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The Degeneracy Loci for Smooth Moduli of Sheaves 剪切的光滑模数的退化位置
arXiv - MATH - Algebraic Geometry Pub Date : 2024-08-26 DOI: arxiv-2408.14021
Yu Zhao
{"title":"The Degeneracy Loci for Smooth Moduli of Sheaves","authors":"Yu Zhao","doi":"arxiv-2408.14021","DOIUrl":"https://doi.org/arxiv-2408.14021","url":null,"abstract":"Let S be a smooth projective surface over $mathbb{C}$. We prove that, under\u0000certain technical assumptions, the degeneracy locus of the universal sheaf over\u0000the moduli space of stable sheaves is either empty or an irreducible\u0000Cohen-Macaulay variety of the expected dimension. We also provide a criterion\u0000for when the degeneracy locus is non-empty. This result generalizes the work of\u0000Bayer, Chen, and Jiang for the Hilbert scheme of points on surfaces. The above result is a special case of a general phenomenon: for a perfect\u0000complex of Tor-amplitude [0,1], the geometry of the degeneracy locus is closely\u0000related to the geometry of the derived Grassmannian. We analyze their\u0000birational geometry and relate it to the incidence varieties of derived\u0000Grassmannians. As a corollary, we prove a statement previously claimed by the\u0000author in arXiv:2408.06860.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220307","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
Distinguishing Calabi-Yau Topology using Machine Learning 利用机器学习区分 Calabi-Yau 拓扑学
arXiv - MATH - Algebraic Geometry Pub Date : 2024-08-09 DOI: arxiv-2408.05076
Yang-Hui He, Zhi-Gang Yao, Shing-Tung Yau
{"title":"Distinguishing Calabi-Yau Topology using Machine Learning","authors":"Yang-Hui He, Zhi-Gang Yao, Shing-Tung Yau","doi":"arxiv-2408.05076","DOIUrl":"https://doi.org/arxiv-2408.05076","url":null,"abstract":"While the earliest applications of AI methodologies to pure mathematics and\u0000theoretical physics began with the study of Hodge numbers of Calabi-Yau\u0000manifolds, the topology type of such manifold also crucially depend on their\u0000intersection theory. Continuing the paradigm of machine learning algebraic\u0000geometry, we here investigate the triple intersection numbers, focusing on\u0000certain divisibility invariants constructed therefrom, using the Inception\u0000convolutional neural network. We find $sim90%$ accuracies in prediction in a\u0000standard fivefold cross-validation, signifying that more sophisticated tasks of\u0000identification of manifold topologies can also be performed by machine\u0000learning.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141968977","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
Quartic surfaces with an outer Galois point and K3 surfaces with an automorphism of order 4 具有外伽罗瓦点的四曲面和具有 4 阶自形性的 K3 曲面
arXiv - MATH - Algebraic Geometry Pub Date : 2024-08-08 DOI: arxiv-2408.04137
Kei Miura, Shingo Taki
{"title":"Quartic surfaces with an outer Galois point and K3 surfaces with an automorphism of order 4","authors":"Kei Miura, Shingo Taki","doi":"arxiv-2408.04137","DOIUrl":"https://doi.org/arxiv-2408.04137","url":null,"abstract":"We prove that there exists a one-to-one correspondence between smooth quartic\u0000surfaces with an outer Galois point and K3 surfaces with a certain automorphism\u0000of order 4. Furthermore, we characterize quartic surfaces with two or more\u0000outer Galois points as K3 surfaces.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141936745","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
$ell$-away ACM line bundles on a nonsingular cubic surface 非正交立方体表面上的 $ell$-away ACM 线束
arXiv - MATH - Algebraic Geometry Pub Date : 2024-08-08 DOI: arxiv-2408.04464
Debojyoti Bhattacharya, A. J. Parameswaran, Jagadish Pine
{"title":"$ell$-away ACM line bundles on a nonsingular cubic surface","authors":"Debojyoti Bhattacharya, A. J. Parameswaran, Jagadish Pine","doi":"arxiv-2408.04464","DOIUrl":"https://doi.org/arxiv-2408.04464","url":null,"abstract":"Let $X subset mathbb P^3$ be a nonsingular cubic hypersurface. Faenzi\u0000(cite{F}) and later Pons-Llopis and Tonini (cite{PLT}) have completely\u0000characterized ACM line bundles over $X$. As a natural continuation of their\u0000study in the non-ACM direction, in this paper, we completely classify\u0000$ell$-away ACM line bundles (introduced recently by Gawron and Genc\u0000(cite{GG})) over $X$, when $ell leq 2$. For $ellgeq 3$, we give examples\u0000of $ell$-away ACM line bundles on $X$ and for each $ell geq 1$, we establish\u0000the existence of smooth hypersurfaces $X^{(d)}$ of degree $d >ell$ in $mathbb\u0000P^3$ admitting $ell$-away ACM line bundles.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"77 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141936808","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
An inductive approach to global generation of adjoint series on irregular varieties 不规则变体上邻接数列全局生成的归纳法
arXiv - MATH - Algebraic Geometry Pub Date : 2024-08-08 DOI: arxiv-2408.04733
Houari Benammar Ammar
{"title":"An inductive approach to global generation of adjoint series on irregular varieties","authors":"Houari Benammar Ammar","doi":"arxiv-2408.04733","DOIUrl":"https://doi.org/arxiv-2408.04733","url":null,"abstract":"In this paper, we show how to prove the global generation of adjoint linear\u0000systems on irregular varieties inductively. For instance, we prove that\u0000Fujita's conjecture holds for irregular varieties of dimension $n$ with nef\u0000anticanonical bundle, assuming it holds for lower-dimensional varieties and\u0000under mild conditions.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"195 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141936807","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
