Lucie Devey, Milena Hering, Katharina Jochemko, Hendrik Süß
We show that for every toric surface apart from the projective plane and a product of two projective lines and every ample line bundle there exists a polarisation such that the syzygy bundle associated to sufficiently high powers of the line bundle is not slope stable.
{"title":"On the instability of syzygy bundles on toric surfaces","authors":"Lucie Devey, Milena Hering, Katharina Jochemko, Hendrik Süß","doi":"arxiv-2409.04666","DOIUrl":"https://doi.org/arxiv-2409.04666","url":null,"abstract":"We show that for every toric surface apart from the projective plane and a\u0000product of two projective lines and every ample line bundle there exists a\u0000polarisation such that the syzygy bundle associated to sufficiently high powers\u0000of the line bundle is not slope stable.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220381","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 survey on algebraically elliptic varieties in the sense of Gromov.
我们研究格罗莫夫意义上的代数椭圆变种。
{"title":"Algebraic Gromov ellipticity: a brief survey","authors":"Mikhail Zaidenberg","doi":"arxiv-2409.04776","DOIUrl":"https://doi.org/arxiv-2409.04776","url":null,"abstract":"We survey on algebraically elliptic varieties in the sense of Gromov.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220378","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 a projective morphism $f:Xto Y$ from a complex space to a complex manifold, we prove the Griffiths semi-positivity and minimal extension property of the direct image sheaf $f_ast(mathscr{F})$. Here, $mathscr{F}$ is a coherent sheaf on $X$, which consists of the Grauert-Riemenschneider dualizing sheaf, a multiplier ideal sheaf, and a variation of Hodge structure (or more generally, a tame harmonic bundle).
{"title":"Minimal extension property of direct images","authors":"Chen Zhao","doi":"arxiv-2409.04754","DOIUrl":"https://doi.org/arxiv-2409.04754","url":null,"abstract":"Given a projective morphism $f:Xto Y$ from a complex space to a complex\u0000manifold, we prove the Griffiths semi-positivity and minimal extension property\u0000of the direct image sheaf $f_ast(mathscr{F})$. Here, $mathscr{F}$ is a\u0000coherent sheaf on $X$, which consists of the Grauert-Riemenschneider dualizing\u0000sheaf, a multiplier ideal sheaf, and a variation of Hodge structure (or more\u0000generally, a tame harmonic bundle).","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"09 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220379","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 article we study the extendability of a smooth projective variety by degenerating it to a ribbon. We apply the techniques to study extendability of Calabi-Yau threefolds $X_t$ that are general deformations of Calabi-Yau double covers of Fano threefolds of Picard rank $1$. The Calabi-Yau threefolds $X_t hookrightarrow mathbb{P}^{N_l}$, embedded by the complete linear series $|lA_t|$, where $A_t$ is the generator of Pic$(X_t)$, $l geq j$ and $j$ is the index of $Y$, are general elements of a unique irreducible component $mathscr{H}_l^Y$ of the Hilbert scheme which contains embedded Calabi-Yau ribbons on $Y$ as a special locus. For $l = j$, using the classification of Mukai varieties, we show that the general Calabi-Yau threefold parameterized by $mathscr{H}_j^Y$ is as many times smoothly extendable as $Y$ itself. On the other hand, we find for each deformation type $Y$, an effective integer $l_Y$ such that for $l geq l_Y$, the general Calabi-Yau threefold parameterized by $mathscr{H}_l^Y$ is not extendable. These results provide a contrast and a parallel with the lower dimensional analogues; namely, $K3$ surfaces and canonical curves, which stems from the following result we prove: for $l geq l_Y$, the general hyperplane sections of elements of $mathscr{H}_l^Y$ fill out an entire irreducible component $mathscr{S}_l^Y$ of the Hilbert scheme of canonical surfaces which are precisely $1-$ extendable with $mathscr{H}^Y_l$ being the unique component dominating $mathscr{S}_l^Y$. The contrast lies in the fact that for polarized $K3$ surfaces of large degree, the canonical curve sections do not fill out an entire component while the parallel is in the fact that the canonical curve sections are exactly one-extendable.
