46,211 results on '"Algebraic Geometry (math.AG)"'
Search Results
152. The integral Chow ring of weighted blow-ups
- Author
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Arena, Veronica, Obinna, Stephen, and Abramovich, Dan
- Subjects
Mathematics - Algebraic Geometry ,FOS: Mathematics ,Algebraic Geometry (math.AG) - Abstract
We give a formula for the integral Chow rings of weighted blow-ups. Along the way, we also compute the integral Chow rings of weighted projective stack bundles, a formula for the Gysin homomorphism of a weighted blow-up, and a generalization of the splitting principle. In addition, in the appendix we compute the Chern class of a weighted blow-up., by: Veronica Arena, Stephen Obinna; with appendix by: Dan Abramovich, Veronica Arena, Stephen Obinna
- Published
- 2023
153. Real Structures on Root Stacks and Parabolic Connections
- Author
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Chakraborty, Sujoy and Paul, Arjun
- Subjects
Mathematics - Algebraic Geometry ,14D23, 14H60, 53B15, 53C05, 14A21 ,FOS: Mathematics ,Algebraic Geometry (math.AG) - Abstract
Let $D$ be a reduced effective strict normal crossing divisor on a smooth complex variety $X$, and let $\mathfrak{X}_D$ be the associated root stack over $\mathbb C$. Suppose that $X$ admits an anti-holomorphic involution (real structure) that keeps $D$ invariant. We show that the root stack $\mathfrak{X}_D$ naturally admits a real structure compatible with $X$. We also establish an equivalence of categories between the category of real logarithmic connections on this root stack and the category of real parabolic connections on $X$., 30 pages, comments are welcome
- Published
- 2023
154. Stability Conditions on Free Abelian Quotients
- Author
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Dell, Hannah
- Subjects
Mathematics - Algebraic Geometry ,FOS: Mathematics ,14F08 (Primary) 14L30, 14J60 (Secondary) ,Algebraic Geometry (math.AG) - Abstract
We study slope-stable vector bundles and Bridgeland stability conditions on varieties which are a quotient of a smooth projective variety by a finite abelian group $G$ acting freely. We show there is a one-to-one correspondence between $\widehat{G}$-invariant geometric stability conditions on the quotient and $G$-invariant geometric stability conditions on the cover. We apply our results to describe a connected component inside the stability manifolds of free abelian quotients when the cover has finite Albanese morphism. This applies to varieties with non-finite Albanese morphism which are free abelian quotients of varieties with finite Albanese morphism, such as Beauville-type and bielliptic surfaces. This gives a partial answer to a question raised by Lie Fu, Chunyi Li, and Xiaolei Zhao: If a variety $X$ has non-finite Albanese morphism, does there always exist a non-geometric stability condition on $X$? We also give counterexamples to a conjecture of Fu-Li-Zhao concerning the Le Potier function, which characterises Chern classes of slope-semistable sheaves. As a result of independent interest, we give a description of the set of geometric stability conditions on an arbitrary surface in terms of a refinement of the Le Potier function. This generalises a result of Fu-Li-Zhao from Picard rank one to arbitrary Picard rank., 34 pages
- Published
- 2023
155. An ML approach to resolution of singularities
- Author
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Bérczi, Gergely, Fan, Honglu, and Zeng, Mingcong
- Subjects
FOS: Computer and information sciences ,Computer Science - Symbolic Computation ,Computer Science - Machine Learning ,Mathematics - Algebraic Geometry ,Artificial Intelligence (cs.AI) ,Computer Science - Artificial Intelligence ,FOS: Mathematics ,Symbolic Computation (cs.SC) ,Algebraic Geometry (math.AG) ,Machine Learning (cs.LG) - Abstract
The solution set of a system of polynomial equations typically contains ill-behaved, singular points. Resolution is a fundamental process in geometry in which we replace singular points with smooth points, while keeping the rest of the solution set unchanged. Resolutions are not unique: the usual way to describe them involves repeatedly performing a fundamental operation known as "blowing-up", and the complexity of the resolution highly depends on certain choices. The process can be translated into various versions of a 2-player game, the so-called Hironaka game, and a winning strategy for the first player provides a solution to the resolution problem. In this paper we introduce a new approach to the Hironaka game that uses reinforcement learning agents to find optimal resolutions of singularities. In certain domains, the trained model outperforms state-of-the-art selection heuristics in total number of polynomial additions performed, which provides a proof-of-concept that recent developments in machine learning have the potential to improve performance of algorithms in symbolic computation., To appear in Proceedings of the 40th International Conference on Machine Learning TAG Workshop (ICML-TAG 2023)
- Published
- 2023
156. Upper bounds for the number of isolated critical points via Thom-Milnor theorem
- Author
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Zolotov, Vladimir
- Subjects
Mathematics - Algebraic Geometry ,Mathematics - Classical Analysis and ODEs ,31B05 ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Algebraic Topology (math.AT) ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Mathematics - Algebraic Topology ,Algebraic Geometry (math.AG) ,Mathematical Physics - Abstract
We apply the Thom-Milnor theorem to obtain the upper bounds on the amount of isolated (1) critical points of a potential generated by several fixed point charges(Maxwell's problem on point charges), (2) critical points of SINR, (3) critical points of a potential generated by several fixed Newtonian point masses augmented with a quadratic term, (4) central configurations in the $n$-body problem. In particular, we get an exponential bound for Maxwell's problem and the polynomial bound for the case of an "even dimensional" potential in Maxwell's problem., A reference to Kuzmina's work is added. The work has a very similar bound for the number of central configurations
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- 2023
157. Abstract Orientable Incidence Structure and Algorithms for Finite Bounded Acyclic Categories. II. Data Structure and Fundamental Operations
- Author
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Huang, Yu-Wei
- Subjects
FOS: Computer and information sciences ,Mathematics - Algebraic Geometry ,Computer Science - Data Structures and Algorithms ,FOS: Mathematics ,Mathematics - Combinatorics ,Data Structures and Algorithms (cs.DS) ,Combinatorics (math.CO) ,Algebraic Geometry (math.AG) - Abstract
A data structure for finite bounded acyclic categories has been built, which is useful to encode and manipulate abstract orientable incidence structure. It can be represented as a directed acyclic multigraph with weighted edges, where the weighs encode the algebraic structure between edges. The fundamental operations on this data structure are investigated from geometrical, categorical and programming perspectives.
