scholarly journals A maximally-graded invertible cubic threefold that does not admit a full exceptional collection of line bundles

2020 ◽  
Vol 8 ◽  
Author(s):  
David Favero ◽  
Daniel Kaplan ◽  
Tyler L. Kelly
Keyword(s):  
Symmetry Group ◽  
Derived Category ◽  
Line Bundles ◽  
Exceptional Collection ◽  
Cubic Threefold

Abstract We show that there exists a cubic threefold defined by an invertible polynomial that, when quotiented by the maximal diagonal symmetry group, has a derived category that does not have a full exceptional collection consisting of line bundles. This provides a counterexample to a conjecture of Lekili and Ueda.

scholarly journals Exceptional collections on toric Fano threefolds and birational geometry

2014 ◽  
Vol 25 (07) ◽  
pp. 1450072 ◽  
Author(s):  
Hokuto Uehara
Keyword(s):  
Toric Varieties ◽  
Derived Category ◽  
Line Bundles ◽  
Birational Geometry ◽  
Fano Threefolds ◽  
Structure Sheaf ◽  
Exceptional Collection ◽  
Push Forward

Bondal's conjecture states that the Frobenius push-forward of the structure sheaf 𝒪X generates the derived category Db(X) for smooth projective toric varieties X. Bernardi and Tirabassi exhibit a full strong exceptional collection consisting of line bundles on smooth toric Fano 3-folds assuming Bondal's conjecture. In this paper, we prove Bondal's conjecture for smooth toric Fano 3-folds and improve upon their result using birational geometry.

scholarly journals A note on thick subcategories of stable derived categories

2013 ◽  
Vol 212 ◽  
pp. 87-96
Author(s):  
Henning Krause ◽  
Greg Stevenson
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Complete Intersections ◽  
Line Bundles ◽  
Finitely Generated ◽  
Exact Category ◽  
Coherent Sheaves ◽  
Noetherian Scheme ◽  
Finitely Generated Modules ◽  
Local Complete Intersections

AbstractFor an exact category having enough projective objects, we establish a bijection between thick subcategories containing the projective objects and thick subcategories of the stable derived category. Using this bijection, we classify thick subcategories of finitely generated modules over strict local complete intersections and produce generators for the category of coherent sheaves on a separated Noetherian scheme with an ample family of line bundles.

Differential Graded Quivers of Smooth Rational Surfaces

2016 ◽  
Vol 60 (4) ◽  
pp. 859-876
Author(s):  
Agnieszka Bodzenta
Keyword(s):  
Minimal Model ◽  
Rational Surface ◽  
Line Bundles ◽  
Toric Surfaces ◽  
Rational Surfaces ◽  
Exceptional Collection ◽  
Dg Algebras

AbstractLetXbe a smooth rational surface. We calculate a differential graded (DG) quiver of a full exceptional collection of line bundles onXobtained by an augmentation from a strong exceptional collection on the minimal model ofX. In particular, we calculate canonical DG algebras of smooth toric surfaces.

Chern number modification in crossing the boundary between different band structures: Three-band models with cubic symmetry

2017 ◽  
Vol 29 (02) ◽  
pp. 1750004
Author(s):  
T Iwai ◽  
B. Zhilinskii
Keyword(s):  
Symmetry Group ◽  
Present Article ◽  
Hermitian Matrix ◽  
Finite Subgroup ◽  
Chern Number ◽  
Line Bundles ◽  
Control Parameters ◽  
Cubic Symmetry ◽  
Chern Numbers ◽  
Two Parameter

A family of [Formula: see text] Hermitian matrix Hamiltonians defined on the sphere [Formula: see text] and depending on extra control parameters in the presence of a finite subgroup of [Formula: see text] as a symmetry group are studied with eigen-line bundles which are constructed by piecing together locally-defined eigenvectors. The condition for degeneracy in eigenvalues splits in general the space of control parameters into distinct iso-Chern domains on each of which the Chern numbers of the associated eigen-line bundles are constant. A Chern number modification or a delta-Chern occurs when crossing the boundary from one iso-Chern domain to another. The present article provides a formula for the delta-Chern on the model of a two-parameter family of [Formula: see text] Hermitian matrix Hamiltonians with cubic symmetry together with the whole sets of Chern numbers on respective iso-Chern domains.

scholarly journals A note on a Brill-Noether locus over a non-hyperelliptic curve of genus 4

