rational singularities. As an easy consequence, we deduce that certain (possibly non-square-free) polynomials associated to pairs of square-free polynomials…

We prove Manin's conjecture over imaginary quadratic number fields for a cubic surface with a singularity of type E_6.

We show that a proof in multiplicative linear logic can be represented as a decorated surface, such that two proofs are logically equivalent just when their surfaces are geometrically equivalent. This is an extended abstract for…

A smooth rational surface X is a Coble surface if the anti-canonical linear system is empty while the anti-bicanonical linear system is non-empty. In this note we shall classify these X and consider the finiteness problem of the number of…

The Eckardt hypersurface in $\mathbb{P}^{19}$ parameterizes smooth cubic surfaces with an Eckardt point, which is a point common to three of the $27$ lines on a smooth cubic surface. We describe the cubic surfaces lying on the singular…

We give an upper bound for the number of rational points of height at most $B$, lying on a surface defined by a quadratic form $Q$. The bound shows an explicit dependence on $Q$. It is optimal with respect to $B$, and is also optimal for…

Let $X$ be a cubic fourfold in $P^5_{C}$. We prove that, assuming the Hodge conjecture for the product $S \times S$, where $S$ is a complex surface, and the finite dimensionality of the Chow motive $h(S)$, there are at most a countable…

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…

We show that there exists a $2$-dimensional family of smooth cubic threefolds admitting unirational parametrizations of coprime degrees. This together with Clemens--Griffiths' work solves the long standing open problem whether there exists…

We establish an analytic Hasse principle for linear spaces of affine dimension m on a complete intersection over an algebraic field extension K of Q. The number of variables required to do this is no larger than what is known for the…

In this article we prove a conjecture formulated by A.B. Soerensen in 1991 on the maximal number of $\mathbb{F}_{q^2}$-rational points on the intersection of a non-degenerate Hermitian surface and a surface of degree $d \le q.$

In this short note we try to generalize the Clemens-Griffiths criterion of non-rationality for smooth cubic threefolds to the case of smooth cubic fourfolds.

The purpose of this note is to give a short, selfcontained proof of the following result: A complex surface which is diffeomeorphic to a rational surface is rational.

We apply a variant of the square-sieve to produce a uniform upper bound for the number of rational points of bounded height on a family of surfaces that admit a fibration over the projective line, whose general fibre is a hyperelliptic…

We investigate the universal Severi variety of rational curves on K3 surfaces, which parametrises irreducible rational curves in a fixed class on varying K3 surfaces of fixed genus. We investigate the conjecuted irreducibility of this space…

Can a smooth plane cubic be defined by the determinant of a square matrix with entries in linear forms in three variables? If we can, we say that it admits a linear determinantal representation. In this paper, we investigate linear…