Pub Date : 2026-02-01DOI: 10.1016/j.jpaa.2026.108193
Zachary Nason
Let R be a commutative noetherian local differential graded (DG) ring. In this paper we propose a definition of a maximal Cohen-Macaulay DG-complex over R that naturally generalizes a maximal Cohen-Macaulay complex over a noetherian local ring, as studied by Iyengar, Ma, Schwede, and Walker. Our proposed definition extends the work of Shaul on Cohen-Macaulay DG-rings and DG-modules, as any maximal Cohen-Macaulay DG-module is a maximal Cohen-Macaulay DG-complex. After proving necessary lemmas in derived commutative algebra, we establish the existence of a maximal Cohen-Macaulay DG-complex for every DG-ring with constant amplitude that admits a dualizing DG-module. We then use the existence of these DG-complexes to establish a derived Improved New Intersection Theorem for all DG-rings with constant amplitude.
{"title":"Maximal Cohen-Macaulay DG-complexes","authors":"Zachary Nason","doi":"10.1016/j.jpaa.2026.108193","DOIUrl":"10.1016/j.jpaa.2026.108193","url":null,"abstract":"<div><div>Let <em>R</em> be a commutative noetherian local differential graded (DG) ring. In this paper we propose a definition of a maximal Cohen-Macaulay DG-complex over <em>R</em> that naturally generalizes a maximal Cohen-Macaulay complex over a noetherian local ring, as studied by Iyengar, Ma, Schwede, and Walker. Our proposed definition extends the work of Shaul on Cohen-Macaulay DG-rings and DG-modules, as any maximal Cohen-Macaulay DG-module is a maximal Cohen-Macaulay DG-complex. After proving necessary lemmas in derived commutative algebra, we establish the existence of a maximal Cohen-Macaulay DG-complex for every DG-ring with constant amplitude that admits a dualizing DG-module. We then use the existence of these DG-complexes to establish a derived Improved New Intersection Theorem for all DG-rings with constant amplitude.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108193"},"PeriodicalIF":0.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174121","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 : 2026-02-01DOI: 10.1016/j.jpaa.2026.108195
Wan Keng Cheong, Ngau Lam
We prove an equivalence between the actions of the Gaudin algebras with irregular singularities for and on the Fock space of bosonic and fermionic oscillators. This establishes a duality of for Gaudin models. As an application, we show that the Gaudin algebra with irregular singularities for acts cyclically on each weight space of a certain class of infinite-dimensional modules over a direct sum of Takiff superalgebras over and that the action is diagonalizable with a simple spectrum under a generic condition. We also study the classical versions of Gaudin algebras with irregular singularities and demonstrate a duality of for classical Gaudin models.
{"title":"Dualities of Gaudin models with irregular singularities for general linear Lie (super)algebras","authors":"Wan Keng Cheong, Ngau Lam","doi":"10.1016/j.jpaa.2026.108195","DOIUrl":"10.1016/j.jpaa.2026.108195","url":null,"abstract":"<div><div>We prove an equivalence between the actions of the Gaudin algebras with irregular singularities for <span><math><msub><mrow><mi>gl</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>gl</mi></mrow><mrow><mi>p</mi><mo>+</mo><mi>m</mi><mo>|</mo><mi>q</mi><mo>+</mo><mi>n</mi></mrow></msub></math></span> on the Fock space of <span><math><mi>d</mi><mo>(</mo><mi>p</mi><mo>+</mo><mi>m</mi><mo>)</mo></math></span> bosonic and <span><math><mi>d</mi><mo>(</mo><mi>q</mi><mo>+</mo><mi>n</mi><mo>)</mo></math></span> fermionic oscillators. This establishes a duality of <span><math><mo>(</mo><msub><mrow><mi>gl</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><msub><mrow><mi>gl</mi></mrow><mrow><mi>p</mi><mo>+</mo><mi>m</mi><mo>|</mo><mi>q</mi><mo>+</mo><mi>n</mi></mrow></msub><mo>)</mo></math></span> for Gaudin models. As an application, we show that the Gaudin algebra with irregular singularities for <span><math><msub><mrow><mi>gl</mi></mrow><mrow><mi>p</mi><mo>+</mo><mi>m</mi><mo>|</mo><mi>q</mi><mo>+</mo><mi>n</mi></mrow></msub></math></span> acts cyclically on each weight space of a certain class of infinite-dimensional