{"title":"Extendability of projective varieties via degeneration to ribbons with applications to Calabi-Yau threefolds","authors":"Purnaprajna Bangere, Jayan Mukherjee","doi":"arxiv-2409.03960","DOIUrl":"https://doi.org/arxiv-2409.03960","url":null,"abstract":"In this article we study the extendability of a smooth projective variety by\u0000degenerating it to a ribbon. We apply the techniques to study extendability of\u0000Calabi-Yau threefolds $X_t$ that are general deformations of Calabi-Yau double\u0000covers of Fano threefolds of Picard rank $1$. The Calabi-Yau threefolds $X_t\u0000hookrightarrow mathbb{P}^{N_l}$, embedded by the complete linear series\u0000$|lA_t|$, where $A_t$ is the generator of Pic$(X_t)$, $l geq j$ and $j$ is the\u0000index of $Y$, are general elements of a unique irreducible component\u0000$mathscr{H}_l^Y$ of the Hilbert scheme which contains embedded Calabi-Yau\u0000ribbons on $Y$ as a special locus. For $l = j$, using the classification of\u0000Mukai varieties, we show that the general Calabi-Yau threefold parameterized by\u0000$mathscr{H}_j^Y$ is as many times smoothly extendable as $Y$ itself. On the\u0000other hand, we find for each deformation type $Y$, an effective integer $l_Y$\u0000such that for $l geq l_Y$, the general Calabi-Yau threefold parameterized by\u0000$mathscr{H}_l^Y$ is not extendable. These results provide a contrast and a\u0000parallel with the lower dimensional analogues; namely, $K3$ surfaces and\u0000canonical curves, which stems from the following result we prove: for $l geq\u0000l_Y$, the general hyperplane sections of elements of $mathscr{H}_l^Y$ fill out\u0000an entire irreducible component $mathscr{S}_l^Y$ of the Hilbert scheme of\u0000canonical surfaces which are precisely $1-$ extendable with $mathscr{H}^Y_l$\u0000being the unique component dominating $mathscr{S}_l^Y$. The contrast lies in\u0000the fact that for polarized $K3$ surfaces of large degree, the canonical curve\u0000sections do not fill out an entire component while the parallel is in the fact\u0000that the canonical curve sections are exactly one-extendable.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220364","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 $mathcal{A}$ be a hyperplane arrangement in a complex projective space. It is an open question if the degree one cohomology jump loci (with complex coefficients) are determined by the combinatorics of $mathcal{A}$. By the work of Falk and Yuzvinsky cite{FY}, all the irreducible components passing through the origin are determined by the multinet structure, which are combinatorially determined. Denham and Suciu introduced the pointed multinet structure to obtain examples of arrangements with translated positive-dimensional components in the degree one cohomology jump loci cite{DS}. Suciu asked the question if all translated positive-dimensional components appear in this manner cite{Suc14}. In this paper, we show that the double star arrangement introduced by Ishibashi, Sugawara and Yoshinaga cite[Example 3.2]{ISY22} gives a negative answer to this question.
让 $mathcal{A}$ 是复投影空间中的一个超平面排列。$mathcal{A}$ 的组合学是否决定了一度同调跃迁位置（具有复系数），这是一个悬而未决的问题。根据 Falk 和 Yuzvinsky cite{FY}的研究，所有通过原点的不可还原成分都是由多网结构决定的，而多网结构是由组合决定的。德纳姆和苏修引入尖多内特结构，以获得在一度同调跃迁位置（the degree one cohomology jump loci cite{DS}）中具有翻译正维成分的排列的例子。Suciu 提出了一个问题：是否所有翻译的正维成分都以这种方式出现？在本文中，我们证明了石桥、菅原和吉永引入的双星排列 （cite[例 3.2]{ISY22} 给出了这个问题的否定答案。
{"title":"Double star arrangement and the pointed multinet","authors":"Yongqiang Liu, Wentao Xie","doi":"arxiv-2409.04032","DOIUrl":"https://doi.org/arxiv-2409.04032","url":null,"abstract":"Let $mathcal{A}$ be a hyperplane arrangement in a complex projective space.\u0000It is an open question if the degree one cohomology jump loci (with complex\u0000coefficients) are determined by the combinatorics of $mathcal{A}$. By the work\u0000of Falk and Yuzvinsky cite{FY}, all the irreducible components passing through\u0000the origin are determined by the multinet structure, which are combinatorially\u0000determined. Denham and Suciu introduced the pointed multinet structure to\u0000obtain examples of arrangements with translated positive-dimensional components\u0000in the degree one cohomology jump loci cite{DS}. Suciu asked the question if\u0000all translated positive-dimensional components appear in this manner\u0000cite{Suc14}. In this paper, we show that the double star arrangement\u0000introduced by Ishibashi, Sugawara and Yoshinaga cite[Example 3.2]{ISY22} gives\u0000a negative answer to this question.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220383","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 study whether a given morphism $f$ from the tangent bundle of $mathbb{P}^1$ to a balanced vector bundle of degree $(n+1)d$ is induced by the restriction of the tangent bundle $T_{mathbb{P}^n}$ to a rational curve of degree $d$ in $mathbb{P}^n$. We propose a conjecture on this problem based on Mathematica computations of some examples and provide computer-assisted proof of the conjecture for certain values of $n$ and $d$.
{"title":"The Syzygy Matrix and the Differential for Rational Curves in Projective Space","authors":"Chen Song","doi":"arxiv-2409.03985","DOIUrl":"https://doi.org/arxiv-2409.03985","url":null,"abstract":"In this paper, we study whether a given morphism $f$ from the tangent bundle\u0000of $mathbb{P}^1$ to a balanced vector bundle of degree $(n+1)d$ is induced by\u0000the restriction of the tangent bundle $T_{mathbb{P}^n}$ to a rational curve of\u0000degree $d$ in $mathbb{P}^n$. We propose a conjecture on this problem based on\u0000Mathematica computations of some examples and provide computer-assisted proof\u0000of the conjecture for certain values of $n$ and $d$.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220323","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}
This paper develops the geometry of rational functions on non-singular real algebraic varieties that are locally bounded. First various basic geometric and algebraic results regarding these functions are established in any dimension, culminating with a version of {L}ojasiewicz's inequality. The geometry is further developed for the case of dimension 2, where it can be shown that there exist many of the usual correspondences between the algebra and geometry of these functions that one expects from complex algebraic geometry and from other classes of functions in real algebraic geometry such as regulous functions.
