We investigate the uniqueness of decomposition of general tensors $Tin {mathbb C}^{n_1+1}otimescdotsotimes{mathbb C}^{n_r+1}$ as a sum of tensors of rank $1$. This is done extending the theory developed in a previous paper by the second author to the framework of non twd varieties. In this way we are able to prove the non generic identifiability of infinitely many partially symmetric tensors.
{"title":"Tangential Weak Defectiveness and Generic Identifiability","authors":"Alex Casarotti, M. Mella","doi":"10.1093/IMRN/RNAB091","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB091","url":null,"abstract":"We investigate the uniqueness of decomposition of general tensors $Tin {mathbb C}^{n_1+1}otimescdotsotimes{mathbb C}^{n_r+1}$ as a sum of tensors of rank $1$. This is done extending the theory developed in a previous paper by the second author to the framework of non twd varieties. In this way we are able to prove the non generic identifiability of infinitely many partially symmetric tensors.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"149 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121771669","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 complex projective plane curves with a given group of automorphisms. Let $G$ be a simple primitive subgroup of $PGL(3, mathbb{C})$, which is isomorphic to $A_{6}$, $A_{5}$ or $PSL(2, mathbb{F}_{7})$. We obtain a necessary and sufficient condition on $d$ for the existence of a nonsingular projective plane curve of degree $d$ invariant under $G$. We also study an analogous problem on integral curves.
{"title":"Projective plane curves whose automorphism groups are simple and primitive","authors":"Yusuke Yoshida","doi":"10.2996/kmj44208","DOIUrl":"https://doi.org/10.2996/kmj44208","url":null,"abstract":"We study complex projective plane curves with a given group of automorphisms. Let $G$ be a simple primitive subgroup of $PGL(3, mathbb{C})$, which is isomorphic to $A_{6}$, $A_{5}$ or $PSL(2, mathbb{F}_{7})$. We obtain a necessary and sufficient condition on $d$ for the existence of a nonsingular projective plane curve of degree $d$ invariant under $G$. We also study an analogous problem on integral curves.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123828870","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}
Pub Date : 2020-08-30DOI: 10.4310/pamq.2020.v16.n4.a10
Thuy Huong Pham, P. Marques
In this note, we give a necessary and sufficient condition for a matrix A in M to be finitely G-determined, where M is the ring of 2 x 2 matrices whose entries are formal power series over an infinite field, and G is a group acting on M by change of coordinates together with multiplication by invertible matrices from both sides.
{"title":"A note on finite determinacy of matrices","authors":"Thuy Huong Pham, P. Marques","doi":"10.4310/pamq.2020.v16.n4.a10","DOIUrl":"https://doi.org/10.4310/pamq.2020.v16.n4.a10","url":null,"abstract":"In this note, we give a necessary and sufficient condition for a matrix A in M to be finitely G-determined, where M is the ring of 2 x 2 matrices whose entries are formal power series over an infinite field, and G is a group acting on M by change of coordinates together with multiplication by invertible matrices from both sides.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"81 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132282146","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 present numerical homotopy continuation algorithms for solving systems of equations on a variety in the presence of a finite Khovanskii basis. These take advantage of Anderson's flat degeneration to a toric variety. When Anderson's degeneration embeds into projective space, our algorithm is a special case of a general toric two-step homotopy algorithm. When Anderson's degeneration is embedded in a weighted projective space, we explain how to lift to a projective space and construct an appropriate modification of the toric homotopy. Our algorithms are illustrated on several examples using Macaulay2.
{"title":"Numerical homotopies from Khovanskii bases","authors":"M. Burr, F. Sottile, Elise Walker","doi":"10.1090/mcom/3689","DOIUrl":"https://doi.org/10.1090/mcom/3689","url":null,"abstract":"We present numerical homotopy continuation algorithms for solving systems of equations on a variety in the presence of a finite Khovanskii basis. These take advantage of Anderson's flat degeneration to a toric variety. When Anderson's degeneration embeds into projective space, our algorithm is a special case of a general toric two-step homotopy algorithm. When Anderson's degeneration is embedded in a weighted projective space, we explain how to lift to a projective space and construct an appropriate modification of the toric homotopy. Our algorithms are illustrated on several examples using Macaulay2.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133210546","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 the Demazure-Lusztig operators induced by the left multiplication on partial flag manifolds $G/P$. We prove that they generate the Chern-Schwartz-MacPherson classes of Schubert cells (in equivariant cohomology), respectively their motivic Chern classes (in equivariant K theory), in any partial flag manifold. Along the way we advertise many properties of the left and right divided difference operators in cohomology and K theory, and their actions on Schubert classes. We apply this to construct left divided difference operators in equivariant quantum cohomology, and equivariant quantum K theory, generating Schubert classes, and satisfying a Leibniz rule compatible with the quantum product.
