Related papers: Full exceptional collections for anticanonical log…

We prove that the integral points are potentially Zariski dense in the complement of a reduced effective singular anticanonical divisor in a smooth del Pezzo surface, with the exception of $\mathbb{P}^2$ minus three concurrent lines (for…

Recently the Euler forms on numerical Grothendieck groups of rank 4 whose properties mimick that of the Euler form of a smooth projective surface have been classified. This classification depends on a natural number $m$, and suggests the…

We classify del Pezzo surfaces of Picard number one with log canonical singularities admitting Q-Gorenstein smoothings.

We state a number of conjectures that together allow one to classify a broad class of del Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We prove our conjectures in the simplest cases. The conjectures relate…

We investigate compactified Deligne-Lusztig varieties whose canonical divisor, when expressed as a linear combination of boundary divisors, has all coefficients strictly negative or zero. In dimension two we obtain explicit descriptions:…

The correspondence between del Pezzo surfaces and field theory models over the complex numbers or for split real forms is extended to other real forms, in particular to those compatible with supersymmetry. Specifically, all theories of the…

We prove a version of the Strominger-Yau-Zaslow mirror symmetry conjecture for non-compact Calabi-Yau surfaces arising from, on the one hand, pairs $(\check{Y},\check{D})$ of a del Pezzo surface $\check{Y}$ and $\check{D}$ a smooth…

We compute global log canonical thresholds of a large class of quasismooth well-formed del Pezzo weighted hypersurfaces in $\mathbb{P}(a_{1},a_{2},a_{3},a_{4})$. As a corollary we obtain the existence of orbifold K\"ahler--Einstein metrics…

We prove that any numerically exceptional collection of maximal length, consisting of line bundles, on a smooth del Pezzo surface is a standard augmentation in the sense of L.Hille and M.Perling. We deduce that any such collection is…

We construct first examples of singular del Pezzo surfaces with Zariski dense exceptional sets in Manin's conjecture, varying in degrees $1, 2$ and $3$. The obstructions arise from accumulating quasi-\'etale covers. We classify all…

Recently, Lehmann, Sengupta, and Tanimoto proposed a conjectural construction of the exceptional set in Manin's Conjecture, which we call the geometric exceptional set. We construct a del Pezzo surface of degree $1$ whose geometric…

In this paper, we explore the derived McKay correspondence for several reflection groups, namely reflection groups of rank two generated by reflections of order two. We prove that for each of the reflection groups $G=G(2m,m,2)$, $G_{12}$,…

We consider singular Q-acyclic surfaces with smooth locus of non-general type. We prove that if the singularities are topologically rational then the smooth locus is C^1- or C*-ruled or the surface is up to isomorphism one of two…

We classify all normal G^2_a-surfaces with Picard number one, and characterize which of these surfaces have at worst log canonical, and which have at worst log terminal singularities, answering a question of Hassett and Tschinkel (Int.…

In order to study integral points of bounded log-anticanonical height on weak del Pezzo surfaces, we classify weak del Pezzo pairs. As a representative example, we consider a quartic del Pezzo surface of singularity type…

In this paper we show that for any affine complete rational surface singularity there is a correspondence between the dual graph of the minimal resolution and the quiver of the endomorphism ring of the special CM modules. We thus call such…

We study the biregular and birational geometry of degree 6 del Pezzo surfaces with Picard number 1, defined over an arbitrary perfect field. Using Galois cohomology techniques, we obtain an explicit description of cocycles for such surfaces…

For any fixed $1 \leq \ell \leq 9$, we characterize all Wahl singularities that appear in degenerations of del Pezzo surfaces of degree $\ell$. This extends the work of Manetti and Hacking-Prokhorov in degree $9$, where Wahl singularities…

We study minimal del Pezzo surfaces of degree 1 with a conic bundle over a finite field $\mathbb{F}_q$ according to the action of the absolute Galois group on the singular fibers (which is known as their type). We give a lower bound on the…

We shall consider minimal analytic compactifications of the affine plane with singularities. In previous work, Kojima and Takahashi proved that any minimal analytic compactification of the affine plane, which has at worse log canonical…