Related papers: Which rational double points occur on del Pezzo su…
We present a dynamical proof of the well-known fact that the Neron-Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field k of transcendence degree 1 over an…
We explore connections between existence of $\Bbbk$-rational points for Fano varieties defined over $\Bbbk$, a subfield of $\mathbb{C}$, and existence of K\"ahler-Einstein metrics on their geometric models. First, we show that geometric…
Take $(R, \mathfrak{m})$ any normal Noetherian domain, either local or $\mathbb{N}$-graded over a field. We study the question of when $R$ satisfies the uniform symbolic topology property (USTP) of Huneke, Katz, and Validashti: namely, that…
We construct jacobians of plane quartics without complex multiplication, using Del Pezzo surfaces of degree 2.
The surface $z^2=ay^2+P(x), \, a \in k, \, P(x) \in k[x]$ is not $k$-rational, if $a \not\in k^2$ and $P(x)$ satisfies some conditions. This result essentially due to Iskovskih but his statement is in terms of algebraic geometry, and not so…
For a dominant rational self-map on a smooth projective variety defined over a number field, Kawaguchi and Silverman conjectured that the (first) dynamical degree is equal to the arithmetic degree at a rational point whose forward orbit is…
A conjecture of Manin predicts the asymptotic distribution of rational points of bounded height on Fano varieties. In this paper we use conic bundles to obtain correct lower bounds or a wide class of surfaces over number fields for which…
In this note we study in detail the geometry of eight rational elliptic surfaces naturally associated to the sixteen reflexive polygons. The elliptic fibrations supported by these surfaces correspond under mirror symmetry to the eight…
We study the failure of the Lipman-Zariski conjecture in positive characteristic. For rational double points, the conjecture holds true except for a short finite list of exceptions. For log canonical surface singularities, the conjecture…
We construct a surface of general type with canonical map of degree 12 which factors as a triple cover and a bidouble cover of $\mathbb P^2$. We also show the existence of a smooth surface with $q=0,$ $\chi=13$ and $K^2=9\chi$ such that its…
We prove that the number of rational points of bounded height on certain del Pezzo surfaces of degree 1 defined over Q grows linearly, as predicted by Manin's conjecture. Along the way, we investigate the average number of integral 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…
Let $\Fq$ be the finite field with $q$ elements. We study the number of $\Fq$-rational points on Danielewski and double Danielewski surfaces. For Danielewski surfaces, the point count is reduced to the number of roots of $P(Z)$ over $\Fq.$…
Mysterious duality is a relationship, described by Vafa in 2000, between $\frac12$-BPS branes in Type II supergravity in dimension $D=d+2$ and rational curves on del Pezzo surfaces of degree $d$. We show that both sides of this…
In this paper we give a complete description of all possible automorphism groups of real $\mathbb{R}$-rational del Pezzo surfaces $X$ of degree $4$, using the description of $X$ as the blow-up of some smooth real quadric surface $Q$ in…
We survey some of the recent works on the geometry of del Pezzo surfaces over imperfect fields, with applications to 3-dimensional del Pezzo fibrations in positive characteristic. We place particular emphasis on cases where the general…
We prove that the rational Picard group of the simple Hurwitz space ${\mathcal H}_{d,g}$ is trivial for $d$ up to five. We also relate the rational Picard groups of the Hurwitz spaces to the rational Picard groups of the Severi varieties of…
We explicitly construct a component of the K-moduli space of K-polystable del Pezzo surfaces which is a smooth rational curve.
We give a classification of toric log del Pezzo surfaces with two or three singular points.
We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups, and describe the connected components of their automorphisms groups.