Classification of arithmetically Gorenstein divisors on some Fano varieties

2008 ◽  
Vol 188 (4) ◽  
pp. 611-617
Author(s):  
Gianni Ciolli ◽  
Alberto Dolcetti
Keyword(s):  
Fano Varieties ◽  
Arithmetically Gorenstein

scholarly journals Betti numbers and pseudoeffective cones in 2-Fano varieties

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Giosuè Emanuele Muratore
Keyword(s):  
Large Class ◽  
Betti Numbers ◽  
Fano Varieties ◽  
Higher Dimensional ◽  
Special Case ◽  
Answering Questions

Abstract The 2-Fano varieties, defined by De Jong and Starr, satisfy some higher-dimensional analogous properties of Fano varieties. We consider (weak) k-Fano varieties and conjecture the polyhedrality of the cone of pseudoeffective k-cycles for those varieties, in analogy with the case k = 1. Then we calculate some Betti numbers of a large class of k-Fano varieties to prove some special case of the conjecture. In particular, the conjecture is true for all 2-Fano varieties of index at least n − 2, and we complete the classification of weak 2-Fano varieties answering Questions 39 and 41 in [2].

scholarly journals Simple normal crossing Fano varieties and log Fano manifolds

2014 ◽  
Vol 214 ◽  
pp. 95-123
Author(s):  
Kento Fujita
Keyword(s):  
Fano Varieties ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Normal Crossing ◽  
Fano Index ◽  
Simple Normal Crossing

AbstractA projective log variety (X, D) is called alog Fano manifoldifXis smooth and ifDis a reduced simple normal crossing divisor onΧwith − (KΧ+D) ample. Then-dimensional log Fano manifolds (X, D) with nonzeroDare classified in this article when the log Fano indexrof (X, D) satisfies eitherr≥n/2withρ(X) ≥ 2 orr≥n− 2. This result is a partial generalization of the classification of logarithmic Fano 3-folds by Maeda.

scholarly journals Canonical Toric Fano Threefolds

2010 ◽  
Vol 62 (6) ◽  
pp. 1293-1309 ◽  
Author(s):  
Alexander M. Kasprzyk
Keyword(s):  
Fano Varieties ◽  
Fano Threefolds ◽  
Inductive Approach ◽  
Canonical Singularities ◽  
Toric Fano Varieties

AbstractAn inductive approach to classifying all toric Fano varieties is given. As an application of this technique, we present a classification of the toric Fano threefolds with at worst canonical singularities. Up to isomorphism, there are 674,688 such varieties.

scholarly journals Simple normal crossing Fano varieties and log Fano manifolds

2014 ◽  
Vol 214 ◽  
pp. 95-123 ◽  
Author(s):  
Kento Fujita
Keyword(s):  
Fano Varieties ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Normal Crossing ◽  
Fano Index ◽  
Simple Normal Crossing

AbstractA projective log variety (X, D) is called a log Fano manifold if X is smooth and if D is a reduced simple normal crossing divisor on Χ with − (KΧ + D) ample. The n-dimensional log Fano manifolds (X, D) with nonzero D are classified in this article when the log Fano index r of (X, D) satisfies either r ≥ n/2 with ρ(X) ≥ 2 or r ≥ n − 2. This result is a partial generalization of the classification of logarithmic Fano 3-folds by Maeda.

scholarly journals CLASSIFICATION OF POLARIZED SYMPLECTIC AUTOMORPHISMS OF FANO VARIETIES OF CUBIC FOURFOLDS

2015 ◽  
Vol 58 (1) ◽  
pp. 17-37 ◽  
Author(s):  
LIE FU
Keyword(s):  
Fano Varieties ◽  
Symplectic Automorphisms

AbstractWe classify the polarized symplectic automorphisms of Fano varieties of smooth cubic fourfolds (equipped with the Plücker polarization) and study the fixed loci.

