Pub Date : 2025-10-20DOI: 10.1016/j.jpaa.2025.108109
Thomas Bouchet
Let K be a field of characteristic 0. We present an explicit algorithm that, given the invariants of a generic homogeneous polynomial f under the linear action of or , returns a polynomial differing from f only by a linear change of variables with coefficients in a finite extension of K. Our approach uses the theory of covariants and the Veronese embeddings to characterize the linear equivalence class of a homogeneous polynomial through equations whose coefficients are invariants. As applications, we derive explicit formulas for reconstructing of a generic non-hyperelliptic curve of genus 4 from its invariants, as well as reconstructing generic non-hyperelliptic curves of genus 3 from their Dixmier-Ohno invariants. Formulas for the reconstruction of cubic surfaces from their Salmon-Clebsch invariants. In all of these cases, the coefficients of the reconstructed object lie in its field of moduli.
{"title":"Reconstruction of hypersurfaces from their invariants","authors":"Thomas Bouchet","doi":"10.1016/j.jpaa.2025.108109","DOIUrl":"10.1016/j.jpaa.2025.108109","url":null,"abstract":"<div><div>Let <em>K</em> be a field of characteristic 0. We present an explicit algorithm that, given the invariants of a generic homogeneous polynomial <em>f</em> under the linear action of <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> or <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, returns a polynomial differing from <em>f</em> only by a linear change of variables with coefficients in a finite extension of <em>K</em>. Our approach uses the theory of covariants and the Veronese embeddings to characterize the linear equivalence class of a homogeneous polynomial through equations whose coefficients are invariants. As applications, we derive explicit formulas for reconstructing of a generic non-hyperelliptic curve of genus 4 from its invariants, as well as reconstructing generic non-hyperelliptic curves of genus 3 from their Dixmier-Ohno invariants. Formulas for the reconstruction of cubic surfaces from their Salmon-Clebsch invariants. In all of these cases, the coefficients of the reconstructed object lie in its field of moduli.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108109"},"PeriodicalIF":0.8,"publicationDate":"2025-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145363615","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 : 2025-10-20DOI: 10.1016/j.jpaa.2025.108114
Andrea Lucchini , Patricia Medina Capilla
Let be the smallest cardinality of a generating set of a finite group G. We give a complete classification of the finite groups with the property that, whenever , for any there exists such that . We also prove that for every finite group G and every maximal subgroup M of G, there exists a generating set for G of minimal size in which at least elements belong to M. We conjecture that the stronger statement holds, that there exists a generating set of size in which only one element does not belong to M, and we prove this conjecture for some suitable choices of M.
{"title":"Finite groups with the minimal generating set exchange property","authors":"Andrea Lucchini , Patricia Medina Capilla","doi":"10.1016/j.jpaa.2025.108114","DOIUrl":"10.1016/j.jpaa.2025.108114","url":null,"abstract":"<div><div>Let <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the smallest cardinality of a generating set of a finite group <em>G</em>. We give a complete classification of the finite groups with the property that, whenever <span><math><mo>〈</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mo>〉</mo><mo>=</mo><mo>〈</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mo>〉</mo><mo>=</mo><mi>G</mi></math></span>, for any <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> there exists <span><math><mn>1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> such that <span><math><mo>〈</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mo>〉</mo><mo>=</mo><mi>G</mi></math></span>. We also prove that for every finite group <em>G</em> and every maximal subgroup <em>M</em> of <em>G</em>, there exists a generating set for <em>G</em> of minimal size in which at least <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>2</mn></math></span> elements belong to <em>M</em>. We conjecture that the stronger statement holds, that there exists a generating set of size <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> in which only one element does not belong to <em>M</em>, and we prove this conjecture for some suitable choices of <em>M</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108114"},"PeriodicalIF":0.8,"publicationDate":"2025-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145363619","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 : 2025-10-10DOI: 10.1016/j.jpaa.2025.108108
Christopher A. Schroeder, Hung P. Tong-Viet
A recurring theme in finite group theory is understanding how the structure of a finite group is determined by the arithmetic properties of group invariants. There are results in the literature determining the structure of finite groups whose irreducible character degrees, conjugacy class sizes or indices of maximal subgroups are odd. These results have been extended to include those finite groups whose character degrees or conjugacy class sizes are not divisible by 4. In this paper, we determine the structure of finite groups whose maximal subgroups have index not divisible by 4. As an application, we obtain some new 2-nilpotency criteria.
