Pub Date : 2025-03-01DOI: 10.1016/j.jpaa.2025.107917
Scott Balchin , Florian Tecklenburg
Building on results of Bazzoni–Št'ovíček, we give a complete classification of the frame of smashing ideals for the derived category of a finite dimensional valuation domain. In particular, we give an explicit construction of an infinite family of commutative rings such that the telescope conjecture fails and which generalise an example of Keller. As a consequence, we deduce that the Krull dimension of the Balmer spectrum and the Krull dimension of the smashing spectrum can differ arbitrarily for rigidly-compactly generated tensor-triangulated categories.
{"title":"Classifying smashing ideals in derived categories of valuation domains","authors":"Scott Balchin , Florian Tecklenburg","doi":"10.1016/j.jpaa.2025.107917","DOIUrl":"10.1016/j.jpaa.2025.107917","url":null,"abstract":"<div><div>Building on results of Bazzoni–Št'ovíček, we give a complete classification of the frame of smashing ideals for the derived category of a finite dimensional valuation domain. In particular, we give an explicit construction of an infinite family of commutative rings such that the telescope conjecture fails and which generalise an example of Keller. As a consequence, we deduce that the Krull dimension of the Balmer spectrum and the Krull dimension of the smashing spectrum can differ arbitrarily for rigidly-compactly generated tensor-triangulated categories.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 3","pages":"Article 107917"},"PeriodicalIF":0.7,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143509541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-28DOI: 10.1016/j.jpaa.2025.107921
Ankita Parashar , Shiv Prakash Patel
Let be a finite principal ideal local ring of length 2. For a representation π of , the degenerate Whittaker space is a representation of . We describe explicitly for an irreducible strongly cuspidal representation π of . This description verifies a special case of a conjecture of Prasad. We also prove that is a multiplicity free representation.
{"title":"On the degenerate Whittaker space for GL4(o2)","authors":"Ankita Parashar , Shiv Prakash Patel","doi":"10.1016/j.jpaa.2025.107921","DOIUrl":"10.1016/j.jpaa.2025.107921","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>o</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> be a finite principal ideal local ring of length 2. For a representation <em>π</em> of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><msub><mrow><mi>o</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>, the degenerate Whittaker space <span><math><msub><mrow><mi>π</mi></mrow><mrow><mi>N</mi><mo>,</mo><mi>ψ</mi></mrow></msub></math></span> is a representation of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>o</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. We describe <span><math><msub><mrow><mi>π</mi></mrow><mrow><mi>N</mi><mo>,</mo><mi>ψ</mi></mrow></msub></math></span> explicitly for an irreducible strongly cuspidal representation <em>π</em> of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><msub><mrow><mi>o</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. This description verifies a special case of a conjecture of Prasad. We also prove that <span><math><msub><mrow><mi>π</mi></mrow><mrow><mi>N</mi><mo>,</mo><mi>ψ</mi></mrow></msub></math></span> is a multiplicity free representation.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 5","pages":"Article 107921"},"PeriodicalIF":0.7,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-21DOI: 10.1016/j.jpaa.2025.107918
Cesar Hilario , Karl-Otto Stöhr
We classify fibrations by integral plane projective rational quartic curves whose generic fibre is regular but admits a non-smooth point that is a canonical divisor. These fibrations can only exist in characteristic two. The geometric generic fibre, which determines the generic behavior of the special fibres, is an integral plane projective rational quartic curve over the algebraic closure of the function field of the base. It has the remarkable property that the tangent lines at the non-singular points are either all bitangents or all non-ordinary inflection tangents; moreover it is strange, that is, all the tangent lines meet in a common point. We construct two fibrations that are universal in the sense that any other fibration with the aforementioned properties can be obtained from one of them by a base extension. Furthermore, among these fibrations we choose a pencil of plane quartic curves and study in detail its geometry. We determine the corresponding minimal regular model and we describe it as a purely inseparable double covering of a quasi-elliptic fibration.
