We construct a linear system on a general curve in a totally geodesic subvariety of the moduli space of curves. As a consequence, one obtains rank bounds for totally geodesic subvarieties of dimension at least two. Furthermore, this leads to a classification of totally geodesic subvarieties of dimension at least two in strata with at most two zeros.
{"title":"Totally Geodesic Subvarieties of the Moduli Space of Curves and Linear Systems","authors":"Frederik Benirschke","doi":"10.1093/imrn/rnae165","DOIUrl":"https://doi.org/10.1093/imrn/rnae165","url":null,"abstract":"We construct a linear system on a general curve in a totally geodesic subvariety of the moduli space of curves. As a consequence, one obtains rank bounds for totally geodesic subvarieties of dimension at least two. Furthermore, this leads to a classification of totally geodesic subvarieties of dimension at least two in strata with at most two zeros.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Freeness is an important property of a hypersurface arrangement, although its presence is not well understood. A hypersurface arrangement in ${mathbb{P}}^{n}$ is free if $S/J$ is Cohen–Macaulay (CM), where $S = K[x_{0},ldots ,x_{n}]$ and $J$ is the Jacobian ideal. We study three related unmixed ideals: $J^{top}$, the intersection of height two primary components, $sqrt{J^{top}}$, the radical of $J^{top}$, and when the $f_{i}$ are smooth we also study $sqrt{J}$. Under mild hypotheses, we show that these ideals are CM. This establishes a full generalization of an earlier result with Schenck from hyperplane arrangements to hypersurface arrangements. If the hypotheses fail for an arrangement in projective $3$-space, the Hartshorne–Rao module measures the failure of CMness. We establish consequences for the even liaison classes of $J^{top}$ and $sqrt{J}$.
{"title":"Jacobian Schemes Arising From Hypersurface Arrangements in ℙn","authors":"Juan Migliore, Uwe Nagel","doi":"10.1093/imrn/rnae164","DOIUrl":"https://doi.org/10.1093/imrn/rnae164","url":null,"abstract":"Freeness is an important property of a hypersurface arrangement, although its presence is not well understood. A hypersurface arrangement in ${mathbb{P}}^{n}$ is free if $S/J$ is Cohen–Macaulay (CM), where $S = K[x_{0},ldots ,x_{n}]$ and $J$ is the Jacobian ideal. We study three related unmixed ideals: $J^{top}$, the intersection of height two primary components, $sqrt{J^{top}}$, the radical of $J^{top}$, and when the $f_{i}$ are smooth we also study $sqrt{J}$. Under mild hypotheses, we show that these ideals are CM. This establishes a full generalization of an earlier result with Schenck from hyperplane arrangements to hypersurface arrangements. If the hypotheses fail for an arrangement in projective $3$-space, the Hartshorne–Rao module measures the failure of CMness. We establish consequences for the even liaison classes of $J^{top}$ and $sqrt{J}$.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce and study the space of homeomorphisms of the circle (up to Möbius transformations), which are in $ell ^{2}$ with respect to modular coordinates called diamond shears along the edges of the Farey tessellation. Diamond shears are related combinatorially to shear coordinates and are also closely related to the $log Lambda $-lengths of decorated Teichmüller space introduced by Penner. We obtain sharp results comparing this new class to the Weil–Petersson class and Hölder classes of circle homeomorphisms. We also express the Weil–Petersson metric tensor and symplectic form in terms of infinitesimal shears and diamond shears.
{"title":"Circle Homeomorphisms with Square Summable Diamond Shears","authors":"Dragomir Šarić, Yilin Wang, Catherine Wolfram","doi":"10.1093/imrn/rnae155","DOIUrl":"https://doi.org/10.1093/imrn/rnae155","url":null,"abstract":"We introduce and study the space of homeomorphisms of the circle (up to Möbius transformations), which are in $ell ^{2}$ with respect to modular coordinates called diamond shears along the edges of the Farey tessellation. Diamond shears are related combinatorially to shear coordinates and are also closely related to the $log Lambda $-lengths of decorated Teichmüller space introduced by Penner. We obtain sharp results comparing this new class to the Weil–Petersson class and Hölder classes of circle homeomorphisms. We also express the Weil–Petersson metric tensor and symplectic form in terms of infinitesimal shears and diamond shears.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141779234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A framework $(G,p)$ in Euclidean space $mathbb{E}^{d}$ is globally rigid if it is the unique realisation, up to rigid congruences, of $G$ with the edge lengths of $(G,p)$. Building on key results of Hendrickson [28] and Connelly [14], Jackson and Jordán [29] gave a complete combinatorial characterisation of when a generic framework is global rigidity in $mathbb{E}^{2}$. We prove an analogous result when the Euclidean norm is replaced by any norm that is analytic on $mathbb{R}^{2} setminus {0}$. Specifically, we show that a graph $G=(V,E)$ has an open set of globally rigid realisations in a non-Euclidean analytic normed plane if and only if $G$ is 2-connected and $G-e$ contains 2 edge-disjoint spanning trees for all $ein E$. We also prove that the analogous necessary conditions hold in $d$-dimensional normed spaces.