modules over a direct sum of Takiff superalgebras over <span><math><msub><mrow><mi>gl</mi></mrow><mrow><mi>p</mi><mo>+</mo><mi>m</mi><mo>|</mo><mi>q</mi><mo>+</mo><mi>n</mi></mrow></msub></math></span> and that the action is diagonalizable with a simple spectrum under a generic condition. We also study the classical versions of Gaudin algebras with irregular singularities and demonstrate a duality of <span><math><mo>(</mo><msub><mrow><mi>gl</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>,</mo><msub><mrow><mi>gl</mi></mrow><mrow><mi>p</mi><mo>+</mo><mi>m</mi><mo>|</mo><mi>q</mi><mo>+</mo><mi>n</mi></mrow></msub><mo>)</mo></math></span> for classical Gaudin models.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108195"},"PeriodicalIF":0.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174122","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 : 2026-02-01DOI: 10.1016/j.jpaa.2026.108184
Enric Nart , Josnei Novacoski
The depth of a simple algebraic extension of valued fields is the minimal length of the Mac Lane-Vaquié chains of the valuations on determined by the choice of different generators of the extension. In [11], we characterized the defectless unibranched extensions of depth one. In this paper, we analyze this problem for towers of Artin-Schreier defect extensions. Under certain conditions on , we prove that the towers obtained as the compositum of linearly disjoint defect Artin-Schreier extensions of K have depth one. We conjecture that these are the only depth one Artin-Schreier defect towers and we present some examples supporting this conjecture.
有值域的简单代数扩展（L/K,v）的深度是K[x]上的赋值的Mac lane - vaqui链的最小长度，该长度由该扩展的不同生成器的选择决定。在[11]中，我们刻画了深度1的无缺陷无分支扩展。本文分析了Artin-Schreier缺陷扩展塔的这一问题。在（K,v）上的一定条件下，证明了由K的线性不相交缺陷Artin-Schreier扩展复合得到的塔深度为1。我们推测这些是唯一深度的阿汀-施赖尔缺陷塔，我们提出了一些例子来支持这一猜想。
{"title":"Depth of Artin-Schreier defect towers","authors":"Enric Nart , Josnei Novacoski","doi":"10.1016/j.jpaa.2026.108184","DOIUrl":"10.1016/j.jpaa.2026.108184","url":null,"abstract":"<div><div>The depth of a simple algebraic extension <span><math><mo>(</mo><mi>L</mi><mo>/</mo><mi>K</mi><mo>,</mo><mi>v</mi><mo>)</mo></math></span> of valued fields is the minimal length of the Mac Lane-Vaquié chains of the valuations on <span><math><mi>K</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> determined by the choice of different generators of the extension. In <span><span>[11]</span></span>, we characterized the defectless unibranched extensions of depth one. In this paper, we analyze this problem for towers of Artin-Schreier defect extensions. Under certain conditions on <span><math><mo>(</mo><mi>K</mi><mo>,</mo><mi>v</mi><mo>)</mo></math></span>, we prove that the towers obtained as the compositum of linearly disjoint defect Artin-Schreier extensions of <em>K</em> have depth one. We conjecture that these are the only depth one Artin-Schreier defect towers and we present some examples supporting this conjecture.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108184"},"PeriodicalIF":0.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079617","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 : 2026-01-12DOI: 10.1016/j.jpaa.2026.108172
Kaiyue He
We introduce a new numerical invariant associated to a finite-length R-module M and an ideal I in an Artinian local ring R. This invariant measures the ratio between and . We establish fundamental relationships between this invariant and the Betti numbers of the module under the assumption of the Tor modules vanishing. In particular, we use this invariant to establish a freeness criterion for modules under certain Tor vanishing conditions. The criterion applies specifically to the class of I-free modules — those modules M for which is isomorphic to a direct sum of copies of . Lastly, we apply these results to the canonical module, proving that, under certain conditions on the ring structure, when the zeroth Betti number is greater than or equal to the first Betti number of the canonical module, then the ring is Gorenstein. This partially answers a question posed by Jorgensen and Leuschke concerning the relationship between Betti numbers of the canonical module and Gorenstein properties.