{"title":"Finite groups all of whose maximal subgroups have almost odd index","authors":"Christopher A. Schroeder, Hung P. Tong-Viet","doi":"10.1016/j.jpaa.2025.108108","DOIUrl":"10.1016/j.jpaa.2025.108108","url":null,"abstract":"<div><div>A recurring theme in finite group theory is understanding how the structure of a finite group is determined by the arithmetic properties of group invariants. There are results in the literature determining the structure of finite groups whose irreducible character degrees, conjugacy class sizes or indices of maximal subgroups are odd. These results have been extended to include those finite groups whose character degrees or conjugacy class sizes are not divisible by 4. In this paper, we determine the structure of finite groups whose maximal subgroups have index not divisible by 4. As an application, we obtain some new 2-nilpotency criteria.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108108"},"PeriodicalIF":0.8,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145333348","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 : 2025-10-06DOI: 10.1016/j.jpaa.2025.108107
Suprajo Das , Saipriya Dubey , Sudeshna Roy , Jugal K. Verma
This article investigates the computational aspects of the ε-multiplicity. Primarily, we show that the ε-multiplicity of a homogeneous ideal I in a two-dimensional standard graded domain of finite type over an algebraically closed field of arbitrary characteristic, is always a rational number. In this situation, we produce a formula for the ε-multiplicity of I in terms of certain mixed multiplicities associated to I. In any dimension, under the assumptions that the saturated Rees algebra of I is finitely generated, we give a different expression of the ε-multiplicity in terms of mixed multiplicities by using the Veronese degree. This enabled us to make various explicit computations of ε-multiplicities. We further write a Macaulay2 algorithm to compute ε-multiplicity (under the Noetherian hypotheses) even when the base ring is not necessarily standard graded.
{"title":"Computing epsilon multiplicities in graded algebras","authors":"Suprajo Das , Saipriya Dubey , Sudeshna Roy , Jugal K. Verma","doi":"10.1016/j.jpaa.2025.108107","DOIUrl":"10.1016/j.jpaa.2025.108107","url":null,"abstract":"<div><div>This article investigates the computational aspects of the <em>ε</em>-multiplicity. Primarily, we show that the <em>ε</em>-multiplicity of a homogeneous ideal <em>I</em> in a two-dimensional standard graded domain of finite type over an algebraically closed field of arbitrary characteristic, is always a rational number. In this situation, we produce a formula for the <em>ε</em>-multiplicity of <em>I</em> in terms of certain mixed multiplicities associated to <em>I</em>. In any dimension, under the assumptions that the saturated Rees algebra of <em>I</em> is finitely generated, we give a different expression of the <em>ε</em>-multiplicity in terms of mixed multiplicities by using the Veronese degree. This enabled us to make various explicit computations of <em>ε</em>-multiplicities. We further write a Macaulay2 algorithm to compute <em>ε</em>-multiplicity (under the Noetherian hypotheses) even when the base ring is not necessarily standard graded.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108107"},"PeriodicalIF":0.8,"publicationDate":"2025-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268345","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 : 2025-10-01DOI: 10.1016/j.jpaa.2025.108105
Sophie Raynor
Circuit algebras are a symmetric analogue of Jones's planar algebras introduced to study finite-type invariants of virtual knotted objects. Circuit algebra structures appear, in different forms, across mathematics. This paper provides a dictionary for translating between their diverse incarnations and describing their wider context. A formal definition of a broad class of circuit algebras is established and three equivalent descriptions of circuit algebras are provided: in terms of operads of wiring diagrams, modular operads and categories of Brauer diagrams. As an application, circuit algebra characterisations of algebras over the orthogonal and symplectic groups are given.
