The present paper is a natural continuation of a previous work where we studied the second syzygy scheme of canonical curves. We find sufficient conditions ensuring that the second syzygy scheme of a genus--$g$ curve of degree at least $2g+2$ coincide with the curve. If the property $(N_2)$ is satisfied, the equality is ensured by a more general fact. If $(N_2)$ fails, then the analysis uses the known case of canonical curves.
{"title":"The second syzygy schemes of curves of large degree","authors":"Marian Aprodu, Andrea Bruno, Edoardo Sernesi","doi":"arxiv-2409.11855","DOIUrl":"https://doi.org/arxiv-2409.11855","url":null,"abstract":"The present paper is a natural continuation of a previous work where we\u0000studied the second syzygy scheme of canonical curves. We find sufficient\u0000conditions ensuring that the second syzygy scheme of a genus--$g$ curve of\u0000degree at least $2g+2$ coincide with the curve. If the property $(N_2)$ is\u0000satisfied, the equality is ensured by a more general fact. If $(N_2)$ fails,\u0000then the analysis uses the known case of canonical curves.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that over an algebraically closed field of characteristic $p>0$ there are exactly, up to isomorphism, $n$ infinitesimal commutative unipotent $k$-group schemes of order $p^n$ with one-dimensional Lie algebra, and we explicitly describe them. We consequently recover an explicit description of the $p^n$-torsion of any supersingular elliptic curve over an algebraically closed field. Finally, we use these results to answer a question of Brion on rational actions of infinitesimal commutative unipotent group schemes on curves.
{"title":"Infinitesimal commutative unipotent group schemes with one-dimensional Lie algebra","authors":"Bianca Gouthier","doi":"arxiv-2409.11997","DOIUrl":"https://doi.org/arxiv-2409.11997","url":null,"abstract":"We prove that over an algebraically closed field of characteristic $p>0$\u0000there are exactly, up to isomorphism, $n$ infinitesimal commutative unipotent\u0000$k$-group schemes of order $p^n$ with one-dimensional Lie algebra, and we\u0000explicitly describe them. We consequently recover an explicit description of\u0000the $p^n$-torsion of any supersingular elliptic curve over an algebraically\u0000closed field. Finally, we use these results to answer a question of Brion on\u0000rational actions of infinitesimal commutative unipotent group schemes on\u0000curves.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Francesco Antonio Denisi, Ángel David Ríos Ortiz, Nikolaos Tsakanikas, Zhixin Xie
We introduce and study the class of primitive Enriques varieties, whose smooth members are Enriques manifolds. We provide several examples and we demonstrate that this class is stable under the operations of the Minimal Model Program (MMP). In particular, given an Enriques manifold $Y$ and an effective $mathbb{R}$-divisor $B_Y$ on $Y$ such that the pair $(Y,B_Y)$ is log canonical, we prove that any $(K_Y+B_Y)$-MMP terminates with a minimal model $(Y',B_{Y'})$ of $(Y,B_Y)$, where $Y'$ is a $mathbb{Q}$-factorial primitive Enriques variety with canonical singularities. Finally, we investigate the asymptotic theory of Enriques manifolds.
{"title":"MMP for Enriques pairs and singular Enriques varieties","authors":"Francesco Antonio Denisi, Ángel David Ríos Ortiz, Nikolaos Tsakanikas, Zhixin Xie","doi":"arxiv-2409.12054","DOIUrl":"https://doi.org/arxiv-2409.12054","url":null,"abstract":"We introduce and study the class of primitive Enriques varieties, whose\u0000smooth members are Enriques manifolds. We provide several examples and we\u0000demonstrate that this class is stable under the operations of the Minimal Model\u0000Program (MMP). In particular, given an Enriques manifold $Y$ and an effective\u0000$mathbb{R}$-divisor $B_Y$ on $Y$ such that the pair $(Y,B_Y)$ is log\u0000canonical, we prove that any $(K_Y+B_Y)$-MMP terminates with a minimal model\u0000$(Y',B_{Y'})$ of $(Y,B_Y)$, where $Y'$ is a $mathbb{Q}$-factorial primitive\u0000Enriques variety with canonical singularities. Finally, we investigate the\u0000asymptotic theory of Enriques manifolds.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given an 'etale endomorphism of a normal irreducible Noetherian and simply connected scheme, we prove that if the endomorphism is surjective then it is injective. The proof is based on Liu's construction of a Galois cover out of a surjective 'etale morphism. If we give up of the surjectivity hypothesis and suppose the endomorphism is separated, then we prove that the induced field extension is Galois. In the case of an 'etale endomorphism of the affine space over an algebraically closed field of characteristic zero, Campbell's theorem implies that the assumption of surjectivity is superfluous.
