Algebra & Number Theory最新文献

{"title":"Moduli of linear slices of high degree smooth hypersurfaces","authors":"Anand Patel, Eric Riedl, Dennis Tseng","doi":"10.2140/ant.2024.18.2133","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2133","url":null,"abstract":"<p>We study the variation of linear sections of hypersurfaces in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℙ</mi></mrow><mrow><mi>n</mi></mrow></msup></math>. We completely classify all plane curves, necessarily singular, whose line sections do not vary maximally in moduli. In higher dimensions, we prove that the family of hyperplane sections of any smooth degree <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi></math> hypersurface in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℙ</mi></mrow><mrow><mi>n</mi></mrow></msup></math> varies maximally for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi>\u0000<mo>≥</mo>\u0000<mi>n</mi>\u0000<mo>+</mo> <mn>3</mn></math>. In the process, we generalize the classical Grauert–Mülich theorem about lines in projective space, both to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>-planes in projective space and to free rational curves on arbitrary varieties. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"225 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142487002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Separating G2-invariants of several octonions","authors":"Artem Lopatin, Alexandr N. Zubkov","doi":"10.2140/ant.2024.18.2157","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2157","url":null,"abstract":"<p>We describe separating <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math>-invariants of several copies of the algebra of octonions over an algebraically closed field of characteristic two. We also obtain a minimal separating and a minimal generating set for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math>-invariants of several copies of the algebra of octonions in case of a field of odd characteristic. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"25 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142486628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Matrix Kloosterman sums","authors":"Márton Erdélyi, Árpád Tóth","doi":"10.2140/ant.2024.18.2247","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2247","url":null,"abstract":"<p>We study a family of exponential sums that arises in the study of expanding horospheres on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi> GL</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--></mrow><mrow><mi>n</mi></mrow></msub></math>. We prove an explicit version of general purity and find optimal bounds for these sums. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"210 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142487037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Scattering diagrams for generalized cluster algebras","authors":"Lang Mou","doi":"10.2140/ant.2024.18.2179","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2179","url":null,"abstract":"<p>We construct scattering diagrams for Chekhov–Shapiro generalized cluster algebras where exchange polynomials are factorized into binomials, generalizing the cluster scattering diagrams of Gross, Hacking, Keel and Kontsevich. They turn out to be natural objects arising in Fock and Goncharov’s cluster duality. Analogous features and structures (such as positivity and the cluster complex structure) in the ordinary case also appear in the generalized situation. With the help of these scattering diagrams, we show that generalized cluster variables are theta functions and hence have certain positivity property with respect to the coefficients in the binomial factors. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"66 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142486664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Rooted tree maps for multiple L-values from a perspective of harmonic algebras","authors":"Hideki Murahara, Tatsushi Tanaka, Noriko Wakabayashi","doi":"10.2140/ant.2024.18.2003","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2003","url":null,"abstract":"<p>We show the image of rooted tree maps forms a subspace of the kernel of the evaluation map of multiple <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>L</mi></math>-values. To prove this, we define the diamond product as a modified harmonic product and describe its properties. We also show that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>τ</mi></math>-conjugate rooted tree maps are their antipodes. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"4 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142449537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The distribution of large quadratic character sums and applications","authors":"Youness Lamzouri","doi":"10.2140/ant.2024.18.2091","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2091","url":null,"abstract":"<p>We investigate the distribution of the maximum of character sums over the family of primitive quadratic characters attached to fundamental discriminants <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo>|</mo><mi>d</mi><mo>|</mo><mo>≤</mo>\u0000<mi>x</mi></math>. In particular, our work improves results of Montgomery and Vaughan, and gives strong evidence that the Omega result of Bateman and Chowla for quadratic character sums is optimal. We also obtain similar results for real characters with prime discriminants up to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>x</mi></math>, and deduce the interesting consequence that almost all primes with large Legendre symbol sums are congruent to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>3</mn></math> modulo <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>4</mn></math>. Our results are motivated by a recent work of Bober, Goldmakher, Granville and Koukoulopoulos, who proved similar results for the family of nonprincipal characters modulo a large prime. However, their method does not seem to generalize to other families of Dirichlet characters. Instead, we use a different and more streamlined approach, which relies mainly on the quadratic large sieve. As an application, we consider a question of Montgomery concerning the positivity of sums of Legendre symbols. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"17 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142449539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Terminal orders on arithmetic surfaces","authors":"Daniel Chan, Colin Ingalls","doi":"10.2140/ant.2024.18.2027","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2027","url":null,"abstract":"<p>The local structure of terminal Brauer classes on arithmetic surfaces was classified (2021), generalising the classification on geometric surfaces (2005). Part of the interest in these classifications is that it enables the minimal model program to be applied to the noncommutative setting of orders on surfaces. We give étale local structure theorems for terminal orders on arithmetic surfaces, at least when the degree is a prime <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>p</mi>\u0000<mo>&gt;</mo> <mn>5</mn></math>. This generalises the structure theorem given in the geometric case. They can all be explicitly constructed as algebras of matrices over symbols. From this description one sees that such terminal orders all have global dimension two, thus generalising the fact that terminal (commutative) surfaces are smooth and hence homologically regular. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"109 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142449538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Word measures on GLn(q) and free group algebras","authors":"Danielle Ernst-West, Doron Puder, Matan Seidel","doi":"10.2140/ant.2024.18.2047","DOIUrl":"https://doi.org/10.2140/ant.2024.18.2047","url":null,"abstract":"&lt;p&gt;Fix a finite field &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/math&gt; of order &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/math&gt; and a word &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;/math&gt; in a free group &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mstyle mathvariant=\"bold-italic\"&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mstyle&gt;&lt;/math&gt; on &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt; generators. A &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;/math&gt;-random element in &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt; GL&lt;/mi&gt;&lt;mo&gt; ⁡&lt;!--FUNCTION APPLICATION--&gt; &lt;/mo&gt;&lt;!--nolimits--&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt; is obtained by sampling &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/math&gt; independent uniformly random elements &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;…&lt;/mi&gt;&lt;mo&gt; ⁡&lt;!--FUNCTION APPLICATION--&gt;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;\u0000&lt;mo&gt;∈&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt; GL&lt;/mi&gt;&lt;mo&gt; ⁡&lt;!--FUNCTION APPLICATION--&gt; &lt;/mo&gt;&lt;!--nolimits--&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt; and evaluating &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;…&lt;/mi&gt;&lt;mo&gt; ⁡&lt;!--FUNCTION APPLICATION--&gt;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;. Consider &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=\"double-struck\"&gt;𝔼&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;mi&gt;fix&lt;/mi&gt;&lt;mo&gt; ⁡&lt;!--FUNCTION APPLICATION--&gt; &lt;/mo&gt;&lt;!--nolimits--&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/math&gt;, the average number of vectors in &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt; fixed by a &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;/math&gt;-random element. We show that &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=\"double-struck\"&gt;𝔼&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;mi&gt;fix&lt;/mi&gt;&lt;mo&gt; ⁡&lt;!--FUNCTION APPLICATION--&gt; &lt;/mo&gt;&lt;!--nolimits--&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/math&gt; is a rational function in &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;. If &lt;math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;w&lt;/mi&gt;\u0000&lt;mo&gt;=&lt;/mo&gt; &lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/ms","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"46 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142449570","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Galois orbits of torsion points near atoral sets","authors":"Vesselin Dimitrov, Philipp Habegger","doi":"10.2140/ant.2024.18.1945","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1945","url":null,"abstract":"<p>We prove that the Galois equidistribution of torsion points of the algebraic torus <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi mathvariant=\"double-struck\">𝔾</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math> extends to the singular test functions of the form <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> log</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo>|</mo><mi>P</mi><mo>|</mo></math>, where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>P</mi></math> is a Laurent polynomial having algebraic coefficients that vanishes on the unit real <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi></math>-torus in a set whose Zariski closure in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi mathvariant=\"double-struck\">𝔾</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math> has codimension at least <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn></math>. Our result includes a power-saving quantitative estimate of the decay rate of the equidistribution. It refines an ergodic theorem of Lind, Schmidt, and Verbitskiy, of which it also supplies a purely Diophantine proof. As an application, we confirm Ih’s integrality finiteness conjecture on torsion points for a class of atoral divisors of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi mathvariant=\"double-struck\">𝔾</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"233 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142449545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A short resolution of the diagonal for smooth projective toric varieties of Picard rank 2","authors":"Michael K. Brown, Mahrud Sayrafi","doi":"10.2140/ant.2024.18.1923","DOIUrl":"https://doi.org/10.2140/ant.2024.18.1923","url":null,"abstract":"<p>Given a smooth projective toric variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi></math> of Picard rank 2, we resolve the diagonal sheaf on <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>X</mi>\u0000<mo>×</mo>\u0000<mi>X</mi></math> by a linear complex of length <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> dim</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mi>X</mi></math> consisting of finite direct sums of line bundles. As applications, we prove a new case of a conjecture of Berkesch, Ermana and Smith that predicts a version of Hilbert’s syzygy theorem for virtual resolutions, and we obtain a Horrocks-type splitting criterion for vector bundles over smooth projective toric varieties of Picard rank 2, extending a result of Eisenbud, Erman and Schreyer. We also apply our results to give a new proof, in the case of smooth projective toric varieties of Picard rank 2, of a conjecture of Orlov concerning the Rouquier dimension of derived categories. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"12 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142384209","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}