{"title":"Functorial, operadic and modular operadic combinatorics of circuit algebras","authors":"Sophie Raynor","doi":"10.1016/j.jpaa.2025.108105","DOIUrl":"10.1016/j.jpaa.2025.108105","url":null,"abstract":"<div><div>Circuit algebras are a symmetric analogue of Jones's planar algebras introduced to study finite-type invariants of virtual knotted objects. Circuit algebra structures appear, in different forms, across mathematics. This paper provides a dictionary for translating between their diverse incarnations and describing their wider context. A formal definition of a broad class of circuit algebras is established and three equivalent descriptions of circuit algebras are provided: in terms of operads of wiring diagrams, modular operads and categories of Brauer diagrams. As an application, circuit algebra characterisations of algebras over the orthogonal and symplectic groups are given.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108105"},"PeriodicalIF":0.8,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268346","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 : 2025-10-01DOI: 10.1016/j.jpaa.2025.108106
Lee Tae Young
We determine precisely which irreducible hypergeometric sheaves have an extraspecial normalizer in characteristic 2 as their geometric monodromy groups. This resolves the last open case of the classification of local monodromy at 0 of irreducible hypergeometric sheaves with finite geometric monodromy group.
{"title":"Hypergeometric sheaves and extraspecial groups in even characteristic","authors":"Lee Tae Young","doi":"10.1016/j.jpaa.2025.108106","DOIUrl":"10.1016/j.jpaa.2025.108106","url":null,"abstract":"<div><div>We determine precisely which irreducible hypergeometric sheaves have an extraspecial normalizer in characteristic 2 as their geometric monodromy groups. This resolves the last open case of the classification of local monodromy at 0 of irreducible hypergeometric sheaves with finite geometric monodromy group.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108106"},"PeriodicalIF":0.8,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145222117","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 : 2025-10-01DOI: 10.1016/j.jpaa.2025.108102
Kiryong Chung , Jaehyun Kim , Jeong-Seop Kim
Let X be a Fano threefold of index one and degree 22 with . Such a threefold X can be realized as the zero locus of a regular section s of over the Grassmannian , where and is the universal subbundle. When the section s is given by the net of the -invariant skew-symmetric forms, we call it the Mukai-Umemura (MU) variety. In this paper, we prove that the Hilbert scheme of rational quartic curves in the MU-variety is smooth, and we compute its Poincaré polynomial by applying Białynicki-Birula's theorem.
{"title":"Rational quartic curves in the Mukai-Umemura variety","authors":"Kiryong Chung , Jaehyun Kim , Jeong-Seop Kim","doi":"10.1016/j.jpaa.2025.108102","DOIUrl":"10.1016/j.jpaa.2025.108102","url":null,"abstract":"<div><div>Let <em>X</em> be a Fano threefold of index one and degree 22 with <span><math><mrow><mi>Pic</mi></mrow><mo>(</mo><mi>X</mi><mo>)</mo><mo>≅</mo><mi>Z</mi></math></span>. Such a threefold <em>X</em> can be realized as the zero locus of a regular section <strong>s</strong> of <span><math><msup><mrow><mo>(</mo><msup><mrow><mo>⋀</mo></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>U</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo></mrow><mrow><mo>⊕</mo><mn>3</mn></mrow></msup></math></span> over the Grassmannian <span><math><mrow><mi>Gr</mi></mrow><mo>(</mo><mn>3</mn><mo>,</mo><mi>V</mi><mo>)</mo></math></span>, where <span><math><mi>dim</mi><mo></mo><mi>V</mi><mo>=</mo><mn>7</mn></math></span> and <span><math><mi>U</mi></math></span> is the universal subbundle. When the section <strong>s</strong> is given by the net of the <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-invariant skew-symmetric forms, we call it the Mukai-Umemura (MU) variety. In this paper, we prove that the Hilbert scheme of rational quartic curves in the MU-variety is smooth, and we compute its Poincaré polynomial by applying Białynicki-Birula's theorem.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108102"},"PeriodicalIF":0.8,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145222114","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 : 2025-10-01DOI: 10.1016/j.jpaa.2025.108104
Justin Fong , Mitsuhiro Miyazaki
The F-pure threshold is the characteristic p counter part of the log canonical threshold in characteristic zero. It is a numerical invariant associated to the singularities of a variety, hence computing its value is important. We give a closed formula for the F-pure threshold of the irrelevant maximal ideal of Schubert cycles, which are the homogeneous coordinate rings of Schubert subvarieties of a Grassmannian. The main point of the computation is to give an explicit formula for the a-invariant of a Schubert cycle. The derivation of both formulas is made possible through the combinatorics of the underlying poset of these rings.
