Gorensteinness and iteration of Cox rings for Fano type varieties

Mathematische Zeitschrift ◽

10.1007/s00209-021-02946-w ◽

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.

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Some non-finitely generated Cox rings

Compositio Mathematica ◽

10.1112/s0010437x15007745 ◽

2015 ◽

Vol 152 (5) ◽

pp. 984-996 ◽

Cited By ~ 17

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$13$.

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Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

Journal of Algebraic Combinatorics ◽

10.1007/s10801-021-01025-x ◽

2021 ◽

Author(s):

Michele Rossi ◽

Lea Terracini

Keyword(s):

Free Group ◽

Toric Variety ◽

Class Group ◽

Abelian Groups ◽

Toric Varieties ◽

Picard Group ◽

Closed Field ◽

Finitely Generated ◽

Free Part ◽

Canonical Injection

AbstractLet X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

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Residual dimension of nilpotent groups

Journal of Group Theory ◽

10.1515/jgth-2019-0117 ◽

2020 ◽

Vol 23 (5) ◽

pp. 801-829

Author(s):

Mark Pengitore

Keyword(s):

New York ◽

Nilpotent Group ◽

Nilpotent Groups ◽

Finitely Generated ◽

Nontrivial Element ◽

Maximum Order ◽

Asymptotic Bounds ◽

Group Morphism ◽

A Finite Group

AbstractThe function {\mathrm{F}_{G}(n)} gives the maximum order of a finite group needed to distinguish a nontrivial element of G from the identity with a surjective group morphism as one varies over nontrivial elements of word length at most n. In previous work [M. Pengitore, Effective separability of finitely generated nilpotent groups, New York J. Math. 24 2018, 83–145], the author claimed a characterization for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and consequently, the statement of the asymptotic characterization of {\mathrm{F}_{N}(n)} is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for {\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of the above article can be improved. Finally, we construct a class of finitely generated nilpotent groups N for which the asymptotic behavior of {\mathrm{F}_{N}(n)} can be fully characterized.

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NORMAL DOMAINS WITH MONOMIAL PRESENTATIONS

International Journal of Algebra and Computation ◽

10.1142/s0218196709005184 ◽

2009 ◽

Vol 19 (03) ◽

pp. 287-303 ◽

Cited By ~ 1

Author(s):

ISABEL GOFFA ◽

ERIC JESPERS ◽

JAN OKNIŃSKI

Keyword(s):

Commutative Algebra ◽

Class Group ◽

Semigroup Algebra ◽

Closed Domain ◽

Finitely Generated ◽

Defining Relations ◽

Integrally Closed ◽

Algebra Over A Field

Let A be a finitely generated commutative algebra over a field K with a presentation A = K 〈X1,…, Xn | R〉, where R is a set of monomial relations in the generators X1,…, Xn. So A = K[S], the semigroup algebra of the monoid S = 〈X1,…, Xn | R〉. We characterize, purely in terms of the defining relations, when A is an integrally closed domain, provided R contains at most two relations. Also the class group of such algebras A is calculated.

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A software package for Mori dream spaces

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157015000212 ◽

2015 ◽

Vol 18 (1) ◽

pp. 647-659 ◽

Cited By ~ 8

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.

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Orbit counting in polarized dynamical systems

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2021112 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Wade Hindes

Keyword(s):

Dynamical Systems ◽

Finitely Generated ◽

Projective Varieties ◽

Infinite Sets ◽

Unicritical Polynomials

<p style='text-indent:20px;'>We extend recent orbit counts for finitely generated semigroups acting on <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{P}^N $\end{document}</tex-math></inline-formula> to certain infinitely generated, polarized semigroups acting on projective varieties. We then apply these results to semigroup orbits generated by some infinite sets of unicritical polynomials.</p>

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Generating infinite monoids of cellular automata

Journal of Algebra and Its Applications ◽

10.1142/s0219498822502152 ◽

2021 ◽

pp. 2250215

Author(s):

Alonso Castillo-Ramirez

Keyword(s):

Cellular Automata ◽

Normal Subgroups ◽

Finitely Generated ◽

Residually Finite ◽

Residually Finite Group ◽

Group Of Units ◽

Finite Dimensional ◽

Index Condition ◽

Indicable Group

For a group [Formula: see text] and a set [Formula: see text], let [Formula: see text] be the monoid of all cellular automata over [Formula: see text], and let [Formula: see text] be its group of units. By establishing a characterization of surjunctive groups in terms of the monoid [Formula: see text], we prove that the rank of [Formula: see text] (i.e. the smallest cardinality of a generating set) is equal to the rank of [Formula: see text] plus the relative rank of [Formula: see text] in [Formula: see text], and that the latter is infinite when [Formula: see text] has an infinite decreasing chain of normal subgroups of finite index, condition which is satisfied, for example, for any infinite residually finite group. Moreover, when [Formula: see text] is a vector space over a field [Formula: see text], we study the monoid [Formula: see text] of all linear cellular automata over [Formula: see text] and its group of units [Formula: see text]. We show that if [Formula: see text] is an indicable group and [Formula: see text] is finite-dimensional, then [Formula: see text] is not finitely generated; however, for any finitely generated indicable group [Formula: see text], the group [Formula: see text] is finitely generated if and only if [Formula: see text] is finite.

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ELEMENTS OF ORDER FOUR IN THE NARROW CLASS GROUP OF REAL QUADRATIC FIELDS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788715000270 ◽

2015 ◽

Vol 100 (1) ◽

pp. 21-32

Author(s):

ELLIOT BENJAMIN ◽

C. SNYDER

Keyword(s):

Class Group ◽

Number Fields ◽

Ideal Class Group ◽

Quadratic Number ◽

Gaussian Integers ◽

Real Quadratic Fields ◽

Order Four ◽

Real Quadratic ◽

Narrow Class

Using the elements of order four in the narrow ideal class group, we construct generators of the maximal elementary $2$-class group of real quadratic number fields with even discriminant which is a sum of two squares and with fundamental unit of positive norm. We then give a characterization of when two of these generators are equal in the narrow sense in terms of norms of Gaussian integers.

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