An Abel-Jacobi theorem for metrized complexes of Riemann surfaces 黎曼曲面元化复数的阿贝尔-雅可比定理
arXiv - MATH - Algebraic Geometry Pub Date : 2024-08-07 DOI: arxiv-2408.03851
Maximilian C. E. Hofmann, Martin Ulirsch
{"title":"An Abel-Jacobi theorem for metrized complexes of Riemann surfaces","authors":"Maximilian C. E. Hofmann, Martin Ulirsch","doi":"arxiv-2408.03851","DOIUrl":"https://doi.org/arxiv-2408.03851","url":null,"abstract":"Motivated by the recent surge of interest in the geometry of hybrid spaces,\u0000we prove an Abel-Jacobi theorem for a metrized complex of Riemann surfaces,\u0000generalizing both the classical Abel-Jacobi theorem and its tropical analogue.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"79 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141936744","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 the genus of projective curves not contained in hypersurfaces of given degree, II 关于不包含在给定度数超曲面中的投影曲线之属，II
arXiv - MATH - Algebraic Geometry Pub Date : 2024-08-07 DOI: arxiv-2408.03715
Vincenzo Di Gennaro, Giambattista Marini
{"title":"On the genus of projective curves not contained in hypersurfaces of given degree, II","authors":"Vincenzo Di Gennaro, Giambattista Marini","doi":"arxiv-2408.03715","DOIUrl":"https://doi.org/arxiv-2408.03715","url":null,"abstract":"Fix integers $rgeq 4$ and $igeq 2$. Let $C$ be a non-degenerate, reduced\u0000and irreducible complex projective curve in $mathbb P^r$, of degree $d$, not\u0000contained in a hypersurface of degree $leq i$. Let $p_a(C)$ be the arithmetic\u0000genus of $C$. Continuing previous research, under the assumption $dgg\u0000max{r,i}$, in the present paper we exhibit a Castelnuovo bound $G_0(r;d,i)$\u0000for $p_a(C)$. In general, we do not know whether this bound is sharp. However,\u0000we are able to prove it is sharp when $i=2$, $r=6$ and $dequiv 0,3,6$ (mod\u0000$9$). Moreover, when $i=2$, $rgeq 9$, $r$ is divisible by $3$, and $dequiv 0$\u0000(mod $r(r+3)/6$), we prove that if $G_0(r;d,i)$ is not sharp, then for the\u0000maximal value of $p_a(C)$ there are only three possibilities. The case in which\u0000$i=2$ and $r$ is not divisible by $3$ has already been examined in the\u0000literature. We give some information on the extremal curves.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141968978","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 dynamics of the Hesse derivative on the $j$-invariant 海瑟导数在不变量 $j$ 上的动态变化
arXiv - MATH - Algebraic Geometry Pub Date : 2024-08-07 DOI: arxiv-2408.04117
Jake Kettinger
{"title":"The dynamics of the Hesse derivative on the $j$-invariant","authors":"Jake Kettinger","doi":"arxiv-2408.04117","DOIUrl":"https://doi.org/arxiv-2408.04117","url":null,"abstract":"The $j$-invariant of a cubic curve is an isomorphism invariant parameterized\u0000by the moduli space of elliptic curves. The Hesse derivative of a curve $V(f)$\u0000given by the homogeneous polynomial $f$ is $V(mathcal{H}(f))$ where\u0000$mathcal{H}(f)$ is a the determinant of the Hesse matrix of $f$. In this\u0000paper, we compute the $j$-invariant of the Hesse derivative of a cubic curve\u0000$C$ in terms of the $j$-invariant of $C$, getting a rational function on the\u0000Riemann sphere. We then analyze the dynamics of this rational function, and\u0000investigate when a cubic curve is isomorphic to its $n$-fold Hesse derivative.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"104 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141936809","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
An abundance-type result for the tangent bundles of smooth Fano varieties 光滑法诺变体切线束的丰度型结果
arXiv - MATH - Algebraic Geometry Pub Date : 2024-08-07 DOI: arxiv-2408.03799
Juanyong Wang
{"title":"An abundance-type result for the tangent bundles of smooth Fano varieties","authors":"Juanyong Wang","doi":"arxiv-2408.03799","DOIUrl":"https://doi.org/arxiv-2408.03799","url":null,"abstract":"In this paper we prove the following abundance-type result: for any smooth\u0000Fano variety $X$, the tangent bundle $T_X$ is nef if and only if it is big and\u0000semiample in the sense that the tautological line bundle\u0000$mathscr{O}_{mathbb{P}T_X}(1)$ is so, by which we establish a weak form of\u0000the Campana-Peternell conjecture (Camapan-Peternell, 1991).","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141936900","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
Symplectic moduli space of 1-dimensional sheaves on Poisson surfaces 泊松曲面上一维剪切的交映模空间
arXiv - MATH - Algebraic Geometry Pub Date : 2024-08-06 DOI: arxiv-2408.02955
Indranil Biswas, Dimitri Markushevich
{"title":"Symplectic moduli space of 1-dimensional sheaves on Poisson surfaces","authors":"Indranil Biswas, Dimitri Markushevich","doi":"arxiv-2408.02955","DOIUrl":"https://doi.org/arxiv-2408.02955","url":null,"abstract":"We show that the Poisson structure on the smooth locus of a moduli space of\u00001-dimensional sheaves on a Poisson projective surface $X$ over $mathbb C$ is a\u0000reduction of a natural symplectic structure.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"90 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141936741","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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