We study Hilbert schemes of quadrics of dimension $k in {0,1,2,3}$ on smooth Gushel-Mukai varieties $X$ of dimension $n in {2,3,4,5,6}$ by relating them to the relative Hilbert schemes of linear subspaces of dimension $k + 1$ of a certain family, naturally associated with $X$, of quadrics of dimension $n - 1$ over the blowup of $mathbf{P}^5$ at a point.
我们研究了维数为 $k in {0,1,2,3}$ 的光滑 Gushel-Mukai varieties $X$ 上维数为 $n in {2,3,4,5、6}$ 的维数为 $k + 1$ 的线性子空间的相对希尔伯特方案相关联，自然与 $X$ 相关联。
{"title":"Quadrics on Gushel-Mukai varieties","authors":"Olivier Debarre, Alexander Kuznetsov","doi":"arxiv-2409.03528","DOIUrl":"https://doi.org/arxiv-2409.03528","url":null,"abstract":"We study Hilbert schemes of quadrics of dimension $k in {0,1,2,3}$ on\u0000smooth Gushel-Mukai varieties $X$ of dimension $n in {2,3,4,5,6}$ by\u0000relating them to the relative Hilbert schemes of linear subspaces of dimension\u0000$k + 1$ of a certain family, naturally associated with $X$, of quadrics of\u0000dimension $n - 1$ over the blowup of $mathbf{P}^5$ at a point.","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":"142220331","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 provide examples of smooth three-dimensional Fano complete intersections of dergee 2, 4, 6, and 8 that have coregularity 0. Considering the main theorem of arXiv:2309.16784 on the remaining 101 families of smooth Fano threefolds, our result implies that each family of smooth Fano threefolds has an element of coregularity zero.
{"title":"On smooth Fano threefolds with coregularity zero","authors":"Olzhas Zhakupov","doi":"arxiv-2409.02523","DOIUrl":"https://doi.org/arxiv-2409.02523","url":null,"abstract":"We provide examples of smooth three-dimensional Fano complete intersections\u0000of dergee 2, 4, 6, and 8 that have coregularity 0. Considering the main theorem\u0000of arXiv:2309.16784 on the remaining 101 families of smooth Fano threefolds,\u0000our result implies that each family of smooth Fano threefolds has an element of\u0000coregularity zero.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220283","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 study locally trivial deformations of toric varieties from a combinatorial point of view. For any fan $Sigma$, we construct a deformation functor $mathrm{Def}_Sigma$ by considering v{C}ech zero-cochains on certain simplicial complexes. We show that under appropriate hypotheses, $mathrm{Def}_Sigma$ is isomorphic to $mathrm{Def}'_{X_Sigma}$, the functor of locally trivial deformations for the toric variety $X_Sigma$ associated to $Sigma$. In particular, for any complete toric variety $X$ that is smooth in codimension $2$ and $mathbb{Q}$-factorial in codimension $3$, there exists a fan $Sigma$ such that $mathrm{Def}_Sigma$ is isomorphic to $mathrm{Def}_X$, the functor of deformations of $X$. We apply these results to give a new criterion for a smooth complete toric variety to have unobstructed deformations, and to compute formulas for higher order obstructions, generalizing a formula of Ilten and Turo for the cup product. We use the functor $mathrm{Def}_Sigma$ to explicitly compute the deformation spaces for a number of toric varieties, and provide examples exhibiting previously unobserved phenomena. In particular, we classify exactly which toric threefolds arising as iterated $mathbb{P}^1$-bundles have unobstructed deformation space.