{"title":"Fibrations by plane quartic curves with a canonical moving singularity","authors":"Cesar Hilario , Karl-Otto Stöhr","doi":"10.1016/j.jpaa.2025.107918","DOIUrl":"10.1016/j.jpaa.2025.107918","url":null,"abstract":"<div><div>We classify fibrations by integral plane projective rational quartic curves whose generic fibre is regular but admits a non-smooth point that is a canonical divisor. These fibrations can only exist in characteristic two. The geometric generic fibre, which determines the generic behavior of the special fibres, is an integral plane projective rational quartic curve over the algebraic closure of the function field of the base. It has the remarkable property that the tangent lines at the non-singular points are either all bitangents or all non-ordinary inflection tangents; moreover it is strange, that is, all the tangent lines meet in a common point. We construct two fibrations that are universal in the sense that any other fibration with the aforementioned properties can be obtained from one of them by a base extension. Furthermore, among these fibrations we choose a pencil of plane quartic curves and study in detail its geometry. We determine the corresponding minimal regular model and we describe it as a purely inseparable double covering of a quasi-elliptic fibration.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 4","pages":"Article 107918"},"PeriodicalIF":0.7,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143487223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-21DOI: 10.1016/j.jpaa.2025.107915
Vasiliki Petrotou
Unprojection theory is a philosophy due to Miles Reid, which becomes a useful tool in algebraic geometry for the construction and the study of new interesting geometric objects such as algebraic surfaces and 3-folds. In this present work we introduce a new format of unprojection, which we call the 4-intersection format. It is specified by a codimension 2 complete intersection ideal I which is contained in four codimension 3 complete intersection ideals and leads to the construction of codimension 6 Gorenstein rings. As an application, we construct three families of codimension 6 Fano 3-folds embedded in weighted projective space which correspond to the entries with identifier numbers 29376, 9176 and 24198 respectively in the Graded Ring Database.
{"title":"The 4-intersection unprojection format","authors":"Vasiliki Petrotou","doi":"10.1016/j.jpaa.2025.107915","DOIUrl":"10.1016/j.jpaa.2025.107915","url":null,"abstract":"<div><div>Unprojection theory is a philosophy due to Miles Reid, which becomes a useful tool in algebraic geometry for the construction and the study of new interesting geometric objects such as algebraic surfaces and 3-folds. In this present work we introduce a new format of unprojection, which we call the 4-intersection format. It is specified by a codimension 2 complete intersection ideal <em>I</em> which is contained in four codimension 3 complete intersection ideals <span><math><msub><mrow><mi>J</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>J</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>J</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>J</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> and leads to the construction of codimension 6 Gorenstein rings. As an application, we construct three families of codimension 6 Fano 3-folds embedded in weighted projective space which correspond to the entries with identifier numbers 29376, 9176 and 24198 respectively in the Graded Ring Database.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 4","pages":"Article 107915"},"PeriodicalIF":0.7,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143478534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-21DOI: 10.1016/j.jpaa.2025.107919
Omer Cantor, Uriya A. First
<div><div>Let <em>A</em> be a finite dimensional algebra (possibly with some extra structure) over an infinite field <em>K</em> and let <span><math><mi>r</mi><mo>∈</mo><mi>N</mi></math></span>. The <em>r</em>-tuples <span><math><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> which fail to generate <em>A</em> are the <em>K</em>-points of a closed subvariety <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> of the affine space underlying <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span>, the codimension of which may be thought of as quantifying how well a generic <em>r</em>-tuple in <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> generates <em>A</em>. Taking this intuition one step further, the second author, Reichstein and Williams showed that lower bounds on the codimension of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> in <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> (for every <em>r</em>) imply upper bounds on the number of generators of <em>forms</em> of the <em>K</em>-algebra <em>A</em> over finitely generated <em>K</em>-rings. That work also demonstrates how finer information on <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> may be used to construct forms of <em>A</em> which require many elements to generate.</div><div>The dimension and irreducible components of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> are known in a few cases, which in particular lead to upper bounds on the number of generators of Azumaya algebras and Azumaya algebras with involution of the first kind (orthogonal or symplectic). This paper treats the case of Azumaya algebras with a unitary involution by finding the dimension and irreducible components of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> when <em>A</em> is the <em>K</em>-algebra with involution <span><math><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>)</mo><mo>×</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>)</mo><mo>,</mo><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo><mo>↦</mo><mo>(</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>,</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>)</mo><mo>)</mo></math></span>. Our analysis implies that every degree-<em>n</em> Azumaya algebra with a unitary involution over a finitely generated <em>K</em>-ring of Krull dimension <em>d</em> can be generated by <span><math><mo>⌊</mo><mfrac><mrow><mi>d</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>−
