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 Some non-finitely generated Cox rings

2015 ◽  
Vol 152 (5) ◽  
pp. 984-996 ◽  
Author(s):  
José Luis González ◽  
Kalle Karu
Keyword(s):  
Moduli Space ◽  
Smooth Point ◽  
Large Family ◽  
Projective Planes ◽  
Finitely Generated ◽  
Genus Zero ◽  
Cox Rings ◽  
Cox Ring

We give a large family of weighted projective planes, blown up at a smooth point, that do not have finitely generated Cox rings. We then use the method of Castravet and Tevelev to prove that the moduli space$\overline{M}_{0,n}$of stable$n$-pointed genus-zero curves does not have a finitely generated Cox ring if$n$is at least$13$.

Positive solutions for a quasilinear system Via blow up

1993 ◽  
Vol 18 (12) ◽  
pp. 2071-2106
Author(s):  
Philippe Clément ◽  
Raúl Manásevich ◽  
Enzo Mitidieri
Keyword(s):  
Positive Solutions ◽  
Blow Up ◽  
Quasilinear System

: The Blow-Up . Michelangelo Antonioni .

1967 ◽  
Vol 20 (3) ◽  
pp. 28-31
Author(s):  
Max Kozloff
Keyword(s):  
Blow Up ◽  
Michelangelo Antonioni

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

Instability and Collapse in Discrete Wave Equations

2005 ◽  
Vol 5 (3) ◽  
pp. 223-241
Author(s):  
A. Carpio ◽  
G. Duro
Keyword(s):  
Continuum Limit ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Blow Up ◽  
Theoretical Predictions ◽  
Initial States ◽  
The Impact ◽  
The Continuum

AbstractUnstable growth phenomena in spatially discrete wave equations are studied. We characterize sets of initial states leading to instability and collapse and obtain analytical predictions for the blow-up time. The theoretical predictions are con- trasted with the numerical solutions computed by a variety of schemes. The behavior of the systems in the continuum limit and the impact of discreteness and friction are discussed.

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