{"title":"Locally Trivial Deformations of Toric Varieties","authors":"Nathan Ilten, Sharon Robins","doi":"arxiv-2409.02824","DOIUrl":"https://doi.org/arxiv-2409.02824","url":null,"abstract":"We study locally trivial deformations of toric varieties from a combinatorial\u0000point of view. For any fan $Sigma$, we construct a deformation functor\u0000$mathrm{Def}_Sigma$ by considering v{C}ech zero-cochains on certain\u0000simplicial complexes. We show that under appropriate hypotheses,\u0000$mathrm{Def}_Sigma$ is isomorphic to $mathrm{Def}'_{X_Sigma}$, the functor\u0000of locally trivial deformations for the toric variety $X_Sigma$ associated to\u0000$Sigma$. In particular, for any complete toric variety $X$ that is smooth in\u0000codimension $2$ and $mathbb{Q}$-factorial in codimension $3$, there exists a\u0000fan $Sigma$ such that $mathrm{Def}_Sigma$ is isomorphic to $mathrm{Def}_X$,\u0000the functor of deformations of $X$. We apply these results to give a new\u0000criterion for a smooth complete toric variety to have unobstructed\u0000deformations, and to compute formulas for higher order obstructions,\u0000generalizing a formula of Ilten and Turo for the cup product. We use the\u0000functor $mathrm{Def}_Sigma$ to explicitly compute the deformation spaces for\u0000a number of toric varieties, and provide examples exhibiting previously\u0000unobserved phenomena. In particular, we classify exactly which toric threefolds\u0000arising as iterated $mathbb{P}^1$-bundles have unobstructed deformation space.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220280","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 there exist only constant morphisms from $mathbb{Q}^{2n+1}(ngeq 1)$ to $mathbb{G}(l,2n+1)$ if $l$ is even $(0
我们证明，如果 $l$ 是偶数 $(0
{"title":"Splitting of uniform bundles on quadrics","authors":"Xinyi Fang, Duo Li, Yanjie Li","doi":"arxiv-2409.02365","DOIUrl":"https://doi.org/arxiv-2409.02365","url":null,"abstract":"We show that there exist only constant morphisms from\u0000$mathbb{Q}^{2n+1}(ngeq 1)$ to $mathbb{G}(l,2n+1)$ if $l$ is even $(0<l<2n)$\u0000and $(l,2n+1)$ is not $ (2,5)$. As an application, we prove on\u0000$mathbb{Q}^{2m+1}$ and $mathbb{Q}^{2m+2}(mgeq 3)$, any uniform bundle of\u0000rank $2m$ splits.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220289","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 perverse-Hodge octahedron is a 3D enhancement of the Hodge diamond of a compact hyperk"{a}hler manifold. Its existence is equivalent to Nagai's conjecture, which holds for all known deformation types. The octahedron appears implicitly in Huybrechts-Mauri and Shen-Yin.
{"title":"Perverse-Hodge octahedron","authors":"Mirko Mauri","doi":"arxiv-2409.01800","DOIUrl":"https://doi.org/arxiv-2409.01800","url":null,"abstract":"The perverse-Hodge octahedron is a 3D enhancement of the Hodge diamond of a\u0000compact hyperk\"{a}hler manifold. Its existence is equivalent to Nagai's\u0000conjecture, which holds for all known deformation types. The octahedron appears\u0000implicitly in Huybrechts-Mauri and Shen-Yin.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220282","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}
A very general hypersurface of dimension $n$ and degree $d$ in complex projective space is rational if $d leq 2$, but is expected to be irrational for all $n, d geq 3$. Hypersurfaces in weighted projective space with degree small relative to the weights are likewise rational. In this paper, we introduce rationality constructions for weighted hypersurfaces of higher degree that provide many new rational examples over any field. We answer in the affirmative a question of T. Okada about the existence of very general terminal Fano rational weighted hypersurfaces in all dimensions $n geq 6$.