{"title":"Uniquely Realisable Graphs in Analytic Normed Planes","authors":"Sean Dewar, John Hewetson, Anthony Nixon","doi":"10.1093/imrn/rnae162","DOIUrl":"https://doi.org/10.1093/imrn/rnae162","url":null,"abstract":"A framework $(G,p)$ in Euclidean space $mathbb{E}^{d}$ is globally rigid if it is the unique realisation, up to rigid congruences, of $G$ with the edge lengths of $(G,p)$. Building on key results of Hendrickson [28] and Connelly [14], Jackson and Jordán [29] gave a complete combinatorial characterisation of when a generic framework is global rigidity in $mathbb{E}^{2}$. We prove an analogous result when the Euclidean norm is replaced by any norm that is analytic on $mathbb{R}^{2} setminus {0}$. Specifically, we show that a graph $G=(V,E)$ has an open set of globally rigid realisations in a non-Euclidean analytic normed plane if and only if $G$ is 2-connected and $G-e$ contains 2 edge-disjoint spanning trees for all $ein E$. We also prove that the analogous necessary conditions hold in $d$-dimensional normed spaces.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141779232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide new criteria for the integrality and birationality of an extension of graded algebras in terms of the general notion of polar multiplicities of Kleiman and Thorup. As an application, we obtain a new criterion for when a module is a reduction of another in terms of certain mixed Buchsbaum–Rim multiplicities. Furthermore, we prove several technical results regarding polar multiplicities.
{"title":"Polar Multiplicities and Integral Dependence","authors":"Yairon Cid-Ruiz","doi":"10.1093/imrn/rnae163","DOIUrl":"https://doi.org/10.1093/imrn/rnae163","url":null,"abstract":"We provide new criteria for the integrality and birationality of an extension of graded algebras in terms of the general notion of polar multiplicities of Kleiman and Thorup. As an application, we obtain a new criterion for when a module is a reduction of another in terms of certain mixed Buchsbaum–Rim multiplicities. Furthermore, we prove several technical results regarding polar multiplicities.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141739975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $X$ be a variety with a stratification ${mathcal{S}}$ into smooth locally closed subvarieties such that $X$ is locally a product along each stratum (e.g., a symplectic singularity). We prove that assigning to each open subset $U subset X$ the set of isomorphism classes of locally projective crepant resolutions of $U$ defines an ${mathcal{S}}$-constructible sheaf of sets. Thus, for each stratum $S$ and basepoint $s in S$, the fundamental group acts on the set of germs of projective crepant resolutions at $s$, leaving invariant the germs extending to the entire stratum. Global locally projective crepant resolutions correspond to compatible such choices for all strata. For example, if the local projective crepant resolutions are unique, they automatically glue uniquely. We give criteria for a locally projective crepant resolution $rho : tilde X to X$ to be globally projective. We show that the sheafification of the presheaf $U mapsto operatorname{Pic}(rho ^{-1}(U)/U)$ of relative Picard classes is also constructible. The resolution is globally projective only if there exist local relatively ample bundles whose classes glue to a global section of this sheaf. The obstruction to lifting this section to a global ample line bundle is encoded by a gerbe on the singularity $X$. We show the gerbes are automatically trivial if $X$ is a symplectic quotient singularity. Our main results hold in the more general setting of partial crepant resolutions, that need not have smooth source. We apply the theory to symmetric powers and Hilbert schemes of surfaces with du Val singularities, finite quotients of tori, multiplicative, and Nakajima quiver varieties, as well as to canonical three-fold singularities.