{"title":"A converse of Ax-Grothendieck theorem for étale endomorphisms of normal schemes","authors":"Lázaro O. Rodríguez Díaz","doi":"arxiv-2409.12163","DOIUrl":"https://doi.org/arxiv-2409.12163","url":null,"abstract":"Given an 'etale endomorphism of a normal irreducible Noetherian and simply\u0000connected scheme, we prove that if the endomorphism is surjective then it is\u0000injective. The proof is based on Liu's construction of a Galois cover out of a\u0000surjective 'etale morphism. If we give up of the surjectivity hypothesis and\u0000suppose the endomorphism is separated, then we prove that the induced field\u0000extension is Galois. In the case of an 'etale endomorphism of the affine space\u0000over an algebraically closed field of characteristic zero, Campbell's theorem\u0000implies that the assumption of surjectivity is superfluous.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we explore the intersection of the Hassett divisor $mathcal C_8$, parametrizing smooth cubic fourfolds $X$ containing a plane $P$ with other divisors $mathcal C_i$. Notably we study the irreducible components of the intersections with $mathcal{C}_{12}$ and $mathcal{C}_{20}$. These two divisors generically parametrize respectively cubics containing a smooth cubic scroll, and a smooth Veronese surface. First, we find all the irreducible components of the two intersections, and describe the geometry of the generic elements in terms of the intersection of $P$ with the other surface. Then we consider the problem of rationality of cubics in these components, either by finding rational sections of the quadric fibration induced by projection off $P$, or by finding examples of reducible one-apparent-double-point surfaces inside $X$. Finally, via some Macaulay computations, we give explicit equations for cubics in each component.
{"title":"Moduli of Cubic fourfolds and reducible OADP surfaces","authors":"Michele Bolognesi, Zakaria Brahimi, Hanine Awada","doi":"arxiv-2409.12032","DOIUrl":"https://doi.org/arxiv-2409.12032","url":null,"abstract":"In this paper we explore the intersection of the Hassett divisor $mathcal\u0000C_8$, parametrizing smooth cubic fourfolds $X$ containing a plane $P$ with\u0000other divisors $mathcal C_i$. Notably we study the irreducible components of\u0000the intersections with $mathcal{C}_{12}$ and $mathcal{C}_{20}$. These two\u0000divisors generically parametrize respectively cubics containing a smooth cubic\u0000scroll, and a smooth Veronese surface. First, we find all the irreducible\u0000components of the two intersections, and describe the geometry of the generic\u0000elements in terms of the intersection of $P$ with the other surface. Then we\u0000consider the problem of rationality of cubics in these components, either by\u0000finding rational sections of the quadric fibration induced by projection off\u0000$P$, or by finding examples of reducible one-apparent-double-point surfaces\u0000inside $X$. Finally, via some Macaulay computations, we give explicit equations\u0000for cubics in each component.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we refine a formula for the Tjurina number of a reducible algebroid plane curve defined over $mathbb C$ obtained in the more general case of complete intersection curves in [1]. As a byproduct, we answer the affirmative to a conjecture proposed by A. Dimca in [7]. Our results are obtained by establishing more manageable formulas to compute the colengths of fractional ideals of the local ring associated with the algebroid (not necessarily a complete intersection) curve with several branches. We then apply these results to the Jacobian ideal of a plane curve over $mathbb C$ to get a new formula for its Tjurina number and a proof of Dimca's conjecture. We end the paper by establishing a connection between the module of K"ahler differentials on the curve modulo its torsion, seen as a fractional ideal, and its Jacobian ideal, explaining the relation between the present approach and that of [1].