{"title":"Betti numbers for modules over Artinian local rings","authors":"Kaiyue He","doi":"10.1016/j.jpaa.2026.108172","DOIUrl":"10.1016/j.jpaa.2026.108172","url":null,"abstract":"<div><div>We introduce a new numerical invariant <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span> associated to a finite-length <em>R</em>-module <em>M</em> and an ideal <em>I</em> in an Artinian local ring <em>R</em>. This invariant measures the ratio between <span><math><mi>λ</mi><mo>(</mo><mi>I</mi><mi>M</mi><mo>)</mo></math></span> and <span><math><mi>λ</mi><mo>(</mo><mi>M</mi><mo>/</mo><mi>I</mi><mi>M</mi><mo>)</mo></math></span>. We establish fundamental relationships between this invariant and the Betti numbers of the module under the assumption of the Tor modules vanishing. In particular, we use this invariant to establish a freeness criterion for modules under certain Tor vanishing conditions. The criterion applies specifically to the class of <em>I</em>-free modules — those modules <em>M</em> for which <span><math><mi>M</mi><mo>/</mo><mi>I</mi><mi>M</mi></math></span> is isomorphic to a direct sum of copies of <span><math><mi>R</mi><mo>/</mo><mi>I</mi></math></span>. Lastly, we apply these results to the canonical module, proving that, under certain conditions on the ring structure, when the zeroth Betti number is greater than or equal to the first Betti number of the canonical module, then the ring is Gorenstein. This partially answers a question posed by Jorgensen and Leuschke concerning the relationship between Betti numbers of the canonical module and Gorenstein properties.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108172"},"PeriodicalIF":0.8,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981621","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 : 2026-01-12DOI: 10.1016/j.jpaa.2026.108171
Donald M. Davis , Douglas C. Ravenel , W. Stephen Wilson
We develop tools for computing the connective n-th Morava K-theory of spaces. Starting with a Universal Coefficient Theorem that computes the cohomology version from the homology version, we show that every step in the process of computing one is mirrored in the other and that this can be used to make computations. As our example, we compute the connective n-th Morava K-theory of the second mod p Eilenberg-MacLane space.
{"title":"The connective Morava K-theory of the second mod p Eilenberg-MacLane space","authors":"Donald M. Davis , Douglas C. Ravenel , W. Stephen Wilson","doi":"10.1016/j.jpaa.2026.108171","DOIUrl":"10.1016/j.jpaa.2026.108171","url":null,"abstract":"<div><div>We develop tools for computing the connective n-th Morava K-theory of spaces. Starting with a Universal Coefficient Theorem that computes the cohomology version from the homology version, we show that every step in the process of computing one is mirrored in the other and that this can be used to make computations. As our example, we compute the connective <em>n</em>-th Morava <em>K</em>-theory of the second mod <em>p</em> Eilenberg-MacLane space.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108171"},"PeriodicalIF":0.8,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981620","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 : 2026-01-12DOI: 10.1016/j.jpaa.2026.108173
Shengding Sun , Aljaž Zalar
The matrix Fejér-Riesz theorem characterizes positive semidefinite matrix polynomials on the real line. In [28] this was extended to the characterization on arbitrary closed semialgebraic sets by using matrix quadratic modules from real algebraic geometry. In the compact case there is a denominator-free characterization, while in the non-compact case denominators are needed except when K is the whole line, an unbounded interval, a union of two unbounded intervals, and according to a conjecture of [28] also when K is a union of an unbounded interval and a point or a union of two unbounded intervals and a point. In this paper, we confirm this conjecture by solving the truncated matrix-valued moment problem on a union of a bounded interval and a point. The presented technique for solving the corresponding moment problem can potentially be used to determine degree bounds in the positivity certificates for matrix polynomials on compact sets K[28, Theorem C].
