scholarly journals Birational geometry of varieties, fibred into complete intersections of codimension two

2022 ◽  
Vol 86 (2) ◽  
Author(s):  
Aleksandr Valentinovich Pukhlikov
Keyword(s):  
Complete Intersections ◽  
Birational Geometry ◽  
Codimension Two

BOUNDEDNESS FOR CODIMENSION TWO SUBVARIETIES

2002 ◽  
Vol 13 (05) ◽  
pp. 479-495 ◽  
Author(s):  
CIRO CILIBERT ◽  
VINCENZO DI GENNARO
Keyword(s):  
Hilbert Scheme ◽  
General Type ◽  
Complete Intersections ◽  
Projective Varieties ◽  
Codimension Two

We prove that for certain projective varieties V ⊂ Pr (e.g. smooth complete intersections with dim (V) ≥ 4, or complete intersections with dim (V) ≥ 7 and codim V ( Sing (V)) ≥ 6), there are only finitely many components of the Hilbert scheme parametrizing irreducible, smooth, projective, codimension two subvarieties of V not of general type.

Birational Mori fiber structures of ℚ-Fano 3-fold weighted complete intersections, II

2018 ◽  
Vol 2018 (738) ◽  
pp. 73-129 ◽  
Author(s):  
Takuzo Okada
Keyword(s):  
Complete Intersection ◽  
Complete Intersections ◽  
Fiber Space ◽  
Codimension Two ◽  
Fano Threefold

Abstract In [T. Okada, Birational Mori fiber structures of \mathbb{Q} -Fano 3-fold weighted complete intersection, Proc. Lond. Math. Soc. (3) 109 2014, 6, 1549–1600], we proved that, among 85 families of \mathbb{Q} -Fano threefold weighted complete intersections of codimension two, 19 families consist of birationally rigid varieties and the remaining families consists of birationally non-rigid varieties. The aim of this paper is to study systematically the remaining families and prove that every quasismooth member of 14 families is birational to another \mathbb{Q} -Fano threefold but not birational to any other Mori fiber space.

scholarly journals GV-spectroscopy for F-theory on genus-one fibrations

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Paul-Konstantin Oehlmann ◽  
Thorsten Schimannek
Keyword(s):  
Complete Intersection ◽  
Complete Intersections ◽  
Detailed Knowledge ◽  
Gauge Groups ◽  
Frobenius Method ◽  
Codimension Two ◽  
Novel Technique ◽  
Genus One

Abstract We present a novel technique to obtain base independent expressions for the matter loci of fibrations of complete intersection Calabi-Yau onefolds in toric ambient spaces. These can be used to systematically construct elliptically and genus one fibered Calabi-Yau d-folds that lead to desired gauge groups and spectra in F-theory. The technique, which we refer to as GV-spectroscopy, is based on the calculation of fiber Gopakumar-Vafa invariants using the Batyrev-Borisov construction of mirror pairs and application of the so-called Frobenius method to the data of a parametrized auxiliary polytope. In particular for fibers that generically lead to multiple sections, only multi-sections or that are complete intersections in higher codimension, our technique is vastly more efficient than classical approaches. As an application we study two Higgs chains of six-dimensional supergravities that are engineered by fibrations of codimension two complete intersection fibers. Both chains end on a vacuum with G = ℤ4 that is engineered by fibrations of bi-quadrics in ℙ3. We use the detailed knowledge of the structure of the reducible fibers that we obtain from GV-spectroscopy to comment on the corresponding Tate-Shafarevich group. We also show that for all fibers the six-dimensional supergravity anomalies including the discrete anomalies generically cancel.

Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

2017 ◽  
Author(s):  
Claire Voisin
Keyword(s):  
Recent Work ◽  
Chow Groups ◽  
Ring Structure ◽  
Complete Intersections ◽  
Algebraic Varieties ◽  
Chow Ring ◽  
Hodge Structures ◽  
Chow Rings ◽  
Geometric Methods ◽  
The Right

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

Constructing symplectic manifolds

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Final Section ◽  
Symplectic Manifolds ◽  
Homotopy Equivalence ◽  
Codimension Two ◽  
Open Manifolds ◽  
Connected Sums ◽  
Symplectic Forms ◽  
Ambient Manifold ◽  
Blowing Up ◽  
Four Manifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

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