在这项工作中，我们完善了[1]中在完全相交曲线的更一般情况下得到的定义在 $mathbb C$ 上的可还原矢平面曲线的 Tjurina 数公式。作为副产品，我们回答了迪姆卡（A. Dimca）在[7]中提出的一个猜想。我们的结果是通过建立更易于管理的公式来计算与有多个分支的整数曲线（不一定是完全相交曲线）相关的局部环的整数理想的长度而得到的。然后，我们将这些结果应用于$mathbb C$上平面曲线的雅各理想，得到其特尤里纳数的新公式和迪姆卡猜想的证明。在论文的最后，我们建立了曲线上的 K"ahlerdifferentials module on the curve modulo its torsion（视为分数理想）与它的雅各理想之间的联系，解释了本方法与 [1] 方法之间的关系。
{"title":"Colengths of fractional ideals and Tjurina number of a reducible plane curve","authors":"Abramo Hefez, Marcelo Escudeiro Hernandes","doi":"arxiv-2409.11153","DOIUrl":"https://doi.org/arxiv-2409.11153","url":null,"abstract":"In this work, we refine a formula for the Tjurina number of a reducible\u0000algebroid plane curve defined over $mathbb C$ obtained in the more general\u0000case of complete intersection curves in [1]. As a byproduct, we answer the\u0000affirmative to a conjecture proposed by A. Dimca in [7]. Our results are\u0000obtained by establishing more manageable formulas to compute the colengths of\u0000fractional ideals of the local ring associated with the algebroid (not\u0000necessarily a complete intersection) curve with several branches. We then apply\u0000these results to the Jacobian ideal of a plane curve over $mathbb C$ to get a\u0000new formula for its Tjurina number and a proof of Dimca's conjecture. We end\u0000the paper by establishing a connection between the module of K\"ahler\u0000differentials on the curve modulo its torsion, seen as a fractional ideal, and\u0000its Jacobian ideal, explaining the relation between the present approach and\u0000that of [1].","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"187 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a class of combinatorial objects that we call ``decorated trees''. These consist of vertices, arrows and edges, where each edge is decorated by two integers (one near each of its endpoints), each arrow is decorated by an integer, and the decorations are required to satisfy certain conditions. The class of decorated trees includes different types of trees used in algebraic geometry, such as the Eisenbud and Neumann diagrams for links of singularities and the Neumann diagrams for links at infinity of algebraic plane curves. By purely combinatorial means, we recover some formulas that were previously understood to be ``topological''. In this way, we extend the generality of those formulas and show that they are in fact ``combinatorial''.
{"title":"Decorated trees","authors":"Pierrette Cassou-Noguès, Daniel Daigle","doi":"arxiv-2409.11559","DOIUrl":"https://doi.org/arxiv-2409.11559","url":null,"abstract":"We study a class of combinatorial objects that we call ``decorated trees''.\u0000These consist of vertices, arrows and edges, where each edge is decorated by\u0000two integers (one near each of its endpoints), each arrow is decorated by an\u0000integer, and the decorations are required to satisfy certain conditions. The\u0000class of decorated trees includes different types of trees used in algebraic\u0000geometry, such as the Eisenbud and Neumann diagrams for links of singularities\u0000and the Neumann diagrams for links at infinity of algebraic plane curves. By\u0000purely combinatorial means, we recover some formulas that were previously\u0000understood to be ``topological''. In this way, we extend the generality of\u0000those formulas and show that they are in fact ``combinatorial''.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Anders S. Buch, Pierre-Emmanuel Chaput, Nicolas Perrin
We prove that Schubert varieties in flag manifolds are uniquely determined by their equivariant cohomology classes, as well as a stronger result that replaces Schubert varieties with closures of Bialynicki-Birula cells under suitable conditions. This is used to prove that any two-pointed curve neighborhood representing a quantum cohomology product with a Seidel class is a Schubert variety.
