The classical statement of Cayley-Salmon that there are 27 lines on every smooth cubic surface in P^3 fails to hold under tropicalization: a tropical cubic surface in TP^3 often contains infinitely many tropical lines. Under mild genericity assumptions, we show that when embedded using the Eckardt triangles in the anticanonical system, tropical cubic del Pezzo surfaces contain exactly 27 tropical lines. In the non-generic case, which we identify explicitly, we find up to 27 extra lines, no multiple of which lifts to a curve on the cubic surface.
We realize the moduli space of stable anticanonical tropical cubics as a four-dimensional fan in R^40 with an action of the Weyl group W(E_6). In the absence of Eckardt points, we show the combinatorial types of these tropical surfaces are determined by the boundary arrangement of 27 metric trees corresponding to the tropicalization of the classical 27 lines on the smooth algebraic cubic surfaces. Tropical convexity and the combinatorics of the root system E_6 play a central role in our analysis.
Characterization of log del Pezzo pairs via anticanonical models
在文献中，log del Pezzo对的定义有几种变化。我们定义了合适的光滑模型，并证明了它们是相同的。特别是，我们得到了一个光滑的log del Pezzo对的anticanonical模型的特征。作为应用，我们对非有理弱对数典型del Pezzo对进行了分类，并证明了所有整体F-正则型曲面都是Fano型曲面。
We construct algebraic geometric codes from del Pezzo surfaces and focus on the ones having Picard rank one and the codes associated to the anticanonical class. We give explicit constructions of del Pezzo surfaces of degree 4, 5 and 6, compute the parameters of the associated anticanonical codes and study their isomorphisms arising from the automorphisms of the surface. We obtain codes with excellent parameters and some of them turn out to beat the best known codes listed on the database codetable.
We completely classify redundant blow-ups appearing in the theory of rational surfaces with big anticanonical divisor due to Sakai. In particular, we construct a rational surface with big anticanonical divisor which is not a minimal resolution of a del Pezzo surface with only rational singularities, which gives a negative answer to a question raised in a paper by Testa, Várilly-Alvarado, and Velasco.
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.
Kavli IPMU (WPI), UTIAS, The University of Tokyo（东京大学）；Vakgroep Wiskunde en Data Science, Vrije Universiteit Brussel（布鲁塞尔自由大学）；Vakgroep Wiskunde, Universiteit Hasselt（哈瑟尔特大学）
We investigate torsion exceptional sheaves on a weak del Pezzo surface of degree greater than two whose anticanonical model has at most $A_n$-singularities. We show that every torsion exceptional sheaf can be obtained from a line bundle on a $(-1)$-curve by acting spherical twist functors successively.
For each del Pezzo surface $S$ with du Val singularities, we determine whether it admits a $(-K_S)$-polar cylinder or not. If it allows one, then we present an effective $\mathbb{Q}$-divisor $D$ that is $\mathbb{Q}$-linearly equivalent to $-K_S$ and such that the open set $S\setminus\mathrm{Supp}(D)$ is a cylinder. As a corollary, we classify all the del Pezzo surfaces with du Val singularities that admit nontrivial $\mathbb{G}_a$-actions on their affine cones defined by their anticanonical divisors.
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 which potential density does not hold). This answers positively a question raised by Hassett and Tschinkel and, combined with previous results, completes the proof of the potential density of integral points for complements of anticanonical divisors in smooth del Pezzo surfaces. We then classify the complements which are simply connected and for these we prove…
Kähler-Einstein Metrics for some quasi smooth log del Pezzo Surfaces
最近Johnson和Kollár确定了加权射影3-空间中反正则嵌入的拟光滑log del Pezzo曲面的完整列表。他们还证明了许多这样的表面都接受凯勒-爱因斯坦度规，而且其中一些表面没有老虎。本文的目的是要解决的问题存在的凯勒-爱因斯坦度量和老虎的表面的问题仍然是开放的。为了做到这一点，我们将使用Nadel、Demailly和Kollár早期开发的技术。
Let $X$ be a del Pezzo surface of degree $2$ or greater over a finite field $\mathbb{F}_q$. The image $Γ$ of the Galois group $\operatorname{Gal}(\overline{\mathbb{F}}_q / \mathbb{F}_q)$ in the group $\operatorname{Aut}(\mathrm{Pic}(\overline{X}))$ is a cyclic subgroup preserving the anticanonical class and the intersection form. The conjugacy class of $Γ$ in the subgroup of $\operatorname{Aut}(\mathrm{Pic}(\overline{X}))$ preserving the anticanonical class and the intersection form is a natural invariant of $X$. We say that the conjugacy class of $Γ$ in $\operatorname{Aut}(\mathrm{Pic}(\overl…
Consider a log Calabi-Yau pair $(X,D)$ consisting of a smooth del Pezzo surface $X$ of degree $\geq 3$ and a smooth anticanonical divisor $D$. We prove a correspondence between genus zero logarithmic Gromov-Witten invariants of $X$ intersecting $D$ in a single point with maximal tangency and the consistent wall structure appearing in the dual intersection complex of $(X,D)$ from the Gross-Siebert reconstruction algorithm. More precisely, the logarithm of the product of functions attached to unbounded walls in the consistent wall structure gives a generating function for these invariants.