{"title":"Matrix Fejér-Riesz type theorem for a union of an interval and a point","authors":"Shengding Sun , Aljaž Zalar","doi":"10.1016/j.jpaa.2026.108173","DOIUrl":"10.1016/j.jpaa.2026.108173","url":null,"abstract":"<div><div>The matrix Fejér-Riesz theorem characterizes positive semidefinite matrix polynomials on the real line. In <span><span>[28]</span></span> this was extended to the characterization on arbitrary closed semialgebraic sets <span><math><mi>K</mi><mo>⊆</mo><mi>R</mi></math></span> by using matrix quadratic modules from real algebraic geometry. In the compact case there is a denominator-free characterization, while in the non-compact case denominators are needed except when <em>K</em> is the whole line, an unbounded interval, a union of two unbounded intervals, and according to a conjecture of <span><span>[28]</span></span> also when <em>K</em> is a union of an unbounded interval and a point or a union of two unbounded intervals and a point. In this paper, we confirm this conjecture by solving the truncated matrix-valued moment problem on a union of a bounded interval and a point. The presented technique for solving the corresponding moment problem can potentially be used to determine degree bounds in the positivity certificates for matrix polynomials on compact sets <em>K</em> <span><span>[28, Theorem C]</span></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108173"},"PeriodicalIF":0.8,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981622","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 : 2026-01-08DOI: 10.1016/j.jpaa.2026.108170
J. Reina
This paper establishes a relation between two invariants of 3-dimensional manifolds: the chromatic spherical invariant and the Hennings-Kauffman-Radford invariant HKR. We show that, for a spherical Hopf algebra H, the invariant associated to the pivotal category of finite-dimensional H-modules is equal to the invariant HKR associated to the Drinfeld double of the same Hopf algebra.
{"title":"Chromatic spherical invariant and Hennings invariant of 3-dimensional manifolds","authors":"J. Reina","doi":"10.1016/j.jpaa.2026.108170","DOIUrl":"10.1016/j.jpaa.2026.108170","url":null,"abstract":"<div><div>This paper establishes a relation between two invariants of 3-dimensional manifolds: the chromatic spherical invariant <span><math><mi>K</mi></math></span> and the Hennings-Kauffman-Radford invariant HKR. We show that, for a spherical Hopf algebra <em>H</em>, the invariant <span><math><mi>K</mi></math></span> associated to the pivotal category of finite-dimensional <em>H</em>-modules is equal to the invariant HKR associated to the Drinfeld double <span><math><mi>D</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> of the same Hopf algebra.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 2","pages":"Article 108170"},"PeriodicalIF":0.8,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145950134","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 : 2026-01-01DOI: 10.1016/j.jpaa.2025.108153
Hans Franzen , Gianni Petrella , Rachel Webb
We give an effective characterization of the walls in the variation of geometric invariant theory problem associated to a quiver and a dimension vector.
