Classification of log del Pezzo surfaces of index $\le 2$
Valery Alexeev, Viacheslav V. Nikulin
备注：
142 pages, many figures. An expanded and improved version of this manuscript was published in Memoirs of the Mathematical Society of Japan, vol. 15 (2006)
This is an expanded version of our work [AN88], 1988, in Russian.
We classify del Pezzo surfaces over C with log terminal singularities of index \le 2. By classification, we understand a description of the intersection graph of all exceptional curves on an appropriate (somewhat stronger than minimal) resolution of singularities together with the subgraph of the curves which are contracted to singular points.
The final results are similar to classical results about classification of non-singular del Pezzo surfaces and use the usual finite root systems. However, the intermediate considerations use the theory of K3 surfaces (especially of K3 surfaces with non-symplectic involutions) and the theory of reflection groups in hyperbolic spaces (especially or reflection groups of hyperbolic lattices).
As an ``elementary'' application, our results permit one to classify sextics in P^2 with simple singularities and a component of geometric genus \ge 2.
In this paper, we investigate quotients of Calabi-Yau manifolds Y embedded in Fano varieties X which are products of two del Pezzo surfaces - with respect to groups G that act freely on Y. In particular, we revisit some known examples and we obtain some new Calabi-Yau varieties with small Hodge numbers. The groups G are subgroups of the automorphism group of X, which is described in terms of the automorphism group of the two del Pezzo surfaces.
Log del Pezzo surfaces play the role of the opposite of surfaces of general type. We will completely classify all the log del Pezzo surfaces of rank 2 and Cartier index 3 with a unique singularity.
Del Pezzo surfaces over C with log terminal singularities of index \le 2 were classified by Alekseev and Nikulin. In this paper, for each of these surfaces, we find an appropriate morphism to projective space. These morphisms enable us to describe del Pezzo surfaces of index \le 2 by equations in some weighted projective space. Obtained results are natural completions of similar results of Du Val, and Hidaka and Watanabe, describing del Pezzo surfaces of index 1.
A normal projective non-Gorenstein log-terminal surface $S$ is called a log del Pezzo surface of index three if the three-times of the anti-canonical divisor $-3K_S$ is an ample Cartier divisor. We classify all of the log del Pezzo surfaces of index three. The technique for the classification based on the argument of Nakayama.
We use classification of non-symplectic automorphisms of K3 surfaces to obtain a partial classification of log del Pezzo surfaces of index three. We can classify those with "Multiple Smooth Divisor Property", whose definition we will give. Our methods include the definition of right resolutions of quotient singularities of index three and some analysis of automorphism-stable elliptic fibrations on K3 surfaces. In particular we find several log del Pezzo surfaces of Picard number one with non-toric singularities of index three.
Toric log Del Pezzo surfaces with Picard number 1 have been completely classified whenever their index is $\le 2$: In this paper we extend the classification for those having index 3: We prove that, up to isomorphism, there are exactly 18 surfaces of this kind.
We study del Pezzo varieties, higher-dimensional analogues of del Pezzo surfaces. In particular, we introduce ADE classification of del Pezzo varieties, show that in type A the dimension of non-conical del Pezzo varieties is bounded by $12 - d - r$, where $d$ is the degree and $r$ is the rank of the class group, and classify maximal del Pezzo varieties.
We address weak approximation for certain del Pezzo surfaces defined over the function field of a curve. We study the rational connectivity of the smooth locus of degree two del Pezzo surfaces with two A1 singularities in order to prove weak approximation for degree two del Pezzo surfaces with square-free discriminant.
We study del Pezzo fibrations of degree 1 with terminal singularities. A connection between singularities on del Pezzo surfaces of degree 1 and Kodaira's classification of elliptic singular fibers will be studied in this paper. By this investigation we obtain several results on birational maps of del Pezzo fibrations of degree 1 with terminal singularities.
We consider the problem of interpolating projective varieties through points and linear spaces. We show that del Pezzo surfaces satisfy weak interpolation.
We discuss the rational points on del Pezzo surface of degree 1 and 2 over any finite field $\mathbb F_q$, and give out the explicit equations of del Pezzo surfaces that have unique rational point.
We consider del Pezzo surfaces with du Val singularities. We'll prove that a del Pezzo surface $X$ with du Val singularities has a $-K_X$-polar cylinder if and only if there exist tiger such that the support of this tiger does not contain anti-canonical divisor. Also we classify all del Pezzo surfaces $X$ such that $X$ has not any cylinders.
We estimate $δ$-invariants of some singular del Pezzo surfaces with quotient singularities, which we studied ten years ago. As a result, we show that each of these surfaces admits an orbifold Kähler--Einstein metric.
We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups, and describe the connected components of their automorphisms groups.