{"title":"Rigidity of equivariant Schubert classes","authors":"Anders S. Buch, Pierre-Emmanuel Chaput, Nicolas Perrin","doi":"arxiv-2409.11387","DOIUrl":"https://doi.org/arxiv-2409.11387","url":null,"abstract":"We prove that Schubert varieties in flag manifolds are uniquely determined by\u0000their equivariant cohomology classes, as well as a stronger result that\u0000replaces Schubert varieties with closures of Bialynicki-Birula cells under\u0000suitable conditions. This is used to prove that any two-pointed curve\u0000neighborhood representing a quantum cohomology product with a Seidel class is a\u0000Schubert variety.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider solutions to parametrized systems of generalized polynomial equations (with real exponents) in $n$ positive variables, involving $m$ monomials with positive parameters; that is, $xinmathbb{R}^n_>$ such that ${A , (c circ x^B)=0}$ with coefficient matrix $Ainmathbb{R}^{l times m}$, exponent matrix $Binmathbb{R}^{n times m}$, parameter vector $cinmathbb{R}^m_>$, and componentwise product $circ$. As our main result, we characterize the existence of a unique solution (modulo an exponential manifold) for all parameters in terms of the relevant geometric objects of the polynomial system, namely the $textit{coefficient polytope}$ and the $textit{monomial dependency subspace}$. We show that unique existence is equivalent to the bijectivity of a certain moment/power map, and we characterize the bijectivity of this map using Hadamard's global inversion theorem. Furthermore, we provide sufficient conditions in terms of sign vectors of the geometric objects, thereby obtaining a multivariate Descartes' rule of signs for exactly one solution.
{"title":"Existence of a unique solution to parametrized systems of generalized polynomial equations","authors":"Abhishek Deshpande, Stefan Müller","doi":"arxiv-2409.11288","DOIUrl":"https://doi.org/arxiv-2409.11288","url":null,"abstract":"We consider solutions to parametrized systems of generalized polynomial\u0000equations (with real exponents) in $n$ positive variables, involving $m$\u0000monomials with positive parameters; that is, $xinmathbb{R}^n_>$ such that ${A\u0000, (c circ x^B)=0}$ with coefficient matrix $Ainmathbb{R}^{l times m}$,\u0000exponent matrix $Binmathbb{R}^{n times m}$, parameter vector\u0000$cinmathbb{R}^m_>$, and componentwise product $circ$. As our main result, we characterize the existence of a unique solution\u0000(modulo an exponential manifold) for all parameters in terms of the relevant\u0000geometric objects of the polynomial system, namely the $textit{coefficient\u0000polytope}$ and the $textit{monomial dependency subspace}$. We show that unique\u0000existence is equivalent to the bijectivity of a certain moment/power map, and\u0000we characterize the bijectivity of this map using Hadamard's global inversion\u0000theorem. Furthermore, we provide sufficient conditions in terms of sign vectors\u0000of the geometric objects, thereby obtaining a multivariate Descartes' rule of\u0000signs for exactly one solution.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"211 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study moduli spaces of Higgs sheaves valued in line bundles and the associated Hitchin maps on surfaces. We first work out Picard groups of generic (very general) spectral varieties which holds for dimension of at least 2, i.e., a Noether--Lefschetz type theorem for spectral varieties. We then apply this to obtain a necessary and sufficient condition for the non-emptyness of generic Hitchin fibers for surfaces cases. Then we move on to detect the geometry of the moduli spaces of Higgs sheaves as the second Chern class varies.
{"title":"Picard Groups of Spectral Varieties and Moduli of Higgs Sheaves","authors":"Xiaoyu Su, Bin Wang","doi":"arxiv-2409.10296","DOIUrl":"https://doi.org/arxiv-2409.10296","url":null,"abstract":"We study moduli spaces of Higgs sheaves valued in line bundles and the\u0000associated Hitchin maps on surfaces. We first work out Picard groups of generic\u0000(very general) spectral varieties which holds for dimension of at least 2,\u0000i.e., a Noether--Lefschetz type theorem for spectral varieties. We then apply\u0000this to obtain a necessary and sufficient condition for the non-emptyness of\u0000generic Hitchin fibers for surfaces cases. Then we move on to detect the\u0000geometry of the moduli spaces of Higgs sheaves as the second Chern class\u0000varies.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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