{"title":"Left Demazure–Lusztig Operators on Equivariant (Quantum) Cohomology and K-Theory","authors":"L. Mihalcea, H. Naruse, C. Su","doi":"10.1093/IMRN/RNAB049","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB049","url":null,"abstract":"We study the Demazure-Lusztig operators induced by the left multiplication on partial flag manifolds $G/P$. We prove that they generate the Chern-Schwartz-MacPherson classes of Schubert cells (in equivariant cohomology), respectively their motivic Chern classes (in equivariant K theory), in any partial flag manifold. Along the way we advertise many properties of the left and right divided difference operators in cohomology and K theory, and their actions on Schubert classes. We apply this to construct left divided difference operators in equivariant quantum cohomology, and equivariant quantum K theory, generating Schubert classes, and satisfying a Leibniz rule compatible with the quantum product.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"146 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123380418","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 note we contribute to the study of Seshadri constants on abelian and bielliptic surfaces. We specifically focus on bounds that hold on all such surfaces, depending only on the self-intersection of the ample line bundle under consideration. Our result improves previous bounds and it provides rational numbers as bounds, which are potential Seshadri constants.
{"title":"Seshadri constants on abelian and bielliptic surfaces–Potential values and lower bounds","authors":"Thomas Bauer, L. Farnik","doi":"10.1090/proc/15893","DOIUrl":"https://doi.org/10.1090/proc/15893","url":null,"abstract":"In this note we contribute to the study of Seshadri constants on abelian and bielliptic surfaces. We specifically focus on bounds that hold on all such surfaces, depending only on the self-intersection of the ample line bundle under consideration. Our result improves previous bounds and it provides rational numbers as bounds, which are potential Seshadri constants.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121751104","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}
Let $C$ be an elliptic curve, $win C$, and let $Ssubset C$ be a finite subset of cardinality at least $3$. We prove a Torelli type theorem for the moduli space of rank two parabolic vector bundles with determinant line bundle $mathcal O_C(w)$ over $(C,S)$ which are semistable with respect to a weight vector $big(frac{1}{2}, dots, frac{1}{2}big)$.
{"title":"A Torelli theorem for moduli spaces of parabolic vector bundles over an elliptic curve","authors":"T. Fassarella, Luana Justo","doi":"10.1090/proc/15937","DOIUrl":"https://doi.org/10.1090/proc/15937","url":null,"abstract":"Let $C$ be an elliptic curve, $win C$, and let $Ssubset C$ be a finite subset of cardinality at least $3$. We prove a Torelli type theorem for the moduli space of rank two parabolic vector bundles with determinant line bundle $mathcal O_C(w)$ over $(C,S)$ which are semistable with respect to a weight vector $big(frac{1}{2}, dots, frac{1}{2}big)$.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124833551","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}
T. Dupuy, E. Katz, J. Rabinoff, D. Zureick-Brown introduced the module of total $p$-differentials for a ring over $Z/p^2Z$. We study the same construction for a ring over $Z_{(p)}$ and prove a regularity criterion. For a local ring, the tensor product with the residue field is constructed in a different way by O. Gabber, L. Ramero. In another article arXiv:2006.00448, we use the sheaf of FW-differentials to define the cotangent bundle and the micro-support of an etale sheaf.
T. Dupuy, E. Katz, J. Rabinoff, D. Zureick-Brown引入了$Z/p^2Z$上环的总$p$微分模。我们研究了$Z_{(p)}$上的环的相同构造，并证明了一个正则性准则。对于局部环，O. Gabber, L. Ramero用不同的方法构造了张量积与剩余场。在另一篇文章[xiv:2006.00448]中，我们用w -微分束定义了一个函数束的共切束和微支撑。
{"title":"Frobenius-Witt differentials and regularity.","authors":"Takeshi Saito","doi":"10.2140/ant.2022.16.369","DOIUrl":"https://doi.org/10.2140/ant.2022.16.369","url":null,"abstract":"T. Dupuy, E. Katz, J. Rabinoff, D. Zureick-Brown introduced the module of total $p$-differentials for a ring over $Z/p^2Z$. We study the same construction for a ring over $Z_{(p)}$ and prove a regularity criterion. For a local ring, the tensor product with the residue field is constructed in a different way by O. Gabber, L. Ramero. In another article arXiv:2006.00448, we use the sheaf of FW-differentials to define the cotangent bundle and the micro-support of an etale sheaf.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114152450","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}
Pub Date : 2020-07-30DOI: 10.1017/9781108877831.002
Asher Auel
We study the Brill-Noether theory of curves on K3 surfaces that are Hodge theoretically associated to cubic fourfolds of discriminant 14. We prove that any smooth curve in the polarization class has maximal Clifford index and deduce that a cubic fourfold contains disjoint planes if and only if it admits a Brill-Noether special associated K3 surface of degree 14. As an application, the complement of the pfaffian locus, inside the Noether-Lefschetz divisor of discriminant 14 in the moduli space of cubic fourfolds, is contained in the irreducible locus of cubic fourfolds containing two disjoint planes.