We classify RDP del Pezzo surfaces with global vector fields over arbitrary algebraically closed fields of characteristic $p \neq 2$. In characteristic $0$, every RDP del Pezzo surface $X$ is equivariant, that is, ${\rm Aut}_X = {\rm Aut}_{\widetilde{X}}$, where $\widetilde{X}$ is the minimal resolution of $X$, hence the classification of RDP del Pezzo surfaces with global vector fields is equivalent to the classification of weak del Pezzo surfaces with global vector fields. In this article, we show that if $p \neq 2,3,5,7$, then it is still true that every RDP del Pezzo surface is equivariant…
In this paper we prove that there are exactly two $G$-minimal surfaces which are $G$-birational to the quintic del Pezzo surface, where $G \cong C_5 \rtimes C_4$. These surfaces are the quintic del Pezzo surface itself and the surface $\mathbb{P}^1 \times \mathbb{P}^1$.
In this paper, we classify del Pezzo foliations on projective manifolds of rank at least 3 and with log canonical singularities in the sense of McQuillan.
In this paper we prove that del Pezzo fibrations admit $\mathbb{G}_{a}^{3}$-structures if and only if they are $\mathbb{P}^2$-bundles over $\mathbb{P}^{1}$.
We classify almost del Pezzo manifolds in arbitrary dimension n, i.e., projective manifolds X with big and nef anticanonical bundle -K_X, such that -K_X is divisible by n-1.
In this paper we classify nodal rational non-$\mathbb{Q}$-factorial del Pezzo threefolds of degree 2 which can be $G$-birationally rigid for some subgroup $G\subset \operatorname{Aut}(X)$.
We classify Galois actions on Picard lattices of del Pezzo surfaces of degrees 1,2, and 3 giving rise to minimal surfaces with no cohomological obstructions to stable rationality.
For a log del Pezzo surface $S$, the fractional index $r(S)\in\mathbb{Q}_{>0}$ is the maximum of $r$ with which $-K_S$ can be written as $r$ times some Cartier divisor. We classify all the log del Pezzo surfaces $S$ with $r(S)\geq 1/2$, after the technique of Nakayama.
We classify all the effective anticanonical divisors on weak del Pezzo surfaces. Through this classification we obtain the smallest number among the log canonical thresholds of effective anticanonical divisors on a given Gorenstein canonical del Pezzo surface.
In this short note, we study the asymptotic Chow polystability of toric Del Pezzo surfaces appear in the moduli space of Kähler-Einstein Fano varieties constructed in [OSS16].
Varieties fibered into del Pezzo surfaces form a class of possible outputs of the minimal model program. It is known that del Pezzo fibrations of degrees $1$ and $2$ over the projective line with smooth total space satisfying the so-called $K^2$-condition are birationally rigid: their Mori fibre space structure is unique. This implies that they are not birational to any Fano varieties, conic bundles or other del Pezzo fibrations. In particular, they are irrational. The families of del Pezzo fibrations with smooth total space of degree $2$ are rather special, as for "most" families a general de…
We study full exceptional collections of line bundles on surfaces. We prove that any full strong exceptional collection of line bundles on a weak del Pezzo surface of degree $\ge 3$ is an augmentation in the sense of L.Hille and M.Perling, while for some weak del Pezzo surfaces of degree $2$ the above is not true. We classify smooth projective surfaces possessing a cyclic strong exceptional collection of line bundles of maximal length: we prove that they are weak del Pezzo surfaces and find all types of weak del Pezzo surfaces admitting such a collection. We find simple criteria of exceptional…
In this paper we give a criterion for birational rigidity of del Pezzo fibrations of degree 1 and 2 with only quotient singularities. As an application, we prove birational rigidity of suitable del Pezzo fibrations admitting an action of the Klein simple group.
The main aim of this article is to show that a very general 3-dimensional del Pezzo fibration of degree 1,2,3 is not stably rational except for a del Pezzo fibration of degree 3 belonging to explicitly described 2 families. Higher dimensional generalizations are also discussed and we prove that a very general del Pezzo fibration of degree 1,2,3 defined over the projective space is not stably rational provided that the anticanonical divisor is not ample.
This is a report of the author's talk at Algebra Symposium 2019 held at Tohoku University. We discuss some improvements of Mori's Bend and Break for sections of del Pezzo fibrations over $\mathbb P^1$.
We describe the effect of rational singularities on the Brauer group of a surface, and compute the Brauer groups of all singular del Pezzo surfaces over an algebraically closed field.
We study the birational rigidity problem for smooth Mori fibrations on del Pezzo surfaces of degree 1 and 2. For degree 1 we obtain a complete description of rigid and non-rigid cases.
To study syzygies of the Cox rings of del Pezzo surfaces, we calculate important syzygetic invariants such as the Hilbert functions, the Green-Lazarsfeld indices, the projective dimensions, and the Castelnuovo-Mumford regularities. Using these computations as well as the natural multigrading structures by the Picard groups of del Pezzo surfaces and Weyl group actions on Picard lattices, we determine the Betti diagrams of the Cox rings of del Pezzo surfaces of degree at most four.