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- 2023
158. On the stability of vanishing cycles of \'etale sheaves in positive characteristic
- Author
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Zhou, Tong
- Subjects
Mathematics - Algebraic Geometry ,FOS: Mathematics ,Algebraic Geometry (math.AG) - Abstract
In positive characteristic, in contrast to the complex analytic case, vanishing cycles are highly sensitive to test functions (the maps to the henselian traits). We study this dependence and show that on a smooth surface, this dependence is generically (in a precise sense) only up to a finite jet of the test functions. We also study the class of sheaves whose vanishing cycles have the strongest stability. Among other things, we show that tame simple normal crossing sheaves belong to this class, and this class is stable under the Radon transform., 28 pages
- Published
- 2023
159. Relative \'etale slices and cohomology of moduli spaces
- Author
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de Cataldo, Mark Andrea, Herrero, Andres Fernandez, and Núñez, Andrés Ibáñez
- Subjects
Mathematics - Algebraic Geometry ,FOS: Mathematics ,Algebraic Geometry (math.AG) ,14D23 (Primary) 14B25, 14J60, 14D07, 14F45 (Secondary) - Abstract
We use techniques of Alper-Hall-Rydh to prove a local structure theorem for smooth morphisms between smooth stacks around points with linearly reductive stabilizers. This implies that the good moduli space of a smooth stack over a base has equisingular fibers. As an application, we show that any two fibres have isomorphic $\ell$-adic cohomology rings and intersection cohomology groups. If we work over the complex numbers, we show that the family is topologically locally trivial on the base, and that the intersection cohomology groups of the fibers fit into a polarizable variation of pure Hodge structures. We apply these results to derive some consequences for the moduli spaces of $G$-bundles on smooth projective curves, and for the moduli spaces of sheaves on "negatively polarized" surfaces and on del Pezzo Gorenstein surfaces for nongeneric stability parameters., 21 pages. Comments welcome
- Published
- 2023
160. Stability conditions and moduli spaces for Kuznetsov components of Gushel–Mukai varieties
- Author
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Perry, Alexander, Pertusi, Laura, and Zhao, Xiaolei
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Mathematics::Category Theory ,FOS: Mathematics ,Geometry and Topology ,Algebraic Geometry (math.AG) - Abstract
We prove the existence of Bridgeland stability conditions on the Kuznetsov components of Gushel-Mukai varieties, and describe the structure of moduli spaces of Bridgeland semistable objects in these categories in the even-dimensional case. As applications, we construct a new infinite series of unirational locally complete families of polarized hyperk\"{a}hler varieties of K3 type, and characterize Hodge-theoretically when the Kuznetsov component of an even-dimensional Gushel-Mukai variety is equivalent to the derived category of a K3 surface., Comment: 47 pages, minor updates, final version
- Published
- 2022
161. Moduli spaces of morphisms into solvable algebraic groups
- Author
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Rosengarten, Zev
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Algebra and Number Theory ,Mathematics::Category Theory ,FOS: Mathematics ,Algebraic Geometry (math.AG) - Abstract
We construct (in significant generality) moduli spaces representing the functor of morphisms from a scheme into a solvable algebraic group., 43 pages
- Published
- 2022
162. Deformation theory of perfect complexes and traces
- Author
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Lieblich, Max and Olsson, Martin
- Subjects
14D15, 14A30, 14D23 ,Mathematics - Algebraic Geometry ,Mathematics - K-Theory and Homology ,FOS: Mathematics ,K-Theory and Homology (math.KT) ,Geometry and Topology ,Algebraic Geometry (math.AG) ,Analysis - Abstract
We show that the deformation theory of a perfect complex and that of its determinant are related by the trace map, in a general setting of sheaves on a site. The key technical step, in passing from the setting of modules over a ring where one has global resolutions to the general setting, is achieved using $K$-theory and higher category theory., Comment: 36 pages, final version, to appear in Annals of K-Theory
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- 2022
163. Tropical ψ classes
- Author
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Cavalieri, Renzo, Gross, Andreas, and Markwig, Hannah
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Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,FOS: Mathematics ,14T05, 14A20 ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,Algebraic Geometry (math.AG) ,Physics::Atmospheric and Oceanic Physics - Abstract
We introduce a tropical geometric framework that allows us to define $\psi$ classes for moduli spaces of tropical curves of arbitrary genus. We prove correspondence theorems between algebraic and tropical $\psi$ classes for some one-dimensional families of genus-one tropical curves.
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- 2022
164. RationalMaps, a package for Macaulay2
- Author
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Bott, C. J., Hassanzadeh, S. Hamid, Schwede, Karl, and Smolkin, Daniel
- Subjects
Mathematics - Algebraic Geometry ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,FOS: Mathematics ,Computer Science::Mathematical Software ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Algebraic Geometry (math.AG) ,14E05, 14E07, 13P99, 13A30 - Abstract
This paper describes the RationalMaps package for Macaulay2. This package provides functionality for computing several aspects of rational maps such as whether a map is birational, or a closed embedding., Comment: 9 pages. The current version of the package (and other necessary files, such as the latest version of FastMinors.m2) can be accessed at https://github.com/Macaulay2/Workshop-2016-Utah/tree/master/RationalMaps
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- 2022
165. On the Bombieri–Lang conjecture over finitely generated fields
- Author
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Bresciani, Giulio
- Subjects
Mathematics - Algebraic Geometry ,Algebra and Number Theory ,Mathematics - Number Theory ,FOS: Mathematics ,Number Theory (math.NT) ,Algebraic Geometry (math.AG) - Abstract
