Big rational surfaces

Abstract

We prove that the Cox ring of a smooth rational surface with big anticanonical class is finitely generated. We classify surfaces of this type that are blow-ups of the plane at distinct points lying on a (possibly reducible) cubic.

Major revision; added examples of big rational surfaces that are not log del Pezzo

References in corpus (2)

• Hilbert's fourteenth problem over finite fields, and a conjecture on the cone of curves
• On the Cox ring of P^2 blown up in points on a line

Cited by in corpus (28)

• Uniform K-stability and plt blowups of log Fano pairs
• The cone of curves and the Cox ring of rational surfaces given by divisorial valuations
• Betti numbers of symmetric shifted ideals
• Non-toric bases for elliptic Calabi-Yau threefolds and 6D F-theory vacua
• Rational points of bounded height on general conic bundle surfaces
• Rationally connected non-Fano type varieties
• Star configurations in ${\mathbb{P}}^n$
• On the anti-canonical ring and varieties of Fano type
• Algebraic surfaces and hyperbolic geometry
• Cox rings over nonclosed fields
• Int-amplified endomorphisms on normal projective surfaces
• Redundant blow-ups of rational surfaces with big anticanonical divisor
• On the Cox ring of P^2 blown up in points on a line
• On the uniform K-stability for some asymptotically log del Pezzo surfaces
• On base point free theorem and Mori dream spaces for log canonical threefolds over the algebraic closure of a finite field
• Geometric properties of projective manifolds of small degree
• Equations of negative curves of blow-ups of Ehrhart rings of rational convex polygons
• Matroid configurations and symbolic powers of their ideals
• Cox rings of rational surfaces and redundant blow-ups
• Characterization of log del Pezzo pairs via anticanonical models
• Volumes of Zariski chambers
• Surfaces of globally $F$-regular and $F$-split type
• Remarks on log Calabi--Yau structure of varieties admitting polarized endomorphism
• Cox rings of surfaces and the anticanonical Iitaka dimension
• The cone of curves and the Cox ring of rational surfaces over Hirzebruch surfaces
• Finitely generated symbolic Rees rings of ideals defining certain finite sets of points in P^2
• Examples of Mori dream spaces with Picard number two
• On K-polystability for log del Pezzo pairs of Maeda type