# Classification Of Exceptional Log Del Pezzo Surfaces With

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### On Birational Morphisms between Pencils of Del Pezzo Surfaces

Here 0 is a sequence of log-flips and p(S/T) = p(S'/T) = 1. We will work with the case of two Del Pezzo fibrations. The simplest possible situation is when X and X' are smooth and the map 0 is a flop. Then our diagram Received by the editors May 27, 1998 and, in revised form, December 23, 1999. 2000 Mathematics Subject Classification.

### Del Pezzo surfaces with Du Val singularities

order to compute the global log canonical thresholds of all Del Pezzo surfaces of degree 1 with Du Val singularities, as well as the global log canonical thresholds of all Del Pezzo surfaces of Picard rank 1 with Du Val singularities. As a consequence, it is proven that Del Pezzo

### Vasilii Alekseevich Iskovskikh

practically definitive results on 3-folds with a pencil of del Pezzo surfaces [80], [84], and Sarkisov s theorem [85] settles the question of birational rigidity for almost all conic bundles. The frontline of research on this topic has come a long way from its starting point in the 1970s but the importance of the first breakthrough shines

### Hodge Theory and Algebraic Geometry

Let S be a weak Del Pezzo surface but a Del Pezzo surface. Then it is a weakened Del Pezzo surface. Thus P1 £ S is a weakened Fano 3-fold. We call such weakened Fano 3-folds product type Theorem 1.4. (H.Sato [Sa]) There are exacty 15 toric weakened Fano 3-folds X up to isomorphism.

### Hokkaido University-KAIST(ASARC) Joint Workshop Algebra and

These surfaces were constructed by logarithmic transformations of rational elliptic surfaces. Dolgachev surfaces can be also constructed via Q-Gorenstein smoothing of singular rational surfaces with two cyclic quotient singularities based on the construction by Lee-Park. In this talk, some exceptional bundles and collections on Dolgachev

### CLASSIFICATION OF LOG DEL PEZZO SURFACES OF arXiv:math

Jun 05, 2020 DEL PEZZO SURFACES OF INDEX ≤2 3 i.e. with PicZ = Z, log del Pezzo surfaces of index ≤ 2 (see Section 4). Of those, 28 are Gorenstein and were previously known (see [BBD84]), and the other 18 are not Gorenstein. Note that minimal log del Pezzo surfaces play a special role in log Minimal Model Program, see f.e. [Sho85, Kaw84, Mor82, Rei83].

### Title Polarized endomorphisms on normal projective varieties

log del Pezzo surfaces and Pi. Similar results are obtained for polarized en-dernorphisms of uniruled threefolds and fourfolds. As a consequence, we show conceptually that every smooth Fano threefold -Fit h a pelarized endomorphism of degree År 1, is rational. l. INTRODUCTION We work over the field C of complex numbers. We study polarized en-

### leq 2$ and applications - Project Euclid

4 Classi cation oflog del Pezzo surfaces ofindex $ leq 2$ and applications 4.1. Classi cation o og del Pezzo surfaces ofindex $ leq 2$ From the results ofChapters 1{3we obtain Theorem 4.1.

### CONTEMPORARY MATHEMATICS - AMS

Exceptional groups and del Pezzo surfaces ROBERT FRIEDMAN AND JOHN W. MORGAN The regulator map for a general curve MARK GREEN AND PHILLIP GRIFFITHS Versal deformations and superpotentials for rational curves in smooth threefolds 9 71 87 101 117 SHELDON KATZ 129 The Nash conjecture for nonprojective threefolds JANOS KOLLAR 137

### Complements on surfaces - maths.ed.ac.uk

COMPLEMENTS ON SURFACES V. V. Shokurov unc 512.774 The main result is a boundedness theorem for n-complements on algebraic surfaces. In addition, this theorem is used in a classification oflog Del Pezzo surfaces and birational contractions for threefolds. Acknowledgment. Partial financial support was provided by the NSF (Grant No. DMS-9500971

### THEOREMS ABOUT GOOD DIVISORS - University of Georgia

ry, if we have the log Del Pezzo surface Y and -Ky= r'H then (n-2)-multiple generalized cone over X 2 (see construction 0-9 below) is a log Fano variety of dimension n and of Fano index r = r'+ (n-2). The following construction is due to T.Fujita, [FI]. Construction 0-9.

### FRACTIONAL INDICE S O F LO G DE L PEZZ O SURFACE S

Let X be a log del Pezzo surface, that is, a singular normal complex surface with ample anticanonical class and log terminal singularities (see §1 for precise defini tions). These surfaces are a generalization of a very well studied class of surfaces, the usual del Pezzo surfaces; their classification is interesting in its own right, and

### Supersymmetric Gauge Theories from D3-branes on Singularities

their features in the language of exceptional Lie algebras. We suggest the possibility 4.2 En symmetries and del Pezzo surfaces the log of energy scale t

### Manin's conjecture for a singular sextic del Pezzo surface

more information on smooth and singular del Pezzo surfaces respectively, and [Bro07] for a general overview of Manin's conjecture for del Pezzo sur-faces. The surface S has one singularity of type A2, which we can resolve using blow-ups to create two exceptional curves on the minimal desingularisation S of S.

### A glimpse at the classification of Orbifold del Pezzo surfaces

Del Pezzo Surfaces The degree of a del Pezzo surface is d = ( K X)2. Theorem (Castelnuovo) Let X be a del Pezzo surface. Then X is either P1 P1 or P2 blown up in 9 d general points (where d 1).)complete birational classi cation of del Pezzo surfaces Remark Every del Pezzo surface is realised as a complete intersection in Fano varieties.

### On Fano foliations

We focus our attention on a special class of Fano foliations, namely del Pezzo foliations on complex projective manifolds. We show that these foliations are algebraically integrable, with one exceptional case when the ambient space is Pn. We also provide a classification of del Pezzo foliations with mild singularities. ⃝c 2013 Elsevier Inc.

### CLASSIFICATION OF LOG DEL PEZZO SURFACES OF arXiv:math

4. Classiﬁcation of log del Pezzo surfaces of index ≤ 2 and applications103 4.1. Classiﬁcation of log del Pezzo surfaces of index ≤ 2 103 4.2. An example: Enumeration of all possible types for N= 20.111 4.3. Application: Minimal projective compactiﬁcations of afﬁne surfaces in P2 by relatively minimal log del Pezzo surfaces of index

### Contents

6. The structure of log del Pezzo surfaces of index two 95 7. Description of log del Pezzo surfaces of index two 133 References 164 1. Introduction In this article, we work in the category of algebraic schemes (or algebraic spaces) over an algebraically closed ﬁeld k. A del Pezzo surface is a non-singular projective surface

### Q-homology projective planes with nodes or cusps

(2) Log del Pezzo surfaces of Picard number 1 with 3 or 4 cusps can be constructed in many ways. One way is to consider a rational elliptic surface V with 4 singular fibres of type h. Such an elliptic surface can be constructed by blowing up IP'2 at the 9 base points of the Hesse pencil. Every section is a ( ~ 1 )-curve. Contracting a section