For all k>0 integer, we consider the regularised I-function of the family of del Pezzo surfaces of degree 8k+4 in P(2,2k+1,2k+1, 4k+1), first constructed by Johnson and Kollár. We show that this function, which is of hypergeometric type, is a period of an explicit pencil of curves. Thus the pencil is a candidate LG mirror of the family of del Pezzo surfaces. The main feature of these surfaces, which makes the mirror construction especially interesting, is that the anticanonical system is empty: because of this, our mirrors are not covered by any other construction known to us. We discuss conne…
We study admissible subcategories of derived categories of coherent sheaves on del Pezzo surfaces and rational elliptic surfaces. Using a relation between admissible subcategories and anticanonical divisors we prove the following results. First, we classify all admissible subcategories of the projective plane by showing that each is generated by a subcollection of a full exceptional collection. Second, we show that the derived categories of del Pezzo surfaces do not contain any phantom subcategories. This provides first examples of varieties of dimension larger than one that have some nontrivi…
Let $X$ be a Fano manifold. While the properties of the anticanonical divisor $-K_X$ and its multiples have been studied by many authors, the positivity of the tangent bundle $T_X$ is much more elusive. We give a complete characterisation of the pseudoeffectivity of $T_X$ for del Pezzo surfaces, hypersurfaces in the projective space and del Pezzo threefolds.
Algebraic Surfaces with Log-Terminal Singularities and nef Anticanonical Class and Reflection Groups in Lobachevsky Spaces. I (Basics of the Diagram Method)
在我们的预印本中：“Algebras surfaces with log-terminal singularities and nef anticanonical class and reflection groups in Lobachevsky spaces”，Preprint Max-Planck-Institut für Mathematik，Bonn，（1989）MPI/89-28（Russian），我们已经将图解方法推广到具有nef anticanonical class的代数曲面（之前，它被应用于Del Pezzo曲面）。这里的主要问题是，森多面体可能不是有限多面体在这种情况下。在本预印本的§4中，我们已经证明了这种方法对具有nef反正则类和对数终端奇点的代数曲面有效。最近，Valery Alexeev（Boundedness and K^2 for log surfaces，Preprint（1994），alg-geom/942007）特别利用我们的这些结果，对具有半对数正则奇点的一般类型曲面获得了强有力的重要结果。原因是德尔佩佐曲面（图方法是特别有效的）和一般类型的表面“连接”通过表面的数字零规范类。这是我们决定将上面预印本的第一部分（§1-§3）翻译成英语的主要原因，其中有一些扩展（我们省略了证明）和小的更正。这一部分包含了罗巴切夫斯基空间中的广义反射群的图解方…
In this article, we consider weak del Pezzo surfaces defined over a finite field, and their associated, singular, anticanonical models. We first define arithmetic types for such surfaces, by considering the Frobenius actions on their Picard groups; this extends the classification of Swinnerton-Dyer and Manin for ordinary del Pezzo surfaces. We also show that some invariants of the surfaces only depend on the above type.Then we study an inverse Galois problem for singular del Pezzo surfaces having degree $3\leq d\leq 6$: we describe which types can occur over a given finite field (of odd charac…
Building on the work of Mukai, we explore the birational geometry of the moduli spaces M_{S,L} of semistable rank two torsion-free sheaves, with c_1=-K_S and c_2=2, on a polarized degree one del Pezzo surface (S,L); this is related to the birational geometry of the blow-up X of P^4 in 8 points. Our analysis is explicit and is obtained by looking at the variation of stability conditions. Then we provide a careful investigation of the blow-up X and of the moduli space Y=M_{S,-K_S}, which is a remarkable family of smooth Fano 4-folds. In particular we describe the relevant cones of divisors of Y,…
Motivated by the relationship between numerical Grothendieck groups induced by the embedding of a smooth anticanonical elliptic curve into a del Pezzo surface, we define the notion of a quasi del Pezzo homomorphism between pseudolattices and establish its basic properties. The primary aim of the paper is then to prove a classification theorem for quasi del Pezzo homomorphisms, using a pseudolattice variant of the minimal model program. Finally, this result is applied to the classification of a certain class of genus one Lefschetz fibrations over discs.
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 consider the problem of interpolating projective varieties through points and linear spaces. We show that del Pezzo surfaces satisfy weak interpolation.
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.
