scholarly journals The Cox ring of a spherical embedding

2014 ◽  
Vol 397 ◽  
pp. 548-569 ◽  
Author(s):  
Giuliano Gagliardi
Keyword(s):  
Cox Ring ◽  
Spherical Embedding

scholarly journals A software package for Mori dream spaces

2015 ◽  
Vol 18 (1) ◽  
pp. 647-659 ◽  
Author(s):  
Jürgen Hausen ◽  
Simon Keicher
Keyword(s):  
Software Package ◽  
Toric Varieties ◽  
Quotient Singularity ◽  
Algebraic Varieties ◽  
Cox Rings ◽  
Cox Ring ◽  
Mori Dream Spaces ◽  
The Common ◽  
Symplectic Resolutions

Mori dream spaces form a large example class of algebraic varieties, comprising the well-known toric varieties. We provide a first software package for the explicit treatment of Mori dream spaces and demonstrate its use by presenting basic sample computations. The software package is accompanied by a Cox ring database which delivers defining data for Cox rings and Mori dream spaces in a suitable format. As an application of the package, we determine the common Cox ring for the symplectic resolutions of a certain quotient singularity investigated by Bellamy–Schedler and Donten-Bury–Wiśniewski.

scholarly journals The Cox ring of the space of complete rank two collineations

2012 ◽  
Vol 285 (8-9) ◽  
pp. 974-980
Author(s):  
Jürgen Hausen ◽  
Michael Liebendörfer
Keyword(s):  
Cox Ring

Deep Supervised Hashing with Spherical Embedding

2019 ◽  
pp. 417-434 ◽  
Author(s):  
Stanislav Pidhorskyi ◽  
Quinn Jones ◽  
Saeid Motiian ◽  
Donald Adjeroh ◽  
Gianfranco Doretto
Keyword(s):  
Spherical Embedding

scholarly journals The Cox ring of an algebraic variety with torus action

2010 ◽  
Vol 225 (2) ◽  
pp. 977-1012 ◽  
Author(s):  
Jürgen Hausen ◽  
Hendrik Süß
Keyword(s):  
Algebraic Variety ◽  
Torus Action ◽  
Cox Ring

scholarly journals On the Cox ring of $\mathbf{P}^2$ blown up in points on a line

2011 ◽  
Vol 109 (1) ◽  
pp. 22 ◽  
Author(s):  
John Christian Ottem
Keyword(s):  
Blow Up ◽  
Finitely Generated ◽  
Cox Ring

We show that the blow-up of $\mathbf{P}^2$ in $n$ points on a line has finitely generated Cox ring. We give explicit generators for the ring and calculate its defining ideal of relations.

scholarly journals Gorensteinness and iteration of Cox rings for Fano type varieties

2022 ◽  
Author(s):  
Lukas Braun
Keyword(s):  
Class Group ◽  
Reductive Group ◽  
Finitely Generated ◽  
Projective Varieties ◽  
Cox Rings ◽  
Cox Ring

AbstractWe show that finitely generated Cox rings are Gorenstein. This leads to a refined characterization of varieties of Fano type: they are exactly those projective varieties with Gorenstein canonical quasicone Cox ring. We then show that for varieties of Fano type and Kawamata log terminal quasicones X, iteration of Cox rings is finite with factorial master Cox ring. In particular, even if the class group has torsion, we can express such X as quotients of a factorial canonical quasicone by a solvable reductive group.

scholarly journals What makes a complex a virtual resolution?

2021 ◽  
Vol 8 (28) ◽  
pp. 885-898
Author(s):  
Michael Loper
Keyword(s):  
Classical Theory ◽  
Toric Variety ◽  
Chain Complex ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Content Type ◽  
Graded Modules ◽  
Cox Ring ◽  
A Chain ◽  
Free Modules

Virtual resolutions are homological representations of finitely generated Pic ( X ) \text {Pic}(X) -graded modules over the Cox ring of a smooth projective toric variety. In this paper, we identify two algebraic conditions that characterize when a chain complex of graded free modules over the Cox ring is a virtual resolution. We then turn our attention to the saturation of Fitting ideals by the irrelevant ideal of the Cox ring and prove some results that mirror the classical theory of Fitting ideals for Noetherian rings.

scholarly journals COX RINGS OF MINIMAL RESOLUTIONS OF SURFACE QUOTIENT SINGULARITIES

2015 ◽  
Vol 58 (2) ◽  
pp. 325-355 ◽  
Author(s):  
MARIA DONTEN-BURY
Keyword(s):  
Toric Variety ◽  
Affine Space ◽  
Explicit Description ◽  
Quotient Singularity ◽  
Coordinate Ring ◽  
Minimal Resolution ◽  
Cox Rings ◽  
Cox Ring ◽  
Quotient Singularities ◽  
Minimal Resolutions

AbstractWe investigate Cox rings of minimal resolutions of surface quotient singularities and provide two descriptions of these rings. The first one is the equation for the spectrum of a Cox ring, which is a hypersurface in an affine space. The second is the set of generators of the Cox ring viewed as a subring of the coordinate ring of a product of a torus and another surface quotient singularity. In addition, we obtain an explicit description of the minimal resolution as a divisor in a toric variety.

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