Varieties with nef diagonal

2019 ◽  
Vol 31 (02) ◽  
pp. 2050011 ◽  
Author(s):  
Taku Suzuki ◽  
Kiwamu Watanabe
Keyword(s):  
Projective Variety ◽  
Smooth Projective Variety ◽  
Complete Intersections ◽  
Spherical Varieties ◽  
Del Pezzo

For a smooth projective variety [Formula: see text], we consider when the diagonal [Formula: see text] is nef as a cycle on [Formula: see text]. In particular, we give a classification of complete intersections and smooth del Pezzo varieties where the diagonal is nef. We also study the nefness of the diagonal for spherical varieties.

scholarly journals On hyperquadrics containing projective varieties

2020 ◽  
Vol 32 (5) ◽  
pp. 1199-1209
Author(s):  
Euisung Park
Keyword(s):  
Projective Variety ◽  
Minimal Degree ◽  
Zero Set ◽  
Projective Varieties ◽  
Quadratic Equations ◽  
Linearly Independent ◽  
Del Pezzo ◽  
Classification Of Varieties ◽  
Irreducible Projective Variety

AbstractClassical Castelnuovo Lemma shows that the number of linearly independent quadratic equations of a nondegenerate irreducible projective variety of codimension c is at most {{{c+1}\choose{2}}} and the equality is attained if and only if the variety is of minimal degree. Also G. Fano’s generalization of Castelnuovo Lemma implies that the next case occurs if and only if the variety is a del Pezzo variety. Recently, these results are extended to the next case in [E. Park, On hypersurfaces containing projective varieties, Forum Math. 27 2015, 2, 843–875]. This paper is intended to complete the classification of varieties satisfying at least {{{c+1}\choose{2}}-3} linearly independent quadratic equations. Also we investigate the zero set of those quadratic equations and apply our results to projective varieties of degree {\geq 2c+1}.

scholarly journals Restrictions on the prime-to- fundamental group of a smooth projective variety

2015 ◽  
Vol 151 (6) ◽  
pp. 1083-1095
Author(s):  
Donu Arapura
Keyword(s):  
Fundamental Group ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Kähler Groups ◽  
One Relator Groups

The goal of this paper is to obtain restrictions on the prime-to-$p$ quotient of the étale fundamental group of a smooth projective variety in characteristic $p\geqslant 0$. The results are analogues of some theorems from the study of Kähler groups. Our first main result is that such groups are indecomposable under coproduct. The second result gives a classification of the pro-$\ell$ parts of one-relator groups in this class.

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 Multiplicative Chow–Künneth decompositions and varieties of cohomological K3 type

2021 ◽  
Author(s):  
Lie Fu ◽  
Robert Laterveer ◽  
Charles Vial
Keyword(s):  
Projective Variety ◽  
Arbitrary Dimension ◽  
Smooth Projective Variety ◽  
The Other ◽  
Fano Variety ◽  
Fano Varieties ◽  
Intersection Product ◽  
Canonical Class ◽  
Main Input

AbstractGiven a smooth projective variety, a Chow–Künneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial conjectured that hyper-Kähler varieties admit a multiplicative Chow–Künneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-Kähler varieties, we ask whether Fano varieties with cohomology of K3 type also admit a multiplicative Chow–Künneth decomposition, and provide evidence by establishing their existence for cubic fourfolds and Küchle fourfolds of type c7. The main input in the cubic hypersurface case is the Franchetta property for the square of the Fano variety of lines; this was established in our earlier work in the fourfold case and is generalized here to arbitrary dimension. On the other end of the spectrum, we also give evidence that varieties with ample canonical class and with cohomology of K3 type might admit a multiplicative Chow–Künneth decomposition, by establishing this for two families of Todorov surfaces.

scholarly journals Codimension One Holomorphic Distributions on the Projective Three-space

2018 ◽  
Vol 2020 (23) ◽  
pp. 9011-9074 ◽  
Author(s):  
Omegar Calvo-Andrade ◽  
Maurício Corrêa ◽  
Marcos Jardim
Keyword(s):  
Projective Space ◽  
Moduli Space ◽  
Tangent Bundle ◽  
Projective Variety ◽  
Arbitrary Degree ◽  
Codimension One ◽  
Quot Scheme ◽  
Space Of Distributions ◽  
Three Space

Abstract We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck’s Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety. Finally, we prove that every rational foliation and certain logarithmic foliations have stable tangent sheaves.

scholarly journals Complete intersections in spherical varieties

2016 ◽  
Vol 22 (4) ◽  
pp. 2099-2141 ◽  
Author(s):  
Kiumars Kaveh ◽  
A. G. Khovanskii
Keyword(s):  
Complete Intersections ◽  
Spherical Varieties

scholarly journals Vector bundles trivialized by proper morphisms and the fundamental group scheme

2010 ◽  
Vol 10 (2) ◽  
pp. 225-234 ◽  
Author(s):  
Indranil Biswas ◽  
João Pedro P. Dos Santos
Keyword(s):  
Finite Group ◽  
Fundamental Group ◽  
Characteristic Zero ◽  
Vector Bundles ◽  
Projective Variety ◽  
Group Scheme ◽  
Smooth Projective Variety ◽  
Closed Field ◽  
Surjective Morphism ◽  
Finite Vector

AbstractLet X be a smooth projective variety defined over an algebraically closed field k. Nori constructed a category of vector bundles on X, called essentially finite vector bundles, which is reminiscent of the category of representations of the fundamental group (in characteristic zero). In fact, this category is equivalent to the category of representations of a pro-finite group scheme which controls all finite torsors. We show that essentially finite vector bundles coincide with those which become trivial after being pulled back by some proper and surjective morphism to X.

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].

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