Related papers: Computing $\alpha$-invariants of singular del Pezz…
We classify all the del Pezzo surfaces with $\frac{1}{3}(1,1)$- and $\frac{1}{4}(1,1)$-singularities having no floating $(-1)$-curves into 39 types.
Using the method of moving frames we analyze the algebra of differential invariants for surfaces in three-dimensional affine geometry. For elliptic, hyperbolic, and parabolic points, we show that if the algebra of differential invariants is…
We study the BPS invariants for local del Pezzo surfaces, which can be obtained as the signed Euler characteristic of the moduli spaces of stable one-dimensional sheaves on the surface $S$. We calculate the Poincare polynomials of the…
The Welschinger invariants of real rational algebraic surfaces count real rational curves which represent a given divisor class and pass through a generic conjugation-invariant configuration of points. No invariants counting real curves of…
We prove that $\delta$-invariants of smooth cubic surfaces are at least $\frac{6}{5}$.
The exceptional log Del Pezzo surfaces with delta=1 are classified.
In earlier joint work with our collaborators Akhtar, Coates, Corti, Heuberger, Kasprzyk, Prince and Tveiten, we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del…
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…
Given a Fano manifold $(X,\omega)$ we develop a variational approach to characterize analytically the existence of K\"ahler-Einstein metrics with prescribed singularities, assuming that these singularities can be approximated algebraically.…
The surfaces considered are real, rational and have a unique smooth real $(-2)$-curve. Their canonical class $K$ is strictly negative on any other irreducible curve in the surface and $K^2>0$. For surfaces satisfying these assumptions, we…
We classify the automorphism groups of del Pezzo surfaces of degrees one and two over an algebraically closed field of characteristic two. This finishes the classification of automorphism groups of del Pezzo surfaces in all characteristics.
In this article, we obtain an upper bound for the number of integral points on the del Pezzo surfaces of degree two.
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…
In an algebro-geometric way, we completely determine whether smooth del Pezzo surfaces are K-(semi)stable or not.
We obtain a formula for the number of genus one curves with a variable complex structure of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This is done using Getzler's…
We study singular del Pezzo surfaces that are quasi-smooth and well-formed weighted hypersurfaces. We give an algorithm how to classify all of them.
In this paper we prove that generic small partial smoothings of Kahler-Einstein (KE) Del Pezzo orbifolds with only nodal singularities, and with no non-zero holomorphic vector fields, admit orbifold KE metrics which are close in the…
The Welschinger invariants of real rational algebraic surfaces are natural analogues of the genus zero Gromov-Witten invariants. We establish a tropical formula to calculate the Welschinger invariants of real toric Del Pezzo surfaces for…
We prove Manin's conjecture for a del Pezzo surface of degree six which has one singularity of type $\mathbf{A}_2$. Moreover, we achieve a meromorphic continuation and explicit expression of the associated height zeta function.
This paper surveys recent progress towards the Manin conjecture for (singular and non-singular) del Pezzo surfaces. To illustrate some of the techniques available, an upper bound of the expected order of magnitude is established for a…