The strong Bombieri-Lang conjecture postulates that, for every variety $X$ of general type over a field $k$ finitely generated over $\mathbb{Q}$, there exists an open subset $U\subset X$ such that $U(K)$ is finite for every finitely generated extension $K/k$. The weak Bombieri-Lang conjecture postulates that, for every positive dimensional variety $X$ of general type over a field $k$ finitely generated over $\mathbb{Q}$, the rational points $X(k)$ are not dense. Furthermore, Lang conjectured that every variety of general type $X$ over a field of characteristic $0$ contains an open subset $U\subset X$ such that every subvariety of $U$ is of general type, this statement is usually called geometric Lang conjecture. We reduce the strong Bombieri-Lang conjecture to the case $k=\mathbb{Q}$. Assuming the geometric Lang conjecture, we reduce the weak Bombieri-Lang conjecture to $k=\mathbb{Q}$, too., Comment: Edited for clarity. Fixed a wrong notation in the proof of Theorem A that confused some readers
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- 2022
166. Volume forms on moduli spaces of d–differentials
- Author
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Nguyen, Duc-Manh
- Subjects
Mathematics - Differential Geometry ,Mathematics - Geometric Topology ,Mathematics - Algebraic Geometry ,Differential Geometry (math.DG) ,FOS: Mathematics ,Geometric Topology (math.GT) ,Geometry and Topology ,Algebraic Geometry (math.AG) ,51H25, 51M05 - Abstract
Given $d\in \mathbb{N}$, $g\in \mathbb{N} \cup\{0\}$, and an integral vector $\kappa=(k_1,\dots,k_n)$ such that $k_i>-d$ and $k_1+\dots+k_n=d(2g-2)$, let $\Omega^d\mathcal{M}_{g,n}(\kappa)$ denote the moduli space of meromorphic $d$-differentials on Riemann surfaces of genus $g$ whose zeros and poles have orders prescribed by $\kappa$. We show that $\Omega^d\mathcal{M}_{g,n}(\kappa)$ carries a canonical volume form that is parallel with respect to its affine complex manifold structure, and that the total volume of $\mathbb{P}\Omega^d\mathcal{M}_{g,n}(\kappa)=\Omega^d\mathcal{M}_{g,n}/\mathbb{C}^*$ with respect to the measure induced by this volume form is finite., Comment: Streamlined, minor corrections added, definition of the volume form independent of the choice of a d-th root of unity
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- 2022
167. The images of multilinear and semihomogeneous polynomials on the algebra of octonions
- Author
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Alexei Kanel-Belov, Sergey Malev, Coby Pines, and Louis Rowen
- Subjects
Mathematics - Algebraic Geometry ,Algebra and Number Theory ,Mathematics::Rings and Algebras ,FOS: Mathematics ,17D05 17D10 14R10 ,Representation Theory (math.RT) ,Algebraic Geometry (math.AG) ,Mathematics - Representation Theory - Abstract
The generalized L'vov-Kaplansky conjecture states that for any finite-dimensional simple algebra $A$ the image of a multilinear polynomial on $A$ is a vector space. In this paper we prove it for the algebra of octonions $\mathbb{O}$ over a field satisfying certain specified conditions (in particular, we prove it for quadratically closed field and for field $\mathbb{R}$). In fact, we prove that the image set must be either $\{0\}$, $F$, the space of pure octonions $V$, or $\mathbb{O}$. We discuss possible evaluations of semihomogeneous polynomials on $\mathbb{O}$ and of arbitrary polynomials on the corresponding Malcev algebra., 14 pages
- Published
- 2022
168. Tannakian reconstruction of reductive group schemes
- Author
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Zhao, Yifei
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Category Theory ,General Mathematics ,FOS: Mathematics ,14L15, 18M25 ,Mathematics - Category Theory ,Category Theory (math.CT) ,Algebraic Geometry (math.AG) - Abstract
We give sharp criteria for when a reductive group scheme satisfies Tannakian reconstruction. When the base scheme is Noetherian, we explicitly identify its Tannaka group scheme., Comment: 9 pages
- Published
- 2022
169. The monodromy pairing for logarithmic 1-motifs
- Author
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Wise, Jonathan
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,General Mathematics ,FOS: Mathematics ,14K05, 14A21, 14H40, 14F42, 14D07, 14C22, 14T10, 14T90 ,Algebraic Geometry (math.AG) - Abstract
We describe a 3-step filtration on all logarithmic abelian varieties with constant degeneration. The obstruction to descending this filtration, as a variegated extension, from logarithmic geometry to algebraic geometry is encoded in a bilinear pairing valued in the characteristic monoid of the base. This pairing is realized as the monodromy pairing in p-adic, l-adic, and Betti cohomolgies, and recovers the Picard-Lefschetz transformation in the case of Jacobians. The Hodge realization of the filtration is the monodromy weight filtration on the limit mixed Hodge structure., 40 pages; this version contains many corrections, as well as new sections about logarithmic tori and duality for logarithmic 1-motifs; comments welcome!
- Published
- 2022
170. Frobenius stable pluricanonical systems on threefolds of general type in positive characteristic
- Author
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Zhang, Lei
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Algebra and Number Theory ,FOS: Mathematics ,Algebraic Geometry (math.AG) - Abstract
This paper aims to investigate effectivity problems of pluricanonical systems on varieties of general type in positive characteristic. In practice, we will consider a sub-linear system $|S^0_{-}(X, K_X + nK_X)| \subseteq |H^0(X, K_X +nK_X)|$ generated by certain Frobenius stable sections, and prove that for a minimal terminal threefold $X$ of general type with either $q(X)>0$ or Gorenstein singularities, if $n\geq 28$ then $|S^0_{-}(X, K_X + nK_X)| \neq \emptyset$; if $n\geq 42$ then the linear system $|S^0_{-}(X, K_X + nK_X)|$ defines a birational map., 34 pages
- Published
- 2022
171. Categorifications of rational Hilbert series and characters of FSop modules
- Author
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Tosteson, Philip
- Subjects
20C30, 06A07, 05E4, 13P10, 16G20 ,Algebra and Number Theory ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Mathematics - Algebraic Geometry ,Mathematics::Category Theory ,FOS: Mathematics ,Mathematics - Combinatorics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,Combinatorics (math.CO) ,Representation Theory (math.RT) ,Algebraic Geometry (math.AG) ,Mathematics - Representation Theory - Abstract
We introduce a method for associating a chain complex to a module over a combinatorial category, such that if the complex is exact then the module has a rational Hilbert series. We prove homology--vanishing theorems for these complexes for several combinatorial categories including: the category of finite sets and injections, the opposite of the category of finite sets and surjections, and the category of finite dimensional vector spaces over a finite field and injections. Our main applications are to modules over the opposite of the category of finite sets and surjections, known as $FS^{op}$ modules. We obtain many constraints on the sequence of symmetric group representations underlying a finitely generated $FS^{op}$ module. In particular, we describe its character in terms of functions that we call character exponentials. Our results have new consequences for the character of the homology of the moduli space of stable marked curves, and for the equivariant Kazhdan-Luzstig polynomial of the braid matroid., Comment: Corrections to definitions in Section 9, to appear in Alg. & Number Thy