{"title":"Rational weighted projective hypersurfaces","authors":"Louis Esser","doi":"arxiv-2409.01333","DOIUrl":"https://doi.org/arxiv-2409.01333","url":null,"abstract":"A very general hypersurface of dimension $n$ and degree $d$ in complex\u0000projective space is rational if $d leq 2$, but is expected to be irrational\u0000for all $n, d geq 3$. Hypersurfaces in weighted projective space with degree\u0000small relative to the weights are likewise rational. In this paper, we\u0000introduce rationality constructions for weighted hypersurfaces of higher degree\u0000that provide many new rational examples over any field. We answer in the\u0000affirmative a question of T. Okada about the existence of very general terminal\u0000Fano rational weighted hypersurfaces in all dimensions $n geq 6$.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220332","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 the theory of complements, introduced by Shokurov, for Calabi-Yau type varieties with the coefficient set $[0,1]$. We show that there exists a finite set of positive integers $mathcal{N}$, such that if a threefold pair $(X/Zni z,B)$ has an $mathbb{R}$-complement which is klt over a neighborhood of $z$, then it has an $n$-complement for some $ninmathcal{N}$. We also show the boundedness of complements for $mathbb{R}$-complementary surface pairs.
{"title":"Boundedness of complements for log Calabi-Yau threefolds","authors":"Guodu Chen, Jingjun Han, Qingyuan Xue","doi":"arxiv-2409.01310","DOIUrl":"https://doi.org/arxiv-2409.01310","url":null,"abstract":"In this paper, we study the theory of complements, introduced by Shokurov,\u0000for Calabi-Yau type varieties with the coefficient set $[0,1]$. We show that\u0000there exists a finite set of positive integers $mathcal{N}$, such that if a\u0000threefold pair $(X/Zni z,B)$ has an $mathbb{R}$-complement which is klt over\u0000a neighborhood of $z$, then it has an $n$-complement for some\u0000$ninmathcal{N}$. We also show the boundedness of complements for\u0000$mathbb{R}$-complementary surface pairs.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220287","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 provide relative hypercohomology version of Serre's GAGA theorem. We prove that relative hypercohomology of a complex of sheaves on complex projective variety with certain conditions and relative hypercohomology of its analytification complex are isomorphic. This implies the original Serre's GAGA theorem.
{"title":"Relative-Hyper GAGA Theorem","authors":"Eita Haibara, Taewan Kim","doi":"arxiv-2409.01481","DOIUrl":"https://doi.org/arxiv-2409.01481","url":null,"abstract":"In this paper, we provide relative hypercohomology version of Serre's GAGA\u0000theorem. We prove that relative hypercohomology of a complex of sheaves on\u0000complex projective variety with certain conditions and relative hypercohomology\u0000of its analytification complex are isomorphic. This implies the original\u0000Serre's GAGA theorem.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220285","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 classical trisecant lemma says that a general chord of a non-degenerate space curve is not a trisecant; that is, the chord only meets the curve in two points. The generalized trisecant lemma extends the result to higher-dimensional varieties. It states that the linear space spanned by general points on a projective variety intersects the variety in exactly these points, provided the dimension of the linear space is smaller than the codimension of the variety and that the variety is irreducible, reduced, and non-degenerate. We prove a real analogue of the generalized trisecant lemma, which takes the form of a trichotomy. Along the way, we characterize the possible numbers of real intersection points between a real projective variety and a complimentary dimension real linear space. We show that any integer of correct parity between a minimum and a maximum number can be achieved. We then specialize to Segre-Veronese varieties, where our results apply to the identifiability of independent component analysis, tensor decomposition and to typical tensor ranks.