{"title":"Brill-Noether special cubic fourfolds of discriminant 14","authors":"Asher Auel","doi":"10.1017/9781108877831.002","DOIUrl":"https://doi.org/10.1017/9781108877831.002","url":null,"abstract":"We study the Brill-Noether theory of curves on K3 surfaces that are Hodge theoretically associated to cubic fourfolds of discriminant 14. We prove that any smooth curve in the polarization class has maximal Clifford index and deduce that a cubic fourfold contains disjoint planes if and only if it admits a Brill-Noether special associated K3 surface of degree 14. As an application, the complement of the pfaffian locus, inside the Noether-Lefschetz divisor of discriminant 14 in the moduli space of cubic fourfolds, is contained in the irreducible locus of cubic fourfolds containing two disjoint planes.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130402810","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}
L. Bossinger, F. Mohammadi, Alfredo N'ajera Ch'avez
Let $V$ be the weighted projective variety defined by a weighted homogeneous ideal $J$ and $C$ a maximal cone in the Grobner fan of $J$ with $m$ rays. We construct a flat family over $mathbb A^m$ that assembles the Grobner degenerations of $V$ associated with all faces of $C$. This is a multi-parameter generalization of the classical one-parameter Grobner degeneration associated to a weight. We show that our family can be constructed from Kaveh-Manon's recent work on the classification of toric flat families over toric varieties: it is the pullback of a toric family defined by a Rees algebra with base $X_C$ (the toric variety associated to $C$) along the universal torsor $mathbb A^m to X_C$. We apply this construction to the Grassmannians ${rm Gr}(2,mathbb C^n)$ with their Plucker embeddings and the Grassmannian ${rm Gr}(3,mathbb C^6)$ with its cluster embedding. In each case there exists a unique maximal Grobner cone whose associated initial ideal is the Stanley-Reisner ideal of the cluster complex. We show that the corresponding cluster algebra with universal coefficients arises as the algebra defining the flat family associated to this cone. Further, for ${rm Gr}(2,mathbb C^n)$ we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be recovered as tropicalized cluster mutation.
设$V$为加权齐次理想$J$所定义的加权射影变，$C$是$J$的Grobner扇形中具有$m$射线的极大锥。我们在$mathbb a ^m$上构造了一个平面族，它集合了与$C$的所有面相关的$V$的Grobner退化。这是与权值相关的经典单参数Grobner退化的多参数推广。我们证明了我们的族可以从Kaveh-Manon最近关于环型平面族的分类的工作中构造出来:它是一个由以$X_C$为基底的Rees代数定义的环型族(与$C$相关的环型族)沿着$mathbb a ^m 到X_C$的回调。我们将这种构造应用于Grassmannian ${rm Gr}(2，mathbb C^n)$及其拔毛器嵌入和Grassmannian ${rm Gr}(3，mathbb C^6)$及其聚类嵌入。在每种情况下，都存在一个唯一的极大Grobner锥，其关联的初始理想是簇复合体的Stanley-Reisner理想。我们证明了相应的具有泛系数的聚类代数作为定义与该锥相关的平面族的代数而产生。此外，对于${rm Gr}(2，mathbb C^n)$，我们展示了如何将牛顿-奥库科夫体的Escobar-Harada突变恢复为热带化簇突变。
{"title":"Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras","authors":"L. Bossinger, F. Mohammadi, Alfredo N'ajera Ch'avez","doi":"10.3842/SIGMA.2021.059","DOIUrl":"https://doi.org/10.3842/SIGMA.2021.059","url":null,"abstract":"Let $V$ be the weighted projective variety defined by a weighted homogeneous ideal $J$ and $C$ a maximal cone in the Grobner fan of $J$ with $m$ rays. We construct a flat family over $mathbb A^m$ that assembles the Grobner degenerations of $V$ associated with all faces of $C$. This is a multi-parameter generalization of the classical one-parameter Grobner degeneration associated to a weight. We show that our family can be constructed from Kaveh-Manon's recent work on the classification of toric flat families over toric varieties: it is the pullback of a toric family defined by a Rees algebra with base $X_C$ (the toric variety associated to $C$) along the universal torsor $mathbb A^m to X_C$. \u0000We apply this construction to the Grassmannians ${rm Gr}(2,mathbb C^n)$ with their Plucker embeddings and the Grassmannian ${rm Gr}(3,mathbb C^6)$ with its cluster embedding. In each case there exists a unique maximal Grobner cone whose associated initial ideal is the Stanley-Reisner ideal of the cluster complex. We show that the corresponding cluster algebra with universal coefficients arises as the algebra defining the flat family associated to this cone. Further, for ${rm Gr}(2,mathbb C^n)$ we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be recovered as tropicalized cluster mutation.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130085817","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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