Equidistribution of Hecke orbits on the Picard group of definite Shimura curves

Picard groups of quasi-Frobenius algebras and a question on Frobenius strongly graded algebras

Abstract We introduce