{"title":"Crepant Resolutions of Stratified Varieties via Gluing","authors":"Daniel Kaplan, Travis Schedler","doi":"10.1093/imrn/rnae135","DOIUrl":"https://doi.org/10.1093/imrn/rnae135","url":null,"abstract":"Let $X$ be a variety with a stratification ${mathcal{S}}$ into smooth locally closed subvarieties such that $X$ is locally a product along each stratum (e.g., a symplectic singularity). We prove that assigning to each open subset $U subset X$ the set of isomorphism classes of locally projective crepant resolutions of $U$ defines an ${mathcal{S}}$-constructible sheaf of sets. Thus, for each stratum $S$ and basepoint $s in S$, the fundamental group acts on the set of germs of projective crepant resolutions at $s$, leaving invariant the germs extending to the entire stratum. Global locally projective crepant resolutions correspond to compatible such choices for all strata. For example, if the local projective crepant resolutions are unique, they automatically glue uniquely. We give criteria for a locally projective crepant resolution $rho : tilde X to X$ to be globally projective. We show that the sheafification of the presheaf $U mapsto operatorname{Pic}(rho ^{-1}(U)/U)$ of relative Picard classes is also constructible. The resolution is globally projective only if there exist local relatively ample bundles whose classes glue to a global section of this sheaf. The obstruction to lifting this section to a global ample line bundle is encoded by a gerbe on the singularity $X$. We show the gerbes are automatically trivial if $X$ is a symplectic quotient singularity. Our main results hold in the more general setting of partial crepant resolutions, that need not have smooth source. We apply the theory to symmetric powers and Hilbert schemes of surfaces with du Val singularities, finite quotients of tori, multiplicative, and Nakajima quiver varieties, as well as to canonical three-fold singularities.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141739976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We continue our study of the monoidal category $D(G)$ begun in [ 12]. At the level of cohomology we transfer the duality functor $Runderline{operatorname{Hom}}(-,k)$ to the derived category of dg-modules $D(H_{U}^{bullet })$. In the process we develop a more general and streamlined approach to the anti-involution $mathscr J$ from [ 8]. We also verify that the tensor product on $D(G)$ corresponds to an operadic tensor product on the dg-side (cf [ 5]). This uses a result of Schnürer on dg-categories with a model structure.
{"title":"dg-Hecke Duality and Tensor Products","authors":"Peter Schneider, Claus Sorensen","doi":"10.1093/imrn/rnae156","DOIUrl":"https://doi.org/10.1093/imrn/rnae156","url":null,"abstract":"We continue our study of the monoidal category $D(G)$ begun in [ 12]. At the level of cohomology we transfer the duality functor $Runderline{operatorname{Hom}}(-,k)$ to the derived category of dg-modules $D(H_{U}^{bullet })$. In the process we develop a more general and streamlined approach to the anti-involution $mathscr J$ from [ 8]. We also verify that the tensor product on $D(G)$ corresponds to an operadic tensor product on the dg-side (cf [ 5]). This uses a result of Schnürer on dg-categories with a model structure.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141722229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given one quasi-smooth derived space cut out of another by a section of a 2-term complex of bundles, we give two formulae for its virtual cycle. They are modelled on the the $p$-fields construction of Chang–Li and the Quantum Lefschetz principle, and recover these when applied to moduli spaces of (stable or quasi-) maps. When the complex is a single bundle we recover the results of Kim–Kresch–Pantev.
{"title":"Quantum Lefschetz Without Curves","authors":"Jeongseok Oh, Richard P Thomas","doi":"10.1093/imrn/rnae158","DOIUrl":"https://doi.org/10.1093/imrn/rnae158","url":null,"abstract":"Given one quasi-smooth derived space cut out of another by a section of a 2-term complex of bundles, we give two formulae for its virtual cycle. They are modelled on the the $p$-fields construction of Chang–Li and the Quantum Lefschetz principle, and recover these when applied to moduli spaces of (stable or quasi-) maps. When the complex is a single bundle we recover the results of Kim–Kresch–Pantev.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note, we give examples that demonstrate a negative answer to the generalized numerical criterion problem for pairs.
在本说明中，我们将举例说明成对的广义数值准则问题的否定答案。
{"title":"On the Generalized Numerical Criterion","authors":"Sean Timothy Paul, Song Sun, Junsheng Zhang","doi":"10.1093/imrn/rnae160","DOIUrl":"https://doi.org/10.1093/imrn/rnae160","url":null,"abstract":"In this note, we give examples that demonstrate a negative answer to the generalized numerical criterion problem for pairs.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $X$ be a connected scheme locally of finite type over ${mathbb{C}}$, and let $X^{textrm{an}}$ be its associated analytic space. In this paper, we define a comparison map from the topological fundamental group of $X^{textrm{an}}$ to the pro-étale fundamental group of $X$.
{"title":"A Comparison Theorem For The Pro-étale Fundamental Group","authors":"Jiu-Kang Yu, Lei Zhang","doi":"10.1093/imrn/rnae150","DOIUrl":"https://doi.org/10.1093/imrn/rnae150","url":null,"abstract":"Let $X$ be a connected scheme locally of finite type over ${mathbb{C}}$, and let $X^{textrm{an}}$ be its associated analytic space. In this paper, we define a comparison map from the topological fundamental group of $X^{textrm{an}}$ to the pro-étale fundamental group of $X$.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141568380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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