相关论文: On the arithmetic of one del Pezzo surface over th…

We present a matrix-based algorithm for deciding if the parametrization of a curve or a surface is invertible or not, and for computing the inverse of the parametrization if it exists.

We prove the "strong form" of the Clemens conjecture in degree 10. Namely, on a general quintic threefold F in P^4, there are only finitely many smooth rational curves of degree 10, and each curve is embedded in F with normal bundle…

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…

In this paper the notion of rational simple connectedness for the quintic Fano threefold $V_5\subset \mathbb{P}^6$ is studied and unirationality of the moduli spaces $\overline{M}_{0,0}^{\text{bir}}(V_5,d)$, with $d \ge 1$, is proved. Many…

Using the homological sieve method developed by Das--Lehmann--Tosteson and the author, we prove Peyre's all height approach to Manin's conjecture for split quintic del Pezzo surfaces defined over $\mathbb F_q(t)$ assuming $q$ is…

The main aim of this article is to show that a very general 3-dimensional del Pezzo fibration of degree 1,2,3 is not stably rational except for a del Pezzo fibration of degree 3 belonging to explicitly described 2 families. Higher…

We revisit a correspondence between toroidal compactifications of M-theory and del Pezzo surfaces, in which rational curves on the del Pezzo are related to ${1\over 2}$-BPS branes of the corresponding compactification. We argue that curves…

Let $\mathbb{F}_q$ denote the finite field with $q$ elements. In this work, we use characters to give the number of rational points on suitable curves of low degree over $\mathbb{F}_q$ in terms of the number of rational points on elliptic…

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…

Let S be a Dedekind scheme with fraction field K. We study the following problem: given a Del Pezzo surface X, defined over K, construct a distinguished integral model of X, defined over all of S. We provide a satisfactory answer if S is a…

A rational triangle is a triangle with rational side lengths. We consider three different families of rational triangles having a fixed side and whose vertices are rational points in the plane. We display a one-to-one correspondence between…

Two classical results in algebraic geometry are that the branch curve of a del Pezzo surface of degree 1 can be embedded as a space sextic curve and that every space sextic curve has exactly 120 tritangents corresponding to its odd theta…

We study a particular plane curve over a finite field whose normalization is of genus 0. The number of rational points of this curve achieves the Aubry-Perret bound for rational curves. The configuration of its rational points and a…

For a smooth Del Pezzo surface the direct sum of global sections of all isomorphism classes of invertible sheaves on it can be almost canonically endowed with a ring structure, called the Cox ring. We show that in characteristic 0 this ring…

We address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the projective plane. We improve the known upper and lower bounds and construct close to optimal…

We exhibit new examples of rational cubic fourfolds, parametrized by a countably infinite union of codimension-two subvarieties in the moduli space. Our examples are fibered in sextic del Pezzo surfaces over the projective plane; they are…

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…

First we solve the problem of finding minimal degree families on toric surfaces by reducing it to lattice geometry. Then we describe how to find minimal degree families on, more generally, rational complex projective surfaces.

We present an approach to a large class of enumerative problems concerning rational curves in projective spaces. This approach uses analysis to obtain topological information about moduli spaces of stable maps. We demonstrate it by…

We address weak approximation for certain del Pezzo surfaces defined over the function field of a curve. We study the rational connectivity of the smooth locus of degree two del Pezzo surfaces with two A1 singularities in order to prove…