- Published
- 2022
172. Severi varieties on blow‐ups of the symmetric square of an elliptic curve
- Author
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Ciro Ciliberto, Thomas Dedieu, Concettina Galati, and Andreas Leopold Knutsen
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,General Mathematics ,FOS: Mathematics ,Algebraic Geometry (math.AG) - Abstract
We prove that certain Severi varieties of nodal curves of positive genus on general blow-ups of the twofold symmetric product of a general elliptic curve are non-empty and smooth of the expected dimension. This result, besides its intrinsic value, is an important preliminary step for the proof of nonemptiness of Severi varieties on general Enriques surfaces in arXiv:2109.10735., Comment: final version; to appear in Math. Nachrichten
- Published
- 2022
173. Synthetic spectra and the cellular motivic category
- Author
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Pstrągowski, Piotr
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,General Mathematics ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,Algebraic Geometry (math.AG) ,Mathematics::Algebraic Topology - Abstract
To an Adams-type homology theory we associate a notion of a synthetic spectrum, this is a product-preserving sheaf on the site of finite spectra with projective $E$-homology. We prove that the $\infty$-category $Syn_{E}$ of synthetic spectra based on $E$ is in a precise sense a deformation of the $\infty$-category of spectra into quasi-coherent sheaves over a certain algebraic stack, and show that this deformation encodes the $E$-based Adams spectral sequence. We describe a symmetric monoidal functor from cellular motivic spectra over the complex numbers into an even variant of synthetic spectra based on $MU$ and show that it induces an equivalence between the $\infty$-categories of $p$-complete objects for all primes $p$. In particular, it follows that the $p$-complete cellular motivic category can be described purely in terms of chromatic homotopy theory., Minor typos corrected
- Published
- 2022
174. Pointwise universal Gysin formulae and applications towards Griffiths’ conjecture
- Author
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Filippo Fagioli and Simone Diverio
- Subjects
Mathematics - Differential Geometry ,Griffiths' conjecture ,Holomorphic function ,Vector bundle ,Theoretical Computer Science ,Combinatorics ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Mathematics (miscellaneous) ,Gysin's formulae ,FOS: Mathematics ,flag bundles ,Complex Variables (math.CV) ,Algebraic Geometry (math.AG) ,positive polynomials for ample vector bundles ,Pointwise ,Physics ,Primary: 14F05, Secondary: 14M15, 57R20 ,Conjecture ,Chern class ,Mathematics - Complex Variables ,Flag (linear algebra) ,Differential Geometry (math.DG) ,Homogeneous polynomial ,Complex manifold - Abstract
Let $X$ be a complex manifold, $(E,h)\to X$ be a rank $r$ holomorphic hermitian vector bundle, and $\rho$ be a sequence of dimensions $0 = \rho_0 < \rho_1 < \cdots < \rho_m = r$. Let $Q_{\rho,j}$, $j=1,\dots,m$, be the tautological line bundles over the (possibly incomplete) flag bundle $\mathbb{F}_{\rho}(E) \to X$ associated to $\rho$, endowed with the natural metrics induced by that of $E$, with Chern curvatures $\Xi_{\rho,j}$. We show that the universal Gysin formula \textsl{\`{a} la} Darondeau--Pragacz for the push-forward of a homogeneous polynomial in the Chern classes of the $Q_{\rho,j}$'s also hold pointwise at the level of the Chern forms $\Xi_{\rho,j}$ in this hermitianized situation. As an application, we show the positivity of several polynomials in the Chern forms of a Griffiths (semi)positive vector bundle not previously known, thus giving some new evidences towards a conjecture by Griffiths, which in turn can be seen as a pointwise hermitianized version of the Fulton--Lazarsfeld Theorem on numerically positive polynomials for ample vector bundles., Comment: 24 pages, no figures, comments are very welcome! v3: several minor corrections, the main application is now stated for strongly positive forms
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- 2022
175. Motivic cohomology and infinitesimal group schemes
- Author
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Primozic, Eric
- Subjects
Mathematics - Algebraic Geometry ,14F42 ,Mathematics::K-Theory and Homology ,FOS: Mathematics ,Geometry and Topology ,Algebraic Geometry (math.AG) ,Analysis - Abstract
For $k$ a perfect field of characteristic $p>0$ and $G/k$ a split reductive group with $p$ a non-torsion prime for $G,$ we compute the mod $p$ motivic cohomology of the geometric classifying space $BG_{(r)}$, where $G_{(r)}$ is the $r$th Frobenius kernel of $G.$ Our main tool is a motivic version of the Eilenberg-Moore spectral sequence, due to Krishna. For a flat affine group scheme $G/k$ of finite type, we define a cycle class map from the mod $p$ motivic cohomology of the classifying space $BG$ to the mod $p$ étale motivic cohomology of the classifying stack $\mathcal{B}G.$ This also gives a cycle class map into the Hodge cohomology of $\mathcal{B}G.$ We study the cycle class map for some examples, including Frobenius kernels., 20 pages, comments welcome
- Published
- 2022
176. NORMAL AND IRREDUCIBLE ADIC SPACES, THE OPENNESS OF FINITE MORPHISMS, AND A STEIN FACTORIZATION
- Author
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Lucas Mann
- Subjects
Mathematics - Algebraic Geometry ,Mathematics - Number Theory ,14G22 (Primary) 11G25 (Secondary) ,General Mathematics ,FOS: Mathematics ,Number Theory (math.NT) ,Mathematics::Representation Theory ,Algebraic Geometry (math.AG) - Abstract
We transfer several elementary geometric properties of rigid-analytic spaces to the world of adic spaces, more precisely to the category of adic spaces which are locally of (weakly) finite type over a non-archimedean field. This includes normality, irreducibility (in particular, irreducible components), and a Stein factorization theorem. Most notably, we show that a finite morphism in our category of adic spaces is automatically open if the target is normal and both source and target are of the same pure dimension. Moreover, our version of the Stein factorization theorem includes a statement about the geometric connectedness of fibers which we have not found in the literature of rigid-analytic or Berkovich spaces.