f -纯阈值是特征0中对数正则阈值的特征p计数器部分。它是一个与各种奇点有关的数值不变量，因此计算它的值是很重要的。我们给出了无关极大理想Schubert环的f -纯阈值的一个封闭公式，Schubert环是一类Grassmannian的Schubert子变量的齐次坐标环。计算的重点是给出舒伯特循环的a不变量的显式公式。这两个公式的推导是通过对这些环的基本偏序集的组合而实现的。
{"title":"The F-pure threshold of a Schubert cycle","authors":"Justin Fong , Mitsuhiro Miyazaki","doi":"10.1016/j.jpaa.2025.108104","DOIUrl":"10.1016/j.jpaa.2025.108104","url":null,"abstract":"<div><div>The <em>F</em>-pure threshold is the characteristic <em>p</em> counter part of the log canonical threshold in characteristic zero. It is a numerical invariant associated to the singularities of a variety, hence computing its value is important. We give a closed formula for the <em>F</em>-pure threshold of the irrelevant maximal ideal of Schubert cycles, which are the homogeneous coordinate rings of Schubert subvarieties of a Grassmannian. The main point of the computation is to give an explicit formula for the <em>a</em>-invariant of a Schubert cycle. The derivation of both formulas is made possible through the combinatorics of the underlying poset of these rings.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108104"},"PeriodicalIF":0.8,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145222116","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 : 2025-10-01DOI: 10.1016/j.jpaa.2025.108101
Elad Paran , Thieu N. Vo
We study Noether's normalization lemma for finitely generated algebras over a division algebra. In its classical form, the lemma states that if I is a proper ideal of the ring of polynomials over a field F, then the quotient ring is a finite extension of a polynomial ring over F. We prove that the lemma holds when is the ring of polynomials in n central variables over a division algebra D. We provide examples demonstrating that Noether's normalization may fail for the skew polynomial ring with respect to commuting automorphisms of D. We give a sufficient condition for under which the normalization lemma holds for such ring. In the case where is a field, this sufficient condition is proved to be necessary.
{"title":"Noether's normalization in skew polynomial rings","authors":"Elad Paran , Thieu N. Vo","doi":"10.1016/j.jpaa.2025.108101","DOIUrl":"10.1016/j.jpaa.2025.108101","url":null,"abstract":"<div><div>We study Noether's normalization lemma for finitely generated algebras over a division algebra. In its classical form, the lemma states that if <em>I</em> is a proper ideal of the ring <span><math><mi>R</mi><mo>=</mo><mi>F</mi><mo>[</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span> of polynomials over a field <em>F</em>, then the quotient ring <span><math><mi>R</mi><mo>/</mo><mi>I</mi></math></span> is a finite extension of a polynomial ring over <em>F</em>. We prove that the lemma holds when <span><math><mi>R</mi><mo>=</mo><mi>D</mi><mo>[</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span> is the ring of polynomials in <em>n</em> central variables over a division algebra <em>D</em>. We provide examples demonstrating that Noether's normalization may fail for the skew polynomial ring <span><math><mi>D</mi><mo>[</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>;</mo><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span> with respect to commuting automorphisms <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of <em>D</em>. We give a sufficient condition for <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> under which the normalization lemma holds for such ring. In the case where <span><math><mi>D</mi><mo>=</mo><mi>F</mi></math></span> is a field, this sufficient condition is proved to be necessary.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108101"},"PeriodicalIF":0.8,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145222115","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 : 2025-10-01DOI: 10.1016/j.jpaa.2025.108103
Natsume Kitagawa
Corti defined the notion of standard models of del Pezzo fibrations, and studied their existence over with a fixed generic fibre in [6]. In this paper, we prove the existence of standard models of del Pezzo fibrations of degree 4 in characteristic >2. To show this, we use the notion of Kollár stability, which was introduced in [12] and [1].
{"title":"On the standard models of del Pezzo fibrations of degree four","authors":"Natsume Kitagawa","doi":"10.1016/j.jpaa.2025.108103","DOIUrl":"10.1016/j.jpaa.2025.108103","url":null,"abstract":"<div><div>Corti defined the notion of standard models of del Pezzo fibrations, and studied their existence over <span><math><mi>C</mi></math></span> with a fixed generic fibre in <span><span>[6]</span></span>. In this paper, we prove the existence of standard models of del Pezzo fibrations of degree 4 in characteristic >2. To show this, we use the notion of Kollár stability, which was introduced in <span><span>[12]</span></span> and <span><span>[1]</span></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108103"},"PeriodicalIF":0.8,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145333342","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}
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