本文给出了一个几何不变理论问题中与一个颤振和一个维向量相关的壁面变化的有效表征。
{"title":"Finding the walls for quiver moduli","authors":"Hans Franzen , Gianni Petrella , Rachel Webb","doi":"10.1016/j.jpaa.2025.108153","DOIUrl":"10.1016/j.jpaa.2025.108153","url":null,"abstract":"<div><div>We give an effective characterization of the walls in the variation of geometric invariant theory problem associated to a quiver and a dimension vector.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 1","pages":"Article 108153"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885249","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 : 2026-01-01DOI: 10.1016/j.jpaa.2025.108168
Ahmed Laghribi, Trisha Maiti
Let F be a field of characteristic 2. The m-Pfister number of a quadratic form is the least number of forms similar to m-fold Pfister forms needed to express φ up to Witt equivalence. Our aim in this note is to discuss the case by giving an inductive formula that explicitly bounds the 3-Pfister number of any form in .
{"title":"On the 3-Pfister number in characteristic 2","authors":"Ahmed Laghribi, Trisha Maiti","doi":"10.1016/j.jpaa.2025.108168","DOIUrl":"10.1016/j.jpaa.2025.108168","url":null,"abstract":"<div><div>Let <em>F</em> be a field of characteristic 2. The <em>m</em>-Pfister number of a quadratic form <span><math><mi>φ</mi><mo>∈</mo><msubsup><mrow><mi>I</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msubsup><mo>(</mo><mi>F</mi><mo>)</mo></math></span> is the least number of forms similar to <em>m</em>-fold Pfister forms needed to express <em>φ</em> up to Witt equivalence. Our aim in this note is to discuss the case <span><math><mi>m</mi><mo>=</mo><mn>3</mn></math></span> by giving an inductive formula that explicitly bounds the 3-Pfister number of any form in <span><math><msubsup><mrow><mi>I</mi></mrow><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mi>F</mi></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 1","pages":"Article 108168"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145926215","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 : 2026-01-01DOI: 10.1016/j.jpaa.2025.108167
Shahriyar Roshan Zamir
Over an algebraically closed field, the double point interpolation problem asks for the vector space dimension of the projective hypersurfaces of degree d singular at a given set of points. After being open for 90 years, a series of papers by J. Alexander and A. Hirschowitz in 1992–1995 settled this question in what is referred to as the Alexander-Hirschowitz theorem. In this paper we primarily use commutative algebra to lay the groundwork necessary to prove analogous statements in the weighted projective space, a natural generalization of the projective space. We prove the Hilbert function of general simple points in any n-dimensional weighted projective space exhibits the expected behavior. We also introduce an inductive procedure for weighted projective space, similar to that originally due to A. Terracini from 1915, to demonstrate an example of a weighted projective plane where the analogue of the Alexander-Hirschowitz theorem holds without exceptions and prove our example is the only such plane. Furthermore, Terracini's lemma regarding secant varieties is adapted to give an interpolation bound for an infinite family of weighted projective planes.
{"title":"Interpolation in weighted projective spaces","authors":"Shahriyar Roshan Zamir","doi":"10.1016/j.jpaa.2025.108167","DOIUrl":"10.1016/j.jpaa.2025.108167","url":null,"abstract":"<div><div>Over an algebraically closed field, the <em>double point interpolation</em> problem asks for the vector space dimension of the projective hypersurfaces of degree <em>d</em> singular at a given set of points. After being open for 90 years, a series of papers by J. Alexander and A. Hirschowitz in 1992–1995 settled this question in what is referred to as the Alexander-Hirschowitz theorem. In this paper we primarily use commutative algebra to lay the groundwork necessary to prove analogous statements in the <em>weighted projective space</em>, a natural generalization of the projective space. We prove the Hilbert function of general simple points in any <em>n</em>-dimensional weighted projective space exhibits the expected behavior. We also introduce an inductive procedure for weighted projective space, similar to that originally due to A. Terracini from 1915, to demonstrate an example of a weighted projective plane where the analogue of the Alexander-Hirschowitz theorem holds without exceptions and prove our example is the only such plane. Furthermore, Terracini's lemma regarding secant varieties is adapted to give an interpolation bound for an infinite family of weighted projective planes.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"230 1","pages":"Article 108167"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145926216","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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