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- 2022
177. Ulrich bundles on cubic fourfolds
- Author
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Faenzi, Daniele, Yeongrak, Kim, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Fachbereich Mathematik Universität des Saarlandess (IAM), Universität des Saarlandes [Saarbrücken], Y.K. was supported by Project I.6 of the SFB-TRR 195 ``Symbolic Tools in Mathematics and their Application' of DFG. Both authors partially supported by Federation de Recherche Bourgogne Franche-Comte Mathematiques (FR CNRS 2011).D.F. partially supported by EUR EIPHI - ANR-17-EURE-0002 and ANR-20-CE40-0023, ANR-20-IDES-0006,IDISITEBFC,Intégration & Développement de l'Initiative pour le SITE Bourgogne Franche-Comté(2020), ANR-17-EURE-0002,EIPHI,Ingénierie et Innovation par les sciences physiques, les savoir-faire technologiques et l'interdisciplinarité(2017), and ANR-20-CE40-0023,FanoHK,Des variétés de Fano aux hyperkählériennes : géométrie et catégories dérivées(2020)
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Mathematics::Commutative Algebra ,General Mathematics ,FOS: Mathematics ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry - Abstract
We show the existence of rank 6 Ulrich bundles on a smooth cubic fourfold. First, we construct a simple sheaf E of rank 6 as an elementary modification of an ACM bundle of rank 6 on a smooth cubic fourfold. Such an E appears as an extension of two Lehn-Lehn-Sorger-van Straten sheaves. Then we prove that a general deformation of E(1) becomes Ulrich. In particular, this says that general cubic fourfolds have Ulrich complexity 6., To appear in Comm. Math. Helv
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- 2022
178. Non-Holomorphic Cycles and Non-BPS Black Branes
- Author
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Long, Cody, Sheshmani, Artan, Vafa, Cumrun, and Yau, Shing-Tung
- Subjects
High Energy Physics - Theory ,Mathematics - Algebraic Geometry ,General Relativity and Quantum Cosmology ,High Energy Physics::Theory ,High Energy Physics - Theory (hep-th) ,FOS: Mathematics ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Mathematics::Symplectic Geometry ,Algebraic Geometry (math.AG) ,Mathematical Physics - Abstract
We study extremal non-BPS black holes and strings arising in M-theory compactifications on Calabi-Yau threefolds, obtained by wrapping M2 branes on non-holomorphic 2-cycles and M5 branes on non-holomorphic 4-cycles. Using the attractor mechanism we compute the black hole mass and black string tension, leading to a conjectural formula for the asymptotic volumes of connected, locally volume-minimizing representatives of non-holomorphic, even-dimensional homology classes in the threefold, without knowledge of an explicit metric. In the case of divisors we find examples where the volume of the representative corresponding to the black string is less than the volume of the minimal piecewise-holomorphic representative, predicting recombination for those homology classes and leading to stable, non-BPS strings. We also compute the central charges of non-BPS strings in F-theory via a near-horizon $AdS_3$ limit in 6d which, upon compactification on a circle, account for the asymptotic entropy of extremal non-supersymmetric 5d black holes (i.e., the asymptotic count of non-holomorphic minimal 2-cycles)., Comment: 57 pages, 15 figures
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- 2022
179. Equations of linear subvarieties of strata of differentials
- Author
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Benirschke, Frederik, Dozier, Benjamin, and Grushevsky, Samuel
- Subjects
Mathematics - Algebraic Geometry ,Mathematics - Geometric Topology ,Mathematics::Algebraic Geometry ,Mathematics::Complex Variables ,FOS: Mathematics ,Geometric Topology (math.GT) ,Dynamical Systems (math.DS) ,Geometry and Topology ,Mathematics - Dynamical Systems ,Algebraic Geometry (math.AG) - Abstract
For a linear subvariety $M$ of a stratum of meromorphic differentials, we investigate its closure in the multi-scale compactification constructed by Bainbridge-Chen-Gendron-Grushevsky-M\"oller. We prove various restrictions on the type of defining linear equations in period coordinates for $M$ near its boundary, and prove that the closure is locally a toric variety. As applications, we give a fundamentally new proof of a generalization of the cylinder deformation theorem of Wright to the case of meromorphic strata, and construct a smooth compactification of the Hurwitz space of covers of the Riemann sphere., Comment: v2: updated version with minor edits, accepted to Geometry&Topology
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- 2022
180. Classifying sections of del Pezzo fibrations, II
- Author
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Brian Lehmann and Sho Tanimoto
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,FOS: Mathematics ,Geometry and Topology ,Algebraic Geometry (math.AG) - Abstract
Let $X$ be a del Pezzo surface over the function field of a complex curve. We study the behavior of rational points on $X$ leading to bounds on the counting function in Geometric Manin's Conjecture. A key tool is the Movable Bend and Break Lemma which yields an inductive approach to classifying relatively free sections for a del Pezzo fibration over a curve. Using this lemma we prove Geometric Manin's Conjecture for certain split del Pezzo surfaces of degree $\geq 2$ admitting a birational morphism to $\mathbb P^2$ over the ground field., 60 pages, to appear in Geometry & Topology
- Published
- 2022
181. Maximal log Fano manifolds are generalized Bott towers
- Author
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Konstantin Loginov and Joaquín Moraga
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Algebra and Number Theory ,FOS: Mathematics ,Condensed Matter::Mesoscopic Systems and Quantum Hall Effect ,Algebraic Geometry (math.AG) ,Mathematics::Algebraic Topology ,Mathematics::Symplectic Geometry - Abstract
We prove that maximal log Fano manifolds are generalized Bott towers. As an application, we prove that in each dimension, there is a unique maximal snc Fano variety satisfying Friedman's d-semistability condition., 27 pages
- Published
- 2022
182. ON THE MILNOR FIBRATION OF CERTAIN NEWTON DEGENERATE FUNCTIONS
- Author
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CHRISTOPHE EYRAL and MUTSUO OKA
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Operator Algebras ,Mathematics::K-Theory and Homology ,14B05, 14M10, 14M25, 32S05, 32S55 ,General Mathematics ,FOS: Mathematics ,Algebraic Geometry (math.AG) - Abstract
It is well known that the diffeomorphism type of the Milnor fibration of a (Newton) nondegenerate polynomial function f is uniquely determined by the Newton boundary of f. In the present paper, we generalize this result to certain degenerate functions, namely we show that the diffeomorphism type of the Milnor fibration of a (possibly degenerate) polynomial function of the form $f=f^1\cdots f^{k_0}$ is uniquely determined by the Newton boundaries of $f^1,\ldots , f^{k_0}$ if $\{f^{k_1}=\cdots =f^{k_m}=0\}$ is a nondegenerate complete intersection variety for any $k_1,\ldots ,k_m\in \{1,\ldots , k_0\}$ .
- Published
- 2022
183. Components of symmetric wide-matrix varieties
- Author
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Draisma, Jan, Eggermont, Rob H., Farooq, Azhar, Discrete Algebra and Geometry, and Coding Theory and Cryptology
- Subjects
Mathematics - Algebraic Geometry ,Applied Mathematics ,General Mathematics ,Mathematics::Category Theory ,13E05, 05E40 ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Algebraic Geometry (math.AG) - Abstract
We show that if X_n is a variety of cxn-matrices that is stable under the group Sym([n]) of column permutations and if forgetting the last column maps X_n into X_{n-1}, then the number of Sym([n])-orbits on irreducible components of X_n is a quasipolynomial in n for all sufficiently large n. To this end, we introduce the category of affine FI^op-schemes of width one, review existing literature on such schemes, and establish several new structural results about them. In particular, we show that under a shift and a localisation, any width-one FI^op-scheme becomes of product form, where X_n=Y^n for some scheme Y in affine c-space. Furthermore, to any FI^op-scheme of width one we associate a component functor from the category FI of finite sets with injections to the category PF of finite sets with partially defined maps. We present a combinatorial model for these functors and use this model to prove that Sym([n])-orbits of components of X_n, for all n, correspond bijectively to orbits of a groupoid acting on the integral points in certain rational polyhedral cones. Using the orbit-counting lemma for groupoids and theorems on quasipolynomiality of lattice point counts, this yields our Main Theorem., final version, 40 pages, added several examples and clarifications and corrected typos---with many thanks to a referee!