{"title":"The Geometry of Locally Bounded Rational Functions","authors":"Victor Delage, Goulwen Fichou, Aftab Patel","doi":"arxiv-2409.04232","DOIUrl":"https://doi.org/arxiv-2409.04232","url":null,"abstract":"This paper develops the geometry of rational functions on non-singular real\u0000algebraic varieties that are locally bounded. First various basic geometric and\u0000algebraic results regarding these functions are established in any dimension,\u0000culminating with a version of {L}ojasiewicz's inequality. The geometry is\u0000further developed for the case of dimension 2, where it can be shown that there\u0000exist many of the usual correspondences between the algebra and geometry of\u0000these functions that one expects from complex algebraic geometry and from other\u0000classes of functions in real algebraic geometry such as regulous functions.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220382","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}
The anticanonical complex is a combinatorial tool that was invented to extend the features of the Fano polytope from toric geometry to wider classes of varieties. In this note we show that the Gorenstein index of Fano varieties with torus action of complexity one (and even more general of the so-called general arrangement varieties) can be read off its anticanonical complex in terms of lattice distances in full analogy to the toric Fano polytope. As an application we give concrete bounds on the defining data of almost homogeneous Fano threefolds of Picard number one having a reductive automorphism group with two-dimensional maximal torus depending on their Gorenstein index.
{"title":"On a combinatorial description of the Gorenstein index for varieties with torus action","authors":"Philipp Iber, Eva Reinert, Milena Wrobel","doi":"arxiv-2409.03649","DOIUrl":"https://doi.org/arxiv-2409.03649","url":null,"abstract":"The anticanonical complex is a combinatorial tool that was invented to extend\u0000the features of the Fano polytope from toric geometry to wider classes of\u0000varieties. In this note we show that the Gorenstein index of Fano varieties\u0000with torus action of complexity one (and even more general of the so-called\u0000general arrangement varieties) can be read off its anticanonical complex in\u0000terms of lattice distances in full analogy to the toric Fano polytope. As an\u0000application we give concrete bounds on the defining data of almost homogeneous\u0000Fano threefolds of Picard number one having a reductive automorphism group with\u0000two-dimensional maximal torus depending on their Gorenstein index.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220278","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}
Rationality is not a constructible property in families. In this article, we consider stronger notions of rationality and study their behavior in families of Fano varieties. We first show that being toric is a constructible property in families of Fano varieties. The second main result of this article concerns an intermediate notion that lies between toric and rational varieties, namely cluster type varieties. A cluster type $mathbb Q$-factorial Fano variety contains an open dense algebraic torus, but the variety does not need to be endowed with a torus action. We prove that, in families of $mathbb Q$-factorial terminal Fano varieties, being of cluster type is a constructible condition. As a consequence, we show that there are finitely many smooth families parametrizing $n$-dimensional smooth cluster type Fano varieties.
{"title":"Toricity in families of Fano varieties","authors":"Lena Ji, Joaquín Moraga","doi":"arxiv-2409.03564","DOIUrl":"https://doi.org/arxiv-2409.03564","url":null,"abstract":"Rationality is not a constructible property in families. In this article, we\u0000consider stronger notions of rationality and study their behavior in families\u0000of Fano varieties. We first show that being toric is a constructible property\u0000in families of Fano varieties. The second main result of this article concerns\u0000an intermediate notion that lies between toric and rational varieties, namely\u0000cluster type varieties. A cluster type $mathbb Q$-factorial Fano variety\u0000contains an open dense algebraic torus, but the variety does not need to be\u0000endowed with a torus action. We prove that, in families of $mathbb\u0000Q$-factorial terminal Fano varieties, being of cluster type is a constructible\u0000condition. As a consequence, we show that there are finitely many smooth\u0000families parametrizing $n$-dimensional smooth cluster type Fano varieties.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220279","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}
Jacobson developed a counterpart of Galois theory for purely inseparable field extensions in positive characteristic. In his theory, a certain type of derivations replace the role of the generators of Galois groups. This article provides a convenient criterion for determining such derivations in dimension two. We also present examples demonstrating the efficiency of our criterion.
{"title":"A criterion for $p$-closedness of derivations in dimension two","authors":"Kentaro Mitsui, Nobuo Sato","doi":"arxiv-2409.03442","DOIUrl":"https://doi.org/arxiv-2409.03442","url":null,"abstract":"Jacobson developed a counterpart of Galois theory for purely inseparable\u0000field extensions in positive characteristic. In his theory, a certain type of\u0000derivations replace the role of the generators of Galois groups. This article\u0000provides a convenient criterion for determining such derivations in dimension\u0000two. We also present examples demonstrating the efficiency of our criterion.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"172 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220281","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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