Related papers: $G$-solid Rational Surfaces
Building on recent work of Bhargava--Elkies--Schnidman and Kriz--Li, we produce infinitely many smooth cubic surfaces defined over the field of rational numbers that contain rational points.
We classify rational surfaces for which the image of the automorphisms group in the group of linear transformations of the Picard group is the largest possible. This answers a question raised by Arthur Coble in 1928, and can be rephrased in…
It is proved that a smooth rational surface in projective four-space, which is ruled by cubics or quartics has degree at most 12. It is also proved that a smooth rational surface in projective four-space which is the image of Fn by a linear…
Rationally convex topological embeddings of compact surfaces (closed or with boundary) into $\mathbb{C}^2$ are constructed.
We show that, for every prime number p, there exist infinitely many K3 surfaces over Q whose rational points lie dense in the space of p-adic points. We also show that there exists a K3 surface over Q whose rational points lie dense in the…
We investigate equivariant birational geometry of rational surfaces and threefolds from the perspective of derived categories.
We classify Galois actions on Picard lattices of del Pezzo surfaces of degrees 1,2, and 3 giving rise to minimal surfaces with no cohomological obstructions to stable rationality.
We classify the possible torsion structures of rational elliptic curves over sextic number fields.
We describe those unipotent representations of a finite group of Lie type which are defined over the rational numbers.
Let $X \subset \mathbb{P}(w_0, w_1, w_2, w_3)$ be a quasismooth well-formed weighted projective hypersurface and let $L = lcm(w_0,w_1,w_2,w_3)$. We characterize when $X$ is rational under the assumption that $L$ divides $deg(X)$ by…
The parametric degree of a rational surface is the degree of the polynomials in the smallest possible proper parametrization. An example shows that the parametric degree is not a geometric but an arithmetic concept, in the sense that it…
In this paper, we study generalized constant ratio surfaces in the Euclidean 4-space. We also obtain a classifications of constant slope surfaces.
Among geometrically rational surfaces, del Pezzo surfaces of degree two over a field k containing at least one point are arguably the simplest that are not known to be unirational over k. Looking for k-rational curves on these surfaces, we…
The genus spectrum of a finite group $G$ is the set of all $g\geq 2$ such that $G$ acts faithfully and orientation-preserving on a closed compact orientable surface of genus $g$. This article is an overview of some results relating the…
We give an upper bound for the degree of rational curves in a family that covers a given birational ruled surface in projective space. The upper bound is stated in terms of the degree, sectional genus and arithmetic genus of the surface. We…
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…
We show that if a compact complex surface admits a locally conformally flat metric, then it cannot contain a smooth rational curve of odd self-intersection. In particular, the surface has to be minimal. Then we give a list of possibilities…
We classify, up to equivalence, all finite-dimensional simple graded division algebras over the field of real numbers. The grading group is any finite abelian group.
We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…
We prove that the complex surfaces parametrizing cuboids and face cuboids, as well as their minimal resolution of singularities, have trivial fundamental group. We then compute the fundamental group of certain open smooth subvarieties of…