- Published
- 2022
184. Curve valuations and mixed volumes in the implicitization of rational varieties
- Author
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Alicia Dickenstein, María Isabel Herrero, Bernard Mourrain, Departamento de Matemàtica, Universidad de Buenos Aires [Buenos Aires] (UBA), AlgebRe, geOmetrie, Modelisation et AlgoriTHmes (AROMATH), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-National and Kapodistrian University of Athens (NKUA), and MIH and AD were supported by ANPCyT PICT 2016-0398, Argentina. AD was also partially supported by UBACYT 20020170100048BA, CONICET PIP 11220150100473, Argentina.
- Subjects
Mathematics - Algebraic Geometry ,Algebra and Number Theory ,[MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC] ,FOS: Mathematics ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Algebraic Geometry (math.AG) - Abstract
International audience; We address the description of the tropicalization of families of rational varieties under parametrizations with prescribed support, via curve valuations. We recover and extend results by Sturmfels, Tevelev and Yu for generic coefficients, considering rational parametrizations with non-trivial denominator. The advantage of our point of view is that it can be generalized to deal with non-generic parametrizations. We provide a detailed analysis of the degree of the closed image, based on combinatorial conditions on the relative positions of the supports of the polynomials defining the parametrization. We obtain a new formula and finer bounds on the degree, when the supports of the polynomials are different. We also present a new formula and bounds for the order at the origin in case the closed image is a hypersurface.
- Published
- 2022
185. Classification of Kuga fiber varieties
- Author
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Salman Abdulali
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Algebra and Number Theory ,Mathematics::Number Theory ,FOS: Mathematics ,Mathematics::Differential Geometry ,Mathematics::Representation Theory ,Algebraic Geometry (math.AG) ,14G35, 14K10, 32M15 - Abstract
We complete Satake's classification of Kuga fiber varieties by showing that if a representation $\rho$ of a hermitian algebraic group satisfies Satake's necessary conditions, then some multiple of $\rho$ defines a Kuga fiber variety.
- Published
- 2022
186. The classification of rigid hyperelliptic fourfolds
- Author
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Andreas Demleitner and Christian Gleissner
- Subjects
Mathematics - Algebraic Geometry ,14J10, 32G05 (Primary) 14L30, 20H15, 20C15, 32Q15 (Secondary) ,Mathematics::Algebraic Geometry ,Mathematics - Complex Variables ,Mathematics::Number Theory ,Applied Mathematics ,FOS: Mathematics ,Mathematics::Differential Geometry ,Complex Variables (math.CV) ,Representation Theory (math.RT) ,Mathematics::Symplectic Geometry ,Algebraic Geometry (math.AG) ,Mathematics - Representation Theory - Abstract
We provide a fine classification of rigid hyperelliptic manifolds in dimension four up to biholomorphism and diffeomorphism. These manifolds are explicitly described as finite \'etale quotients of a product of four Fermat elliptic curves., Comment: 22 pages, MAGMA code on personal homepage
- Published
- 2022
187. Rational function semifields of tropical curves are finitely generated over the tropical semifield
- Author
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Song, JuAe
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::General Mathematics ,Computer Science::Systems and Control ,General Mathematics ,Mathematics::Rings and Algebras ,FOS: Mathematics ,Mathematics::Optimization and Control ,14T10(Primary), 14T20(Secondary) ,Algebraic Geometry (math.AG) - Abstract
We prove that the rational function semifield of a tropical curve is finitely generated as a semifield over the tropical semifield $\boldsymbol{T} := ( \boldsymbol{R} \cup \{ - \infty \}, \operatorname{max}, +)$ by giving a specific finite generating set. Also, we show that for a finite harmonic morphism between tropical curves $\varphi : \varGamma \to \varGamma^{\prime}$, the rational function semifield of $\varGamma$ is finitely generated as a $\varphi^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$-algebra, where $\varphi^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$ stands for the pull-back of the rational function semifield of $\varGamma^{\prime}$ by $\varphi$., Comment: 16 pages. arXiv admin note: text overlap with arXiv:2110.08091
- Published
- 2022
188. Cones of lines having high contact with general hypersurfaces and applications
- Author
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Francesco Bastianelli, Ciro Ciliberto, Flaminio Flamini, Paola Supino, Bastianelli, Francesco, Ciliberto, Ciro, Flamini, Flaminio, and Supino, Paola
- Subjects
Settore MAT/03 ,Mathematics - Algebraic Geometry ,General Mathematics ,FOS: Mathematics ,Hypersurfaces families ,irrationality degrees ,Algebraic Geometry, high contact, hypersurfaces, curve gonality ,Algebraic Geometry (math.AG) - Abstract
Given a smooth hypersurface $X\subset \mathbb{P}^{n+1}$ of degree $d\geqslant 2$, we study the cones $V^h_p\subset \mathbb{P}^{n+1}$ swept out by lines having contact order $h\geqslant 2$ at a point $p\in X$. In particular, we prove that if $X$ is general, then for any $p\in X$ and $2 \leqslant h\leqslant \min\{ n+1,d\}$, the cone $V^h_p$ has dimension exactly $n+2-h$. Moreover, when $X$ is a very general hypersurface of degree $d\geqslant 2n+2$, we describe the relation between the cones $V^h_p$ and the degree of irrationality of $k$--dimensional subvarieties of $X$ passing through a general point of $X$. As an application, we give some bounds on the least degree of irrationality of $k$--dimensional subvarieties of $X$ passing through a general point of $X$, and we prove that the connecting gonality of $X$ satisfies $d-\left\lfloor\frac{\sqrt{16n+25}-3}{2}\right\rfloor\leqslant\conngon(X)\leqslant d-\left\lfloor\frac{\sqrt{8n+1}+1}{2}\right\rfloor$., Comment: 15 pages; v2: minor changes
- Published
- 2022
189. Mathematical Structures of Non-perturbative Topological String Theory: From GW to DT Invariants
- Author
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Murad Alim, Arpan Saha, Jörg Teschner, and Iván Tulli
- Subjects
High Energy Physics - Theory ,Mathematics - Differential Geometry ,expansion: strong coupling ,FOS: Physical sciences ,nonperturbative ,strong coupling [expansion] ,Mathematics - Algebraic Geometry ,string model: topological ,FOS: Mathematics ,topological [string model] ,Gromov-Witten theory ,ddc:510 ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Mathematical Physics ,partition function [string] ,asymptotic expansion ,mathematical methods ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Borel transformation ,BPS [spectrum] ,string: partition function ,High Energy Physics - Theory (hep-th) ,Differential Geometry (math.DG) ,spectrum: BPS - Abstract
Communications in mathematical physics 399(2), 1039 - 1101 (2023). doi:10.1007/s00220-022-04571-y, We study the Borel summation of the Gromov-Witten potential for the resolved conifold. The Stokes phenomena associated to this Borel summation are shown to encode the Donaldson-Thomas invariants of the resolved conifold, having a direct relation to the Riemann-Hilbert problem formulated by T. Bridgeland. There exist distinguished integration contours for which the Borel summation reproduces previous proposals for the non-perturbative topological string partition functions of the resolved conifold. These partition functions are shown to have another asymptotic expansion at strong topological string coupling. We demonstrate that the Stokes phenomena of the strong-coupling expansion encode the DT invariants of the resolved conifold in a second way. Mathematically, one finds a relation to Riemann-Hilbert problems associated to DT invariants which is different from the one found at weak coupling. The Stokes phenomena of the strong-coupling expansion turn out to be closely related to the wall-crossing phenomena in the spectrum of BPS states on the resolved conifold studied in the context of supergravity by D. Jafferis and G. Moore., Published by Springer, Heidelberg
- Published
- 2022
190. A new line-symmetric mobile infinity-pod
- Author
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Matteo Gallet, Josef Schicho, Gallet, M., and Schicho, J.