{"title":"Spaces of generators for Azumaya algebras with unitary involution","authors":"Omer Cantor, Uriya A. First","doi":"10.1016/j.jpaa.2025.107919","DOIUrl":"10.1016/j.jpaa.2025.107919","url":null,"abstract":"<div><div>Let <em>A</em> be a finite dimensional algebra (possibly with some extra structure) over an infinite field <em>K</em> and let <span><math><mi>r</mi><mo>∈</mo><mi>N</mi></math></span>. The <em>r</em>-tuples <span><math><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>A</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> which fail to generate <em>A</em> are the <em>K</em>-points of a closed subvariety <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> of the affine space underlying <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span>, the codimension of which may be thought of as quantifying how well a generic <em>r</em>-tuple in <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> generates <em>A</em>. Taking this intuition one step further, the second author, Reichstein and Williams showed that lower bounds on the codimension of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> in <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> (for every <em>r</em>) imply upper bounds on the number of generators of <em>forms</em> of the <em>K</em>-algebra <em>A</em> over finitely generated <em>K</em>-rings. That work also demonstrates how finer information on <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> may be used to construct forms of <em>A</em> which require many elements to generate.</div><div>The dimension and irreducible components of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> are known in a few cases, which in particular lead to upper bounds on the number of generators of Azumaya algebras and Azumaya algebras with involution of the first kind (orthogonal or symplectic). This paper treats the case of Azumaya algebras with a unitary involution by finding the dimension and irreducible components of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> when <em>A</em> is the <em>K</em>-algebra with involution <span><math><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>)</mo><mo>×</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>)</mo><mo>,</mo><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo><mo>↦</mo><mo>(</mo><msup><mrow><mi>b</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>,</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>)</mo><mo>)</mo></math></span>. Our analysis implies that every degree-<em>n</em> Azumaya algebra with a unitary involution over a finitely generated <em>K</em>-ring of Krull dimension <em>d</em> can be generated by <span><math><mo>⌊</mo><mfrac><mrow><mi>d</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>−","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 4","pages":"Article 107919"},"PeriodicalIF":0.7,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143479280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-19DOI: 10.1016/j.jpaa.2025.107920
Ahmet I. Seven, İbrahim Ünal
Motivated by the recent work of R. Casals on binary invariants for matrix mutation, we study the matrix congruence relation on quasi-Cartan matrices. In particular, we obtain a classification and determine normal forms modulo 4. As an application, we obtain new mutation invariants, which include the one obtained by R. Casals.
{"title":"Congruence invariants of matrix mutation","authors":"Ahmet I. Seven, İbrahim Ünal","doi":"10.1016/j.jpaa.2025.107920","DOIUrl":"10.1016/j.jpaa.2025.107920","url":null,"abstract":"<div><div>Motivated by the recent work of R. Casals on binary invariants for matrix mutation, we study the matrix congruence relation on quasi-Cartan matrices. In particular, we obtain a classification and determine normal forms modulo 4. As an application, we obtain new mutation invariants, which include the one obtained by R. Casals.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 3","pages":"Article 107920"},"PeriodicalIF":0.7,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-14DOI: 10.1016/j.jpaa.2025.107910
Divya Ahuja, Surjeet Kour
Let be a Grothendieck category and U be a monad on that is exact and preserves colimits. In this article, we prove that every hereditary torsion theory on the Eilenberg-Moore category of modules over a monad U is differential. Furthermore, if denotes a derivation on a monad U, then we show that every δ-derivation on a U-module M extends uniquely to a δ-derivation on the module of quotients of M.
{"title":"Differential torsion theories on Eilenberg-Moore categories of monads","authors":"Divya Ahuja, Surjeet Kour","doi":"10.1016/j.jpaa.2025.107910","DOIUrl":"10.1016/j.jpaa.2025.107910","url":null,"abstract":"<div><div>Let <span><math><mi>C</mi></math></span> be a Grothendieck category and <em>U</em> be a monad on <span><math><mi>C</mi></math></span> that is exact and preserves colimits. In this article, we prove that every hereditary torsion theory on the Eilenberg-Moore category of modules over a monad <em>U</em> is differential. Furthermore, if <span><math><mi>δ</mi><mo>:</mo><mi>U</mi><mo>⟶</mo><mi>U</mi></math></span> denotes a derivation on a monad <em>U</em>, then we show that every <em>δ</em>-derivation on a <em>U</em>-module <em>M</em> extends uniquely to a <em>δ</em>-derivation on the module of quotients of <em>M</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 3","pages":"Article 107910"},"PeriodicalIF":0.7,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143446072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-14DOI: 10.1016/j.jpaa.2025.107909
Amnon Yekutieli
Weak proregularity of an ideal in a commutative ring is a subtle generalization of the noetherian property of the ring. Weak proregularity is of special importance for the study of derived completion, and it occurs quite often in non-noetherian rings arising in Hochschild and prismatic cohomologies.
This paper is about several related topics: adically flat modules, recognizing derived complete complexes, the structure of the category of derived complete complexes, and a derived complete Nakayama theorem – all with respect to a weakly proregular ideal; and the preservation of weak proregularity under completion of the ring.
