scholarly journals Classification of Fano 4-folds with Lefschetz defect 3 and Picard number 5

2021 ◽  
pp. 106864
Author(s):  
Cinzia Casagrande ◽  
Eleonora A. Romano
Keyword(s):  
Picard Number

scholarly journals Quotients of del Pezzo surfaces

2019 ◽  
Vol 30 (12) ◽  
pp. 1950068
Author(s):  
Andrey Trepalin
Keyword(s):  
Complete Classification ◽  
Finite Subgroup ◽  
Del Pezzo Surfaces ◽  
Picard Number ◽  
Del Pezzo Surface ◽  
Cyclic Groups ◽  
Trivial Group ◽  
Quotient Surface ◽  
Del Pezzo

Let [Formula: see text] be any field of characteristic zero, [Formula: see text] be a del Pezzo surface and [Formula: see text] be a finite subgroup in [Formula: see text]. In this paper, we study when the quotient surface [Formula: see text] can be non-rational over [Formula: see text]. Obviously, if there are no smooth [Formula: see text]-points on [Formula: see text] then it is not [Formula: see text]-rational. Therefore, under assumption that the set of smooth [Formula: see text]-points on [Formula: see text] is not empty we show that there are few possibilities for non-[Formula: see text]-rational quotients. The quotients of del Pezzo surfaces of degree [Formula: see text] and greater are considered in the author’s previous papers. In this paper, we study the quotients of del Pezzo surfaces of degree [Formula: see text]. We show that they can be non-[Formula: see text]-rational only for the trivial group or cyclic groups of order [Formula: see text], [Formula: see text] and [Formula: see text]. For the trivial group and the group of order [Formula: see text], we show that both [Formula: see text] and [Formula: see text] are not [Formula: see text]-rational if the [Formula: see text]-invariant Picard number of [Formula: see text] is [Formula: see text]. For the groups of order [Formula: see text] and [Formula: see text], we construct examples of both [Formula: see text]-rational and non-[Formula: see text]-rational quotients of both [Formula: see text]-rational and non-[Formula: see text]-rational del Pezzo surfaces of degree [Formula: see text] such that the [Formula: see text]-invariant Picard number of [Formula: see text] is [Formula: see text]. As a result of complete classification of non-[Formula: see text]-rational quotients of del Pezzo surfaces we classify surfaces that are birationally equivalent to quotients of [Formula: see text]-rational surfaces, and obtain some corollaries concerning fields of invariants of [Formula: see text].

scholarly journals On the classification of 3-dimensionalSL2(ℂ)-varieties

2000 ◽  
Vol 157 ◽  
pp. 129-147 ◽  
Author(s):  
Stefan Kebekus
Keyword(s):  
Automorphism Group ◽  
Semisimple Group ◽  
Picard Number ◽  
Projective Varieties ◽  
3 Dimensional

In the present work we describe 3-dimensional complexSL2-varieties where the genericSL2-orbit is a surface. We apply this result to classify the minimal 3-dimensional projective varieties with Picard-number 1 where a semisimple group acts such that the generic orbits are 2-dimensional.This is an ingredient of the classification [Keb99] of the 3-dimensional relatively minimal quasihomogeneous varieties where the automorphism group is not solvable.

scholarly journals Fibration and classification of smooth projective toric varieties of low Picard number

2020 ◽  
Vol 31 (06) ◽  
pp. 2050043
Author(s):  
Michele Rossi ◽  
Lea Terracini
Keyword(s):  
Toric Variety ◽  
Fiber Type ◽  
Toric Varieties ◽  
Computational Techniques ◽  
Picard Number ◽  
Geometric Problem ◽  
Dimension Formula ◽  
Number Formula ◽  
Effective Divisors

In this paper, we show that a smooth toric variety [Formula: see text] of Picard number [Formula: see text] always admits a nef primitive collection supported on a hyperplane admitting non-trivial intersection with the cone [Formula: see text] of numerically effective divisors and cutting a facet of the pseudo-effective cone [Formula: see text], that is [Formula: see text]. In particular, this means that [Formula: see text] admits non-trivial and non-big numerically effective divisors. Geometrically, this guarantees the existence of a fiber type contraction morphism over a smooth toric variety of dimension and Picard number lower than those of [Formula: see text], so giving rise to a classification of smooth and complete toric varieties with [Formula: see text]. Moreover, we revise and improve results of Oda–Miyake by exhibiting an extension of the above result to projective, toric, varieties of dimension [Formula: see text] and Picard number [Formula: see text], allowing us to classifying all these threefolds. We then improve results of Fujino–Sato, by presenting sharp (counter)examples of smooth, projective, toric varieties of any dimension [Formula: see text] and Picard number [Formula: see text] whose non-trivial nef divisors are big, that is [Formula: see text]. Producing those examples represents an important goal of computational techniques in definitely setting an open geometric problem. In particular, for [Formula: see text], the given example turns out to be a weak Fano toric fourfold of Picard number 4.

scholarly journals Wedge Operations and Torus Symmetries II

2017 ◽  
Vol 69 (4) ◽  
pp. 767-789 ◽  
Author(s):  
Suyoung Choi ◽  
Hanchul Park
Keyword(s):  
Positive Integer ◽  
Simplicial Complex ◽  
Main Idea ◽  
Picard Number ◽  
Fundamental Idea ◽  
Combinatorial Objects ◽  
Toric Topology ◽  
Simplicial Sphere ◽  
Fixed Positive Integer

