Manin's conjecture for a cubic surface with D_5 singularity

The Manin conjecture is established for a split singular cubic surface in P^3, with singularity type D_5.Comment: 26 pages, 1 figur

Manin's conjecture for a cubic Surface with D5 singularity

The Manin conjecture is established for a split singular cubic surface in Formula, with singularity type D

Singular del Pezzo surfaces that are equivariant compactifications

We determine which singular del Pezzo surfaces are equivariant compactifications of G_a^2, to assist with proofs of Manin's conjecture for such surfaces. Additionally, we give an example of a singular quartic del Pezzo surface that is an equivariant compactification of a semidirect product of G_a and G_m.Comment: 14 pages, main result extended to non-closed ground field

Campana points of bounded height on vector group compactifications

We initiate a systematic quantitative study of subsets of rational points that are integral with respect to a weighted boundary divisor on Fano orbifolds. We call the points in these sets Campana points. Earlier work of Campana and subsequently Abramovich shows that there are several reasonable competing definitions for Campana points. We use a version that delineates well different types of behaviour of points as the weights on the boundary divisor vary. This prompts a Manin-type conjecture on Fano orbifolds for sets of Campana points that satisfy a klt (Kawamata log terminal) condition. By importing work of Chambert-Loir and Tschinkel to our set-up, we prove a log version of Manin's conjecture for klt Campana points on equivariant compactifications of vector groups.Comment: 52 pages; minor revision, changes in the definition of Campana point

On a constant arising in Manin's conjecture for del Pezzo surfaces

For split smooth Del Pezzo surfaces, we analyse the structure of the effective cone and prove a recursive formula for the value of alpha, appearing in the leading constant as predicted by Peyre of Manin's conjecture on the number of rational points of bounded height on the surface. Furthermore, we calculate alpha for all singular Del Pezzo surfaces of degree at least 3

Manin's conjecture for a cubic Surface with D5 singularity

The Manin conjecture is established for a split singular cubic surface in Formula, with singularity type D5

Manin's conjecture for a quartic del Pezzo surface with A 4 singularity

The Manin conjecture is established for a split singular del Pezzo surface of degree four, with singularity type A 4