2009 ◽  
Vol 7 (4) ◽  
Author(s):  
Sukmoon Huh
Keyword(s):  
Hyperelliptic Curve ◽  
Line Bundles ◽  
Cubic Threefold

AbstractWe prove that a certain Brill-Noether locus over a non-hyperelliptic curve C of genus 4, is isomorphic to the Donagi-Izadi cubic threefold in the case when the pencils of the two trigonal line bundles of C coincide.

scholarly journals The small quantum cohomology of the Cayley Grassmannian

2020 ◽  
Vol 31 (03) ◽  
pp. 2050019
Author(s):  
Vladimiro Benedetti ◽  
Laurent Manivel
Keyword(s):  
Cohomology Ring ◽  
Quantum Cohomology ◽  
Derived Category ◽  
Multiplication Table ◽  
Maximal Length ◽  
Small Quantum ◽  
Exceptional Collection ◽  
Schubert Classes

We compute the small cohomology ring of the Cayley Grassmannian, that parametrizes four-dimensional subalgebras of the complexified octonions. We show that all the Gromov–Witten invariants in the multiplication table of the Schubert classes are nonnegative and deduce Golyshev’s conjecture [Formula: see text] holds true for this variety. We also check that the quantum cohomology is semisimple and that there exists, as predicted by Dubrovin’s conjecture, an exceptional collection of maximal length in the derived category.

scholarly journals Fourier–Mukai and autoduality for compactified Jacobians. I

2019 ◽  
Vol 2019 (755) ◽  
pp. 1-65 ◽  
Author(s):  
Margarida Melo ◽  
Antonio Rapagnetta ◽  
Filippo Viviani
Keyword(s):  
Derived Category ◽  
Line Bundles ◽  
Higgs Bundles ◽  
Connected Component ◽  
Generalized Jacobian ◽  
Langlands Duality ◽  
Integral Curves ◽  
Numerical Equivalence ◽  
Algebraic Equivalence ◽  
Compactified Jacobian

AbstractTo every singular reduced projective curve X one can associate, following Esteves, many fine compactified Jacobians, depending on the choice of a polarization on X, each of which yields a modular compactification of a disjoint union of the generalized Jacobian of X. We prove that, for a reduced curve with locally planar singularities, the integral (or Fourier–Mukai) transform with kernel the Poincaré sheaf from the derived category of the generalized Jacobian of X to the derived category of any fine compactified Jacobian of X is fully faithful, generalizing a previous result of Arinkin in the case of integral curves. As a consequence, we prove that there is a canonical isomorphism (called autoduality) between the generalized Jacobian of X and the connected component of the identity of the Picard scheme of any fine compactified Jacobian of X and that algebraic equivalence and numerical equivalence of line bundles coincide on any fine compactified Jacobian, generalizing previous results of Arinkin, Esteves, Gagné, Kleiman, Rocha, and Sawon.The paper contains an Appendix in which we explain how our work can be interpreted in view of the Langlands duality for the Higgs bundles as proposed by Donagi–Pantev.

scholarly journals A note on thick subcategories of stable derived categories

2013 ◽  
Vol 212 ◽  
pp. 87-96 ◽  
Author(s):  
Henning Krause ◽  
Greg Stevenson
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Complete Intersections ◽  
Line Bundles ◽  
Finitely Generated ◽  
Exact Category ◽  
Coherent Sheaves ◽  
Noetherian Scheme ◽  
Finitely Generated Modules ◽  
Local Complete Intersections

AbstractFor an exact category having enough projective objects, we establish a bijection between thick subcategories containing the projective objects and thick subcategories of the stable derived category. Using this bijection, we classify thick subcategories of finitely generated modules over strict local complete intersections and produce generators for the category of coherent sheaves on a separated Noetherian scheme with an ample family of line bundles.

scholarly journals Deformation classes in generalized Kähler geometry

2020 ◽  
Vol 7 (1) ◽  
pp. 241-256
Author(s):  
Matthew Gibson ◽  
Jeffrey Streets
Keyword(s):  
Symmetry Group ◽  
Ricci Flow ◽  
Kahler Geometry ◽  
Poisson Tensor ◽  
Natural Extensions ◽  
Generalized Kähler Geometry ◽  
Kähler Structures

AbstractWe describe natural deformation classes of generalized Kähler structures using the Courant symmetry group, which determine natural extensions of the notions of Kähler class and Kähler cone to generalized Kähler geometry. We show that the generalized Kähler-Ricci flow preserves this generalized Kähler cone, and the underlying real Poisson tensor.

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