Detecting non-displaceable toric fibers on compact toric manifolds via tropicalizations

2019 ◽  
Vol 30 (01) ◽  
pp. 1950003
Author(s):  
Yoosik Kim ◽  
Jaeho Lee ◽  
Fumihiko Sanda
Keyword(s):  
Discrete Comput Geom ◽  
Duke Math ◽  
Fano Varieties ◽  
Toric Manifold ◽  
Floer Cohomology ◽  
Floer Theory ◽  
Toric Manifolds ◽  
Moment Polytope ◽  
Bulk Deformations

We provide a combinatorial way to locate non-displaceable Lagrangian toric fibers on any compact toric manifold. By taking the intersection of certain tropicalizations coming from its moment polytope, one can detect all Lagrangian toric fibers having non-vanishing Floer cohomology ([K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian Floer theory on compact toric manifolds, I, Duke Math. J. 151(1) (2010) 23–174; K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian Floer theory on compact toric manifolds II: bulk deformations, Selecta Math. (N.S.) 17(3) (2011) 609–711.]). The intersection completely characterizes all non-displaceable toric fibers, in some cases including pseudo symmetric smooth Fano varieties ([G. Ewald, On the classification of toric Fano varieties, Discrete Comput. Geom. 3(1) (1988) 49–54.]).

scholarly journals Maximally mutable Laurent polynomials

2021 ◽  
Vol 477 (2254) ◽  
Author(s):  
Tom Coates ◽  
Alexander M. Kasprzyk ◽  
Giuseppe Pitton ◽  
Ketil Tveiten
Keyword(s):  
Mirror Symmetry ◽  
Three Dimensional ◽  
Computer Assisted ◽  
Del Pezzo Surfaces ◽  
Fano Varieties ◽  
Laurent Polynomials ◽  
Fano Manifold ◽  
One To One ◽  
Del Pezzo

We introduce a class of Laurent polynomials, called maximally mutable Laurent polynomials (MMLPs), which we believe correspond under mirror symmetry to Fano varieties. A subclass of these, called rigid, are expected to correspond to Fano varieties with terminal locally toric singularities. We prove that there are exactly 10 mutation classes of rigid MMLPs in two variables; under mirror symmetry these correspond one-to-one with the 10 deformation classes of smooth del Pezzo surfaces. Furthermore, we give a computer-assisted classification of rigid MMLPs in three variables with reflexive Newton polytope; under mirror symmetry these correspond one-to-one with the 98 deformation classes of three-dimensional Fano manifolds with very ample anti-canonical bundle. We compare our proposal to previous approaches to constructing mirrors to Fano varieties, and explain why mirror symmetry in higher dimensions necessarily involves varieties with terminal singularities. Every known mirror to a Fano manifold, of any dimension, is a rigid MMLP.

scholarly journals Positivity of the CM line bundle for families of K-stable klt Fano varieties

2020 ◽  
Author(s):  
Giulio Codogni ◽  
Zsolt Patakfalvi
Keyword(s):  
Probability Theory ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Line Bundle ◽  
Central Limit ◽  
Moduli Space ◽  
Fano Varieties ◽  
Real Numbers ◽  
The Central Limit Theorem

AbstractThe Chow–Mumford (CM) line bundle is a functorial line bundle on the base of any family of klt Fano varieties. It is conjectured that it yields a polarization on the moduli space of K-poly-stable klt Fano varieties. Proving ampleness of the CM line bundle boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with K-semi-stable/K-polystable fibers. We prove the necessary semi-positivity statements in the K-semi-stable situation, and the necessary positivity statements in the uniform K-stable situation, including in both cases variants assuming K-stability only for general fibers. Our statements work in the most general singular situation (klt singularities), and the proofs are algebraic, except the computation of the limit of a sequence of real numbers via the central limit theorem of probability theory. We also present an application to the classification of Fano varieties. Additionally, our semi-positivity statements work in general for log-Fano pairs.

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