Del Pezzo surface fibrations obtained by blow-up of a smooth curve in a projective manifold

2005 ◽  
Vol 340 (8) ◽  
pp. 581-586 ◽  
Author(s):  
Toru Tsukioka
Keyword(s):  
Blow Up ◽  
Smooth Curve ◽  
Pezzo Surface ◽  
Projective Manifold ◽  
Del Pezzo Surface ◽  
Del Pezzo

scholarly journals $${\mathfrak {S}}_5$$-equivariant syzygies for the Del Pezzo surface of degree 5

2020 ◽  
Author(s):  
Ingrid Bauer ◽  
Fabrizio Catanese
Keyword(s):  
Automorphism Group ◽  
Symmetric Group ◽  
Blow Up ◽  
Pezzo Surface ◽  
Irreducible Representations ◽  
Del Pezzo Surface ◽  
Del Pezzo ◽  
Arithmetically Gorenstein ◽  
Pfaffian Equations

Abstract The Del Pezzo surface Y of degree 5 is the blow up of the plane in 4 general points, embedded in $${\mathbb {P}}^5$$P5 by the system of cubics passing through these points. It is the simplest example of the Buchsbaum–Eisenbud theorem on arithmetically-Gorenstein subvarieties of codimension 3 being Pfaffian. Its automorphism group is the symmetric group $${\mathfrak {S}}_5$$S5. We give canonical explicit $${\mathfrak {S}}_5$$S5-invariant Pfaffian equations through a 6$$\times $$×6 antisymmetric matrix. We give concrete geometric descriptions of the irreducible representations of $${\mathfrak {S}}_5$$S5. Finally, we give $${\mathfrak {S}}_5$$S5-invariant equations for the embedding of Y inside $$({\mathbb {P}}^1)^5$$(P1)5, and show that they have the same Hilbert resolution as for the Del Pezzo of degree 4.

scholarly journals Genus Six Curves, K3 Surfaces, and Stable Pairs

2020 ◽  
Author(s):  
J Ross Goluboff
Keyword(s):  
Smooth Curve ◽  
Explicit Description ◽  
Pezzo Surface ◽  
Del Pezzo Surfaces ◽  
Del Pezzo Surface ◽  
Double Covers ◽  
Del Pezzo ◽  
Stable Pairs ◽  
Special Genus ◽  
Surface Curve

Abstract A general smooth curve of genus six lies on a quintic del Pezzo surface. Artebani and Kondō [ 4] construct a birational period map for genus six curves by taking ramified double covers of del Pezzo surfaces. The map is not defined for special genus six curves. In this paper, we construct a smooth Deligne–Mumford stack ${\mathfrak{P}}_0$ parametrizing certain stable surface-curve pairs, which essentially resolves this map. Moreover, we give an explicit description of pairs in ${\mathfrak{P}}_0$ containing special curves.

scholarly journals Affine cones over cubic surfaces are flexible in codimension one

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Alexander Perepechko
Keyword(s):  
Automorphism Group ◽  
Open Subset ◽  
Pezzo Surface ◽  
Ample Divisor ◽  
Transitive Action ◽  
Del Pezzo Surface ◽  
Codimension One ◽  
Cubic Surfaces ◽  
Special Automorphism ◽  
Del Pezzo

AbstractLet Y be a smooth del Pezzo surface of degree 3 polarized by a very ample divisor that is not proportional to the anticanonical one. Then the affine cone over Y is flexible in codimension one. Equivalently, such a cone has an open subset with an infinitely transitive action of the special automorphism group on it.

scholarly journals An Extension of BCOV Invariant

2020 ◽  
Author(s):  
Yeping Zhang
Keyword(s):  
Kähler Manifold ◽  
Pezzo Surface ◽  
Del Pezzo Surface ◽  
Kahler Manifold ◽  
Del Pezzo ◽  
Compact Kähler Manifold ◽  
Calabi Yau Manifolds

Abstract Bershadsky, Cecotti, Ooguri, and Vafa constructed a real-valued invariant for Calabi–Yau manifolds, which is called the BCOV invariant. In this paper, we consider a pair $(X,Y)$, where $X$ is a compact Kähler manifold and $Y\in \big |K_X^m\big |$ with $m\in{\mathbb{Z}}\backslash \{0,-1\}$. We extend the BCOV invariant to such pairs. If $m=-2$ and $X$ is a rigid del Pezzo surface, the extended BCOV invariant is equivalent to Yoshikawa’s equivariant BCOV invariant. If $m=1$, the extended BCOV invariant is well behaved under blowup. It was conjectured that birational Calabi–Yau three-folds have the same BCOV invariant. As an application of our extended BCOV invariant, we show that this conjecture holds for Atiyah flops.

On the Quartic Del Pezzo Surface

1941 ◽  
Vol 63 (2) ◽  
pp. 256
Author(s):  
C. Ronald Cassity
Keyword(s):  
Pezzo Surface ◽  
Del Pezzo Surface ◽  
Del Pezzo

scholarly journals Cylinders in singular del Pezzo surfaces

2016 ◽  
Vol 152 (6) ◽  
pp. 1198-1224 ◽  
Author(s):  
Ivan Cheltsov ◽  
Jihun Park ◽  
Joonyeong Won
Keyword(s):  
Pezzo Surface ◽  
Del Pezzo Surfaces ◽  
Del Pezzo Surface ◽  
Open Set ◽  
Del Pezzo

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 $S\setminus \text{Supp}(D)$ is a cylinder. As a corollary, we classify all the del Pezzo surfaces with du Val singularities that admit non-trivial $\mathbb{G}_{a}$-actions on their affine cones defined by their anticanonical divisors.

scholarly journals Kummer surfaces associated to (1, 2)-polarized abelian surfaces

2011 ◽  
Vol 202 ◽  
pp. 127-143
Author(s):  
Afsaneh Mehran
Keyword(s):  
Double Cover ◽  
Pezzo Surface ◽  
Abelian Surface ◽  
Kummer Surface ◽  
Abelian Surfaces ◽  
Del Pezzo Surface ◽  
Singular Fibers ◽  
Kummer Surfaces ◽  
Del Pezzo

AbstractThe aim of this paper is to describe the geometry of the generic Kummer surface associated to a (1, 2)-polarized abelian surface. We show that it is the double cover of a weak del Pezzo surface and that it inherits from the del Pezzo surface an interesting elliptic fibration with twelve singular fibers of typeI2.

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