- Subjects
Computer Science::Robotics ,FOS: Computer and information sciences ,Computer Science - Robotics ,Mathematics - Algebraic Geometry ,Mathematics (miscellaneous) ,Applied Mathematics ,FOS: Mathematics ,Robotics (cs.RO) ,Algebraic Geometry (math.AG) ,Parallel manipulator ,Mathematical Physics - Abstract
We construct parallel manipulators with one degree of freedom and admitting infinitely many legs lying on a curve of degree ten and genus six. Our technique relies upon a duality between the spaces parametrizing all the possible legs and all the possible configurations of a manipulator. Before describing our construction, we show how this duality helps explaining several known phenomena regarding mobility of parallel manipulators., Comment: 14 pages
- Published
- 2022
191. Quiver Symmetries and Wall-Crossing Invariance
- Author
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Fabrizio Del Monte and Pietro Longhi
- Subjects
High Energy Physics - Theory ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,High Energy Physics - Theory (hep-th) ,FOS: Mathematics ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Mathematics::Symplectic Geometry ,Algebraic Geometry (math.AG) ,Mathematical Physics - Abstract
We study the BPS particle spectrum of five-dimensional superconformal field theories (SCFTs) on $\mathbb{R}^4\times S^1$ with one-dimensional Coulomb branch, by means of their associated BPS quivers. By viewing these theories as arising from the geometric engineering within M-theory, the quivers are naturally associated to the corresponding local Calabi-Yau threefold. We show that the symmetries of the quiver, descending from the symmetries of the Calabi-Yau geometry, together with the affine root lattice structure of the flavor charges, provide equations for the Kontsevich-Soibelman wall-crossing invariant. We solve these equations iteratively: the pattern arising from the solution is naturally extended to an exact conjectural expression, that we provide for the local Hirzebruch $\mathbb{F}_0$, and local del Pezzo $dP_3$ and $dP_5$ geometries. Remarkably, the BPS spectrum consists of two copies of suitable $4d$ $\mathcal{N}=2$ spectra, augmented by Kaluza-Klein towers., Comment: 48 pages; v2 minor corrections
- Published
- 2022
192. Vologodsky integration on curves with semi-stable reduction
- Author
-
Besser, Amnon and Zerbes, Sarah Livia
- Subjects
Mathematics - Algebraic Geometry ,Mathematics - Number Theory ,Mathematics::Number Theory ,General Mathematics ,FOS: Mathematics ,Number Theory (math.NT) ,11S80, 11G20 (Primary) 14G22, 14F40 (Secondary) ,Algebraic Geometry (math.AG) - Abstract
We prove that the Vologodsky integral of a mermorphic one-form on a curve over a $p$-adic field with semi-stable reduction restrict to Coleman integrals on the rigid subdomains reducing to the components of the smooth part of the special fiber and that on the connecting annuli the differences of these Coleman integrals form a harmonic cochain on the edges of the dual graph of the special fiber. This determines the Vologodsky integral completely. We analyze the behavior of the integral on the connecting annuli and we explain the results in the case of a Tate elliptic curve., 6 pages
- Published
- 2022
193. Local-global principles for homogeneous spaces of reductive groups over global function fields
- Author
-
Demarche, Cyril and Harari, David
- Subjects
Mathematics - Algebraic Geometry ,14G12, 11G35, 20G30 ,Mathematics - Number Theory ,FOS: Mathematics ,Ocean Engineering ,Number Theory (math.NT) ,Algebraic Geometry (math.AG) - Abstract
Let $K$ be a global field of positive characteristic. We prove that the Brauer-Manin obstructions to the Hasse principle, to weak approximation and to strong approximation are the only ones for homogeneous spaces of reductive groups with reductive stabilizers. The methods involve abelianization techniques and arithmetic duality theorems for complexes of tori over K., Comment: 31 pages
- Published
- 2022
194. Maximally Twisted Eleven-Dimensional Supergravity
- Author
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Richard Eager and Fabian Hahner
- Subjects
High Energy Physics - Theory ,High Energy Physics::Theory ,Mathematics - Algebraic Geometry ,High Energy Physics - Theory (hep-th) ,FOS: Mathematics ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Mathematics::Differential Geometry ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Mathematics::Algebraic Topology ,Mathematical Physics - Abstract
We perform the maximal twist of eleven-dimensional supergravity. This twist is partially topological and exists on manifolds of $G_2 \times SU(2)$ holonomy. Our derivation starts with an explicit description of the Batalin-Vilkovisky complex associated to the three-form multiplet in the pure spinor superfield formalism. We then determine the $L_\infty$ module structure of the supersymmetry algebra on the component fields. We twist the theory by modifying the differential of the Batalin-Vilkovisky complex to incorporate the action of a scalar supercharge. We find that the resulting free twisted theory is given by the tensor product of the de Rham and Dolbeault complexes of the respective $G_2$ and $SU(2)$ holonomy manifolds as conjectured by Costello., 40 pages, 10 tables
- Published
- 2022
195. Non-degeneracy of Cohomological Traces for General Landau–Ginzburg Models
- Author
-
Dmitry Doryn and Calin Iuliu Lazaroiu
- Subjects
High Energy Physics - Theory ,Mathematics - Algebraic Geometry ,High Energy Physics - Theory (hep-th) ,Mathematics::K-Theory and Homology ,Mathematics - Complex Variables ,32C37, 32C81, 2S20, 32Q99 ,FOS: Mathematics ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Complex Variables (math.CV) ,Algebraic Geometry (math.AG) ,Mathematical Physics - Abstract
We prove non-degeneracy of the cohomological bulk and boundary traces for general open-closed Landau-Ginzburg models associated to a pair $(X,W)$, where $X$ is a non-compact complex manifold with trivial canonical line bundle and $W$ is a complex-valued holomorphic function defined on $X$, assuming only that the critical locus of $W$ is compact (but may not consist of isolated points). These results can be viewed as certain "deformed" versions of Serre duality. The first amounts to a duality property for the hypercohomology of the sheaf Koszul complex of $W$, while the second is equivalent with the statement that a certain power of the shift functor is a Serre functor on the even subcategory of the $\mathbb{Z}_2$-graded category of topological D-branes of such models., 29 pages
