Related papers: Big rational surfaces
We prove that the Cox ring of the blowing-up of a minimal toric surface of Picard rank two is finitely generated. As part of our proof of this result we provide a necessary and sufficient condition for finite generation of Cox rings of…
The aim is to give a geometric characterization of the finite generation of the Cox ring of anticanonical rational surfaces. This characterization is encoded in the finite generation of the effective monoid. Furthermore, we prove that in…
A determination of the fixed components, base points and irregularity is made for arbitrary numerically effective divisors on any smooth projective rational surface having an effective anticanonical divisor. All of the results are proven…
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…
We prove that the redundant blow-up preserves the finite generation of the Cox ring of a rational surface under a suitable assumption, and we study the birational structure of Mori dream rational surfaces via redundant blow-ups. It turns…
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…
We give generators for the nef cone and the cone of curves of rational surfaces obtained by blowing-up the complex projective plane at a set of points $\mathcal{B} \cup \mathcal{D}$, where $\mathcal{B}$ is the set of (proper and infinitely…
Let $X$ be a rational surface obtained by blowing up at a configuration $\mathcal{C}$ of infinitely near points over a Hirzebruch surface $\mathbb{F}_\delta$. We prove that there exist two positive integers $a \leq b$ such that the cone of…
We show that the Okounkov body of a big divisor with finitely generated section ring is a rational simplex, for an appropriate choice of flag; furthermore, when the ambient variety is a surface, the same holds for every big divisor. Under…
We determine the fixed locus of the anticanonical complete linear system of a given anticanonical rational surface. The case of a geometrically ruled rational surface is fully studied, e.g., the monoid of numerically effective divisor…
This paper deals with the problem of computing a generating set for the Cox ring $R(X)$ of a smooth projective rational surface $X$ with nef anticanonical class. In case $R(X)$ is finitely generated, we show that the degrees of its…
We investigate the structure of geometrically ruled surfaces whose anti-canonical class is big. As an application we show that the Picard group of a normal projective surface whose anti-canonical class is nef and big is a free abelian group…
We classify minimal pairs (X, G) for smooth rational projective surface X and finite group G of automorphisms on X. We also determine the fixed locus X^G and the quotient surface Y = X/G as well as the fundamental group of the smooth part…
We exhibit a smooth complex rational affine surface with uncountably many nonisomorphic real forms.
We completely classify redundant blow-ups appearing in the theory of rational surfaces with big anticanonical divisor due to Sakai. In particular, we construct a rational surface with big anticanonical divisor which is not a minimal…
We classify the singularities of a surface ruled by conics: they are rational double points of type $A_n$ or $D_n$. This is proved by showing that they arise from a precise series of blow-ups of a suitable surface geometrically ruled by…
In this paper we investigate the relation between the finite generation of the Cox ring R(X) of a smooth projective surface X and its anticanonical Iitaka dimension k(-K_X).
We determine the Cox rings of the minimal resolutions of cubic surfaces with at most rational double points, of blow ups of the projective plane at non-general configurations of six points and of three dimensional smooth Fano varieties of…
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 construct the first examples of rationally convex surfaces in the complex plane with hyperbolic complex tangencies. In fact, we give two very different types of rationally convex surfaces: those that admit analytic fillings by…