AbstractA fundamental idea in toric topology is that classes of manifolds with well-behaved torus actions (simply, toric spaces) are classified by pairs of simplicial complexes and (non-singular) characteristic maps. In a previous paper, the authors provided a new way to find all characteristic maps on a simplicial complex K(J) obtainable by a sequence of wedgings from K.The main idea was that characteristic maps on K theoretically determine all possible characteristic maps on a wedge of K.We further develop our previous work for classification of toric spaces. For a star-shaped simplicial sphere K of dimension n-1 with m vertices, the Picard number Pic(K) of K is m-n. We call K a seed if K cannot be obtained by wedgings. First, we show that for a fixed positive integer 𝓁, there are at most finitely many seeds of Picard 𝓁 number supporting characteristic maps. As a corollary, the conjecture proposed by V. V. Batyrev in is solved affirmatively.Secondly, we investigate a systematicmethod to find all characteristic maps on K(J) using combinatorial objects called (realizable) puzzles that only depend on a seed K. These two facts lead to a practical way to classify the toric spaces of fixed Picard number.

scholarly journals Smooth Fano Intrinsic Grassmannians of Type 2,n with Picard Number Two

2021 ◽  
Author(s):  
Muhammad Imran Qureshi ◽  
Milena Wrobel
Keyword(s):  
Explicit Formula ◽  
Projective Variety ◽  
Complete Classification ◽  
Picard Number ◽  
Cox Ring

Abstract We introduce the notion of intrinsic Grassmannians that generalizes the well-known weighted Grassmannians. An intrinsic Grassmannian is a normal projective variety whose Cox ring is defined by the Plucker ideal $I_{d,n}$ of the Grassmannian $\textrm{Gr}(d,n)$. We give a complete classification of all smooth Fano intrinsic Grassmannians of type $(2,n)$ with Picard number two and prove an explicit formula to compute the total number of such varieties for an arbitrary $n$. We study their geometry and show that they satisfy Fujita’s freeness conjecture.

scholarly journals Codimension one Fano distributions on Fano manifolds

2018 ◽  
Vol 20 (05) ◽  
pp. 1750058 ◽  
Author(s):  
Carolina Araujo ◽  
Mauricio Corrêa ◽  
Alex Massarenti
Keyword(s):  
Moduli Spaces ◽  
Projective Spaces ◽  
Complete Intersections ◽  
Picard Number ◽  
Contact Manifolds ◽  
Fano Manifolds ◽  
Codimension One ◽  
Linear Sections ◽  
Del Pezzo

In this paper, we investigate codimension one Fano distributions on Fano manifolds with Picard number one. We classify Fano distributions of maximal index on complete intersections in weighted projective spaces, Fano contact manifolds, Grassmannians of lines and their linear sections, and describe their moduli spaces. As a consequence, we obtain a classification of codimension one del Pezzo distributions on Fano manifolds with Picard number one.

scholarly journals Fano manifolds containing a negative divisor isomorphic to a rational homogeneous space of Picard number one

2020 ◽  
Vol 31 (09) ◽  
pp. 2050066
Author(s):  
Jie Liu
Keyword(s):  
Homogeneous Space ◽  
Complete Classification ◽  
Picard Number ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Conormal Bundle ◽  
Rational Homogeneous Space

Let [Formula: see text] be an [Formula: see text]-dimensional complex Fano manifold [Formula: see text]. Assume that [Formula: see text] contains a divisor [Formula: see text], which is isomorphic to a rational homogeneous space with Picard number one, such that the conormal bundle [Formula: see text] is ample over [Formula: see text]. Building on the works of Tsukioka, Watanabe and Casagrande–Druel, we give a complete classification of such pairs [Formula: see text].

Towards the classification of weak Fano threefolds with ρ = 2

2013 ◽  
Vol 11 (9) ◽  
Author(s):  
Joseph Cutrone ◽  
Nicholas Marshburn
Keyword(s):  
Finite Number ◽  
Type Ii ◽  
Numerical Classification ◽  
Picard Number ◽  
Fano Threefolds ◽  
Smooth Complex ◽  
Complex Projective ◽  
Extremal Ray ◽  
Weak Fano

AbstractIn this paper, examples of type II Sarkisov links between smooth complex projective Fano threefolds with Picard number one are provided. To show examples of these links, we study smooth weak Fano threefolds X with Picard number two and with a divisorial extremal ray. We assume that the pluri-anticanonical morphism of X contracts only a finite number of curves. The numerical classification of these particular smooth weak Fano threefolds is completed and the geometric existence of some numerical cases is proven.

Note on the spectral classification of carbon stars in the photographic infrared region

1966 ◽  
Vol 24 ◽  
pp. 21-23
Author(s):  
Y. Fujita
Keyword(s):  
Infrared Region ◽  
Spectral Classification ◽  
Carbon Stars

We have investigated the spectrograms (dispersion: 8Å/mm) in the photographic infrared region fromλ7500 toλ9000 of some carbon stars obtained by the coudé spectrograph of the 74-inch reflector attached to the Okayama Astrophysical Observatory. The names of the stars investigated are listed in Table 1.

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