Related papers: Irrational HK multiplicities are possible for trin…

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 define a categorical birational invariant for minimal geometrically rational surfaces with a conic bundle structure over a perfect field via components of a natural semiorthogonal decomposition. Together with the similar known result on…

We provide new examples of integrable rational maps in four dimensions with two rational invariants, which have unexpected geometric properties, as for example orbits confined to non algebraic varieties, and fall outside classes studied by…

Yanchevski\u{i} had asked whether conic bundle surfaces over $\mathbf{P}^1_k$ are unirational when $k$ is a finite field. We give a partial answer to his question by showing that for quasi-finite fields $k$ (e.g. finite fields) a regular…

We exhibit a smooth complex rational affine surface with uncountably many nonisomorphic real forms.

Let k be a subfield of a p-adic field of odd residue characteristic, and let L be the function field of a variety of dimension n >= 1 over k. Then Hilbert's Tenth Problem for L is undecidable. In particular, Hilbert's Tenth Problem for…

We show that among simply connected surfaces of general type unirationality is a common feature, even when fixing the positive characteristic or numerical invariants. To do so, we construct unirational Horikawa surfaces in abundance.

The more recent paper "Generic strange duality for K3 surfaces" by the authors contains stronger results.

We classify, up to some lattice-theoretic equivalence, all possible configurations of rational double points that can appear on a surface whose minimal resolution is a complex Enriques surface.

The nonsingular Hermitian surface of degree $\sqrt{q} +1$ is characterized by its number of $\Bbb{F}_q$-points among the irreducible surfaces over $\Bbb{F}_q$ of degree $\sqrt{q} +1$ in the projective 3-space.

In this paper, we check that Fano schemes of lines on certain rational cubic fourfolds are birational to Hilbert schemes of two points on K3 surfaces.

We show the birational boundedness of anti-canonical irreducible hypersurfaces which form 3-fold plt pairs. We also treat a collection of Du Val K3 surfaces which is birationally bounded but unbounded.

We prove an upper bound for the number of rational points of bounded height on irreducible affine hypersurfaces. More precisely, given an irreducible polynomial $f \in \mathbb{Z}[X_1, \dots, X_n]$, we prove an upper bound on the number of…

The developable surface is an important surface in computer aided design, geometric modeling and industrial manufactory. It is often given in the stan- dard parametric form, but it can also be in the implicit form which is commonly used in…

We explicit some general properties regarding surfaces with Prym-canonical hyperplane sections and the geometric genus of their possible singularities. Moreover, we construct new examples of this type of surfaces.

For any number field k, upper bounds are established for the number of k-rational points of bounded height on non-singular del Pezzo surfaces defined over k, which are equipped with suitable conic bundle structures over k.

We provide a criterion for when Hilbert schemes of points on K3 surfaces are birational. In particular, this allows us to generate a plethora of examples of non-birational Hilbert schemes which are derived equivalent.

We prove that, under certain assumptions, generalized Hilbert-Kunz multiplicities can be expressed as linear combinations of classical Hilbert-Kunz multiplicities.

We associate to triangulations of infinite type surface a type of flip graph where simultaneous flips are allowed. Our main focus is on understanding exactly when two triangulations can be related by a sequence of flips. A consequence of…

We show that the maximal number of singular points of a normal quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically closed field $K$ of characteristic 2 is at most 12, if the minimal resolution of $X$ is not a…