- Published
- 2022
196. Two-dimensional categorified Hall algebras
- Author
-
Mauro Porta, Francesco Sala, Institut de Recherche Mathématique Avancée (IRMA), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
High Energy Physics - Theory ,General Mathematics ,[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] ,FOS: Physical sciences ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Mathematics::Category Theory ,FOS: Mathematics ,surface ,Algebraic Topology (math.AT) ,14A20 (Primary), 17B37, 55P99 (Secondary) ,Mathematics - Algebraic Topology ,moduli ,Representation Theory (math.RT) ,dimension: 2 ,Mathematics::Representation Theory ,enhancement ,Algebraic Geometry (math.AG) ,[PHYS.HTHE]Physics [physics]/High Energy Physics - Theory [hep-th] ,Applied Mathematics ,algebra: Higgs ,stability ,coherence ,High Energy Physics - Theory (hep-th) ,category ,cohomology ,Mathematics - Representation Theory - Abstract
In the present paper, we introduce two-dimensional categorified Hall algebras of smooth curves and smooth surfaces. A categorified Hall algebra is an associative monoidal structure on the stable $\infty$-category $\mathsf{Coh}^{\mathsf{b}}(\mathbb{R}\mathsf{M})$ of complexes of sheaves with bounded coherent cohomology on a derived moduli stack $\mathbb{R}\mathsf{M}$. In the surface case, $\mathbb{R}\mathsf{M}$ is a suitable derived enhancement of the moduli stack $\mathsf{M}$ of coherent sheaves on the surface. This construction categorifies the K-theoretical and cohomological Hall algebras of coherent sheaves on a surface of Zhao and Kapranov-Vasserot. In the curve case, we define three categorified Hall algebras associated with suitable derived enhancements of the moduli stack of Higgs sheaves on a curve $X$, the moduli stack of vector bundles with flat connections on $X$, and the moduli stack of finite-dimensional local systems on $X$, respectively. In the Higgs sheaves case we obtain a categorification of the K-theoretical and cohomological Hall algebras of Higgs sheaves on a curve of Minets and Sala-Schiffmann, while in the other two cases our construction yields, by passing to $\mathsf K_0$, new K-theoretical Hall algebras, and by passing to $\mathsf H_\ast^{\mathsf{BM}}$, new cohomological Hall algebras. Finally, we show that the Riemann-Hilbert and the non-abelian Hodge correspondences can be lifted to the level of our categorified Hall algebras of a curve., Comment: v4: 71 pages; Final version. v3: 69 pages; introduction expanded; proofs in Sections 2 and 3 strengthened; new appendix about ind quasi-compact stacks and their correspondences included. The main results are unchanged
- Published
- 2022
197. Non-torsion Brauer groups in positive characteristic
- Author
-
Esser, Louis
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::K-Theory and Homology ,Applied Mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,FOS: Mathematics ,14F22 (primary) ,Mathematics::Representation Theory ,Algebraic Geometry (math.AG) - Abstract
Unlike the classical Brauer group of a field, the Brauer-Grothendieck group of a singular scheme need not be torsion. We show that there exist integral normal projective surfaces over a large field of positive characteristic with non-torsion Brauer group. In contrast, we demonstrate that such examples cannot exist over the algebraic closure of a finite field., Comment: 5 pages. Comments welcome! v3: final version, to appear in Proceedings of the American Mathematical Society
- Published
- 2022
198. Maximal variation of curves on K3 surfaces
- Author
-
Dutta, Yajnaseni and Huybrechts, Daniel
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,General Mathematics ,FOS: Mathematics ,Algebraic Geometry (math.AG) - Abstract
We prove that curves in a non-primitive, base point free, ample linear system on a K3 surface have maximal variation. The result is deduced from general restriction theorems applied to the tangent bundle. We also show how to use specialisation to spectral curves to deduce information about the variation of curves contained in a K3 surface more directly. The situation for primitive linear systems is not clear at the moment. However, the maximal variation holds in genus two and can, in many cases, be deduced from a recent result of van Geemen and Voisin confirming a conjecture due to Matsushita., 19 pages. Comments welcome. Proper credit to the work of Ciliberto, Dedieu and Sernesi added
- Published
- 2022
199. The spectrum of a well‐generated tensor‐triangulated category
- Author
-
Krause, Henning and Letz, Janina C.
- Subjects
Mathematics::Logic ,Mathematics - Algebraic Geometry ,Mathematics::Category Theory ,18G80 (primary), 18F70, 18C35 (secondary) ,General Mathematics ,FOS: Mathematics ,Category Theory (math.CT) ,Mathematics - Category Theory ,Representation Theory (math.RT) ,Algebraic Geometry (math.AG) ,Mathematics - Representation Theory - Abstract
For a tensor triangulated category and any regular cardinal $\alpha$ we study the frame of $\alpha$-localizing tensor ideals and its associated space of points. For a well-generated category and its frame of localizing tensor ideals we provide conditions such that the associated space is obtained by refining the topology of the corresponding space for the triangulated subcategory of $\alpha$-compact objects. This is illustrated by several known examples for $\alpha=\aleph_0$, and new spaces arise for $\alpha>\aleph_0$., Comment: 20 pages. Revisions. To appear in Bulletin of the London Mathematical Society
- Published
- 2022
200. On canonical Fano intrinsic quadrics
- Author
-
Hische, Christoff
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,General Mathematics ,FOS: Mathematics ,14J45, 14J30, 14L30 ,Algebraic Geometry (math.AG) - Abstract
We classify all $\mathbb{Q}$-factorial Fano intrinsic quadrics of dimension three and Picard number one having at most canonical singularities., Comment: 17 pages
- Published
- 2022
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