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On the instability of syzygy bundles on toric surfaces 论环状曲面上对称束的不稳定性
arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-07 DOI: arxiv-2409.04666
Lucie Devey, Milena Hering, Katharina Jochemko, Hendrik Süß
{"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":null,"pages":null},"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}
引用次数: 0
Algebraic Gromov ellipticity: a brief survey 代数格罗莫夫椭圆性：简要考察
arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-07 DOI: arxiv-2409.04776
Mikhail Zaidenberg
{"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":null,"pages":null},"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}
引用次数: 0
Minimal extension property of direct images 直接图像的最小扩展特性
arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-07 DOI: arxiv-2409.04754
Chen Zhao
{"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":null,"pages":null},"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}
引用次数: 0
Extendability of projective varieties via degeneration to ribbons with applications to Calabi-Yau threefolds 投影变种通过退化到带的可扩展性及其在 Calabi-Yau 三折中的应用
arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-06 DOI: arxiv-2409.03960
Purnaprajna Bangere, Jayan Mukherjee
{"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":null,"pages":null},"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}
引用次数: 0
Double star arrangement and the pointed multinet 双星排列和尖顶多网
arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-06 DOI: arxiv-2409.04032
Yongqiang Liu, Wentao Xie
{"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":null,"pages":null},"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}
引用次数: 0
The Syzygy Matrix and the Differential for Rational Curves in Projective Space 投影空间有理曲线的对称矩阵和微分
arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-06 DOI: arxiv-2409.03985
Chen Song
{"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":null,"pages":null},"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}
引用次数: 0
The Geometry of Locally Bounded Rational Functions 局部有界有理函数的几何学
arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-06 DOI: arxiv-2409.04232
Victor Delage, Goulwen Fichou, Aftab Patel
{"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":null,"pages":null},"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}
引用次数: 0
On a combinatorial description of the Gorenstein index for varieties with torus action 关于具有环作用的品种的戈伦斯坦指数的组合描述
arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-05 DOI: arxiv-2409.03649
Philipp Iber, Eva Reinert, Milena Wrobel
{"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":null,"pages":null},"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}
引用次数: 0
Toricity in families of Fano varieties 法诺变体族的遍历性
arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-05 DOI: arxiv-2409.03564
Lena Ji, Joaquín Moraga
{"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":null,"pages":null},"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}
引用次数: 0
A criterion for $p$-closedness of derivations in dimension two 二维导数的$p$封闭性标准
arXiv - MATH - Algebraic Geometry Pub Date : 2024-09-05 DOI: arxiv-2409.03442
Kentaro Mitsui, Nobuo Sato
{"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":null,"pages":null},"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}
引用次数: 0
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