{"title":"Derived complete complexes at weakly proregular ideals","authors":"Amnon Yekutieli","doi":"10.1016/j.jpaa.2025.107909","DOIUrl":"10.1016/j.jpaa.2025.107909","url":null,"abstract":"<div><div>Weak proregularity of an ideal in a commutative ring is a subtle generalization of the noetherian property of the ring. Weak proregularity is of special importance for the study of derived completion, and it occurs quite often in non-noetherian rings arising in Hochschild and prismatic cohomologies.</div><div>This paper is about several related topics: adically flat modules, recognizing derived complete complexes, the structure of the category of derived complete complexes, and a derived complete Nakayama theorem – all with respect to a weakly proregular ideal; and the preservation of weak proregularity under completion of the ring.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 3","pages":"Article 107909"},"PeriodicalIF":0.7,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143474589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-14DOI: 10.1016/j.jpaa.2025.107911
Alvaro Liendo , Charlie Petitjean
In this paper, we classify smooth, contractible affine varieties equipped with faithful torus actions of complexity two, having a unique fixed point and a two-dimensional algebraic quotient isomorphic to a toric blow-up of a toric surface. These varieties are of particular interest as they represent the simplest candidates for potential counterexamples to the linearization conjecture in affine geometry.
{"title":"Codimension two torus actions on the affine space","authors":"Alvaro Liendo , Charlie Petitjean","doi":"10.1016/j.jpaa.2025.107911","DOIUrl":"10.1016/j.jpaa.2025.107911","url":null,"abstract":"<div><div>In this paper, we classify smooth, contractible affine varieties equipped with faithful torus actions of complexity two, having a unique fixed point and a two-dimensional algebraic quotient isomorphic to a toric blow-up of a toric surface. These varieties are of particular interest as they represent the simplest candidates for potential counterexamples to the linearization conjecture in affine geometry.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 5","pages":"Article 107911"},"PeriodicalIF":0.7,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-12DOI: 10.1016/j.jpaa.2025.107907
Riccardo Camerlo , Carla Massaza
We study the preorder (and the associated equivalence relation ) on the family of subsets of an algebraically closed field k of characteristic 0, defined by letting iff there exists a polynomial P such that . We concentrate mainly on the finite subsets of k and prove that the -equivalence classes of sets of a given finite cardinality form an affine algebraic variety; inside these varieties, we compute in particular the dimension of the set of -classes that have less representatives than the generic ones and the dimension of the set of -classes that are comparable with a given -class.
We also show that, in a specified sense, very many -classes are -maximal (or -maximal under the class of singletons, for -classes of finite sets).
{"title":"Reducibility by polynomial functions","authors":"Riccardo Camerlo , Carla Massaza","doi":"10.1016/j.jpaa.2025.107907","DOIUrl":"10.1016/j.jpaa.2025.107907","url":null,"abstract":"<div><div>We study the preorder <span><math><msub><mrow><mo>≤</mo></mrow><mrow><mi>Pol</mi></mrow></msub></math></span> (and the associated equivalence relation <span><math><msub><mrow><mo>≡</mo></mrow><mrow><mi>Pol</mi></mrow></msub></math></span>) on the family of subsets of an algebraically closed field <em>k</em> of characteristic 0, defined by letting <span><math><mi>A</mi><msub><mrow><mo>≤</mo></mrow><mrow><mi>Pol</mi></mrow></msub><mi>B</mi></math></span> iff there exists a polynomial <em>P</em> such that <span><math><mi>A</mi><mo>=</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>B</mi><mo>)</mo></math></span>. We concentrate mainly on the finite subsets of <em>k</em> and prove that the <span><math><msub><mrow><mo>≡</mo></mrow><mrow><mi>Pol</mi></mrow></msub></math></span>-equivalence classes of sets of a given finite cardinality form an affine algebraic variety; inside these varieties, we compute in particular the dimension of the set of <span><math><msub><mrow><mo>≡</mo></mrow><mrow><mi>Pol</mi></mrow></msub></math></span>-classes that have less representatives than the generic ones and the dimension of the set of <span><math><msub><mrow><mo>≡</mo></mrow><mrow><mi>Pol</mi></mrow></msub></math></span>-classes that are comparable with a given <span><math><msub><mrow><mo>≡</mo></mrow><mrow><mi>Pol</mi></mrow></msub></math></span>-class.</div><div>We also show that, in a specified sense, very many <span><math><msub><mrow><mo>≡</mo></mrow><mrow><mi>Pol</mi></mrow></msub></math></span>-classes are <span><math><msub><mrow><mo>≤</mo></mrow><mrow><mi>Pol</mi></mrow></msub></math></span>-maximal (or <span><math><msub><mrow><mo>≤</mo></mrow><mrow><mi>Pol</mi></mrow></msub></math></span>-maximal under the class of singletons, for <span><math><msub><mrow><mo>≡</mo></mrow><mrow><mi>Pol</mi></mrow></msub></math></span>-classes of finite sets).</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 3","pages":"Article 107907"},"PeriodicalIF":0.7,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143446073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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