Let S be a smooth projective surface over $mathbb{C}$. We prove that, under certain technical assumptions, the degeneracy locus of the universal sheaf over the moduli space of stable sheaves is either empty or an irreducible Cohen-Macaulay variety of the expected dimension. We also provide a criterion for when the degeneracy locus is non-empty. This result generalizes the work of Bayer, 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 complex of Tor-amplitude [0,1], the geometry of the degeneracy locus is closely related to the geometry of the derived Grassmannian. We analyze their birational geometry and relate it to the incidence varieties of derived Grassmannians. As a corollary, we prove a statement previously claimed by the author in arXiv:2408.06860.
设 S 是$mathbb{C}$上的光滑投影面。我们证明，在某些技术假设下，稳定剪子模空间上的普遍剪子的退化位点要么是空的，要么是预期维数的不可还原的科恩-麦考莱（Cohen-Macaulay）簇。我们还提供了一个判据来判定何时退化位置是非空的。这一结果推广了拜尔、陈和江对曲面上点的希尔伯特方案的研究。上述结果是一个普遍现象的特例：对于 Tor 振幅 [0,1] 的完美复数，退化位点的几何与衍生格拉斯曼几何密切相关。我们分析了它们的配位几何，并将其与派生格拉斯曼的入射品种联系起来。作为推论，我们证明了作者之前在 arXiv:2408.06860 中提出的一个声明。
{"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}
While the earliest applications of AI methodologies to pure mathematics and theoretical physics began with the study of Hodge numbers of Calabi-Yau manifolds, the topology type of such manifold also crucially depend on their intersection theory. Continuing the paradigm of machine learning algebraic geometry, we here investigate the triple intersection numbers, focusing on certain divisibility invariants constructed therefrom, using the Inception convolutional neural network. We find $sim90%$ accuracies in prediction in a standard fivefold cross-validation, signifying that more sophisticated tasks of identification of manifold topologies can also be performed by machine learning.
{"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}
We prove that there exists a one-to-one correspondence between smooth quartic surfaces with an outer Galois point and K3 surfaces with a certain automorphism of order 4. Furthermore, we characterize quartic surfaces with two or more outer Galois points as K3 surfaces.
{"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}
Debojyoti Bhattacharya, A. J. Parameswaran, Jagadish Pine
Let $X subset mathbb P^3$ be a nonsingular cubic hypersurface. Faenzi (cite{F}) and later Pons-Llopis and Tonini (cite{PLT}) have completely characterized ACM line bundles over $X$. As a natural continuation of their study in the non-ACM direction, in this paper, we completely classify $ell$-away ACM line bundles (introduced recently by Gawron and Genc (cite{GG})) over $X$, when $ell leq 2$. For $ellgeq 3$, we give examples of $ell$-away ACM line bundles on $X$ and for each $ell geq 1$, we establish the existence of smooth hypersurfaces $X^{(d)}$ of degree $d >ell$ in $mathbb P^3$ admitting $ell$-away ACM line bundles.
{"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}
In this paper, we show how to prove the global generation of adjoint linear systems on irregular varieties inductively. For instance, we prove that Fujita's conjecture holds for irregular varieties of dimension $n$ with nef anticanonical bundle, assuming it holds for lower-dimensional varieties and under mild conditions.
{"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}
Motivated by the recent surge of interest in the geometry of hybrid spaces, we prove an Abel-Jacobi theorem for a metrized complex of Riemann surfaces, generalizing both the classical Abel-Jacobi theorem and its tropical analogue.
{"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}
Fix integers $rgeq 4$ and $igeq 2$. Let $C$ be a non-degenerate, reduced and irreducible complex projective curve in $mathbb P^r$, of degree $d$, not contained in a hypersurface of degree $leq i$. Let $p_a(C)$ be the arithmetic genus of $C$. Continuing previous research, under the assumption $dgg max{r,i}$, in the present paper we exhibit a Castelnuovo bound $G_0(r;d,i)$ for $p_a(C)$. In general, we do not know whether this bound is sharp. However, we are able to prove it is sharp when $i=2$, $r=6$ and $dequiv 0,3,6$ (mod $9$). Moreover, when $i=2$, $rgeq 9$, $r$ is divisible by $3$, and $dequiv 0$ (mod $r(r+3)/6$), we prove that if $G_0(r;d,i)$ is not sharp, then for the maximal value of $p_a(C)$ there are only three possibilities. The case in which $i=2$ and $r$ is not divisible by $3$ has already been examined in the literature. We give some information on the extremal curves.
{"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}
The $j$-invariant of a cubic curve is an isomorphism invariant parameterized by the moduli space of elliptic curves. The Hesse derivative of a curve $V(f)$ given by the homogeneous polynomial $f$ is $V(mathcal{H}(f))$ where $mathcal{H}(f)$ is a the determinant of the Hesse matrix of $f$. In this paper, we compute the $j$-invariant of the Hesse derivative of a cubic curve $C$ in terms of the $j$-invariant of $C$, getting a rational function on the Riemann sphere. We then analyze the dynamics of this rational function, and investigate when a cubic curve is isomorphic to its $n$-fold Hesse derivative.
{"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}
In this paper we prove the following abundance-type result: for any smooth Fano variety $X$, the tangent bundle $T_X$ is nef if and only if it is big and semiample in the sense that the tautological line bundle $mathscr{O}_{mathbb{P}T_X}(1)$ is so, by which we establish a weak form of the Campana-Peternell conjecture (Camapan-Peternell, 1991).
{"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}
We show that the Poisson structure on the smooth locus of a moduli space of 1-dimensional sheaves on a Poisson projective surface $X$ over $mathbb C$ is a reduction of a natural symplectic structure.
{"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}
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