{"title":"A Real Generalized Trisecant Trichotomy","authors":"Kristian Ranestad, Anna Seigal, Kexin Wang","doi":"arxiv-2409.01356","DOIUrl":"https://doi.org/arxiv-2409.01356","url":null,"abstract":"The classical trisecant lemma says that a general chord of a non-degenerate\u0000space curve is not a trisecant; that is, the chord only meets the curve in two\u0000points. The generalized trisecant lemma extends the result to\u0000higher-dimensional varieties. It states that the linear space spanned by\u0000general points on a projective variety intersects the variety in exactly these\u0000points, provided the dimension of the linear space is smaller than the\u0000codimension of the variety and that the variety is irreducible, reduced, and\u0000non-degenerate. We prove a real analogue of the generalized trisecant lemma,\u0000which takes the form of a trichotomy. Along the way, we characterize the\u0000possible numbers of real intersection points between a real projective variety\u0000and a complimentary dimension real linear space. We show that any integer of\u0000correct parity between a minimum and a maximum number can be achieved. We then\u0000specialize to Segre-Veronese varieties, where our results apply to the\u0000identifiability of independent component analysis, tensor decomposition and to\u0000typical tensor ranks.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220284","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 mirror symmetry for flag varieties and Grassmannians has been extensively studied, Schubert varieties in the Grassmannian are singular, and hence standard mirror symmetry statements are not well-defined. Nevertheless, in this article we introduce a ``superpotential'' $W^{lambda}$ for each Grassmannian Schubert variety $X_{lambda}$, generalizing the Marsh-Rietsch superpotential for Grassmannians, and we show that $W^{lambda}$ governs many toric degenerations of $X_{lambda}$. We also generalize the ``polytopal mirror theorem'' for Grassmannians from our previous work: namely, for any cluster seed $G$ for $X_{lambda}$, we construct a corresponding Newton-Okounkov convex body $Delta_G^{lambda}$, and show that it coincides with the superpotential polytope $Gamma_G^{lambda}$, that is, it is cut out by the inequalities obtained by tropicalizing an associated Laurent expansion of $W^{lambda}$. This gives us a toric degeneration of the Schubert variety $X_{lambda}$ to the (singular) toric variety $Y(mathcal{N}_{lambda})$ of the Newton-Okounkov body. Finally, for a particular cluster seed $G=G^lambda_{mathrm{rec}}$ we show that the toric variety $Y(mathcal{N}_{lambda})$ has a small toric desingularisation, and we describe an intermediate partial desingularisation $Y(mathcal{F}_lambda)$ that is Gorenstein Fano. Many of our results extend to more general varieties in the Grassmannian.
{"title":"A superpotential for Grassmannian Schubert varieties","authors":"Konstanze Rietsch, Lauren Williams","doi":"arxiv-2409.00734","DOIUrl":"https://doi.org/arxiv-2409.00734","url":null,"abstract":"While mirror symmetry for flag varieties and Grassmannians has been\u0000extensively studied, Schubert varieties in the Grassmannian are singular, and\u0000hence standard mirror symmetry statements are not well-defined. Nevertheless,\u0000in this article we introduce a ``superpotential'' $W^{lambda}$ for each\u0000Grassmannian Schubert variety $X_{lambda}$, generalizing the Marsh-Rietsch\u0000superpotential for Grassmannians, and we show that $W^{lambda}$ governs many\u0000toric degenerations of $X_{lambda}$. We also generalize the ``polytopal mirror\u0000theorem'' for Grassmannians from our previous work: namely, for any cluster\u0000seed $G$ for $X_{lambda}$, we construct a corresponding Newton-Okounkov convex\u0000body $Delta_G^{lambda}$, and show that it coincides with the superpotential\u0000polytope $Gamma_G^{lambda}$, that is, it is cut out by the inequalities\u0000obtained by tropicalizing an associated Laurent expansion of $W^{lambda}$.\u0000This gives us a toric degeneration of the Schubert variety $X_{lambda}$ to the\u0000(singular) toric variety $Y(mathcal{N}_{lambda})$ of the Newton-Okounkov\u0000body. Finally, for a particular cluster seed $G=G^lambda_{mathrm{rec}}$ we\u0000show that the toric variety $Y(mathcal{N}_{lambda})$ has a small toric\u0000desingularisation, and we describe an intermediate partial desingularisation\u0000$Y(mathcal{F}_lambda)$ that is Gorenstein Fano. Many of our results extend to\u0000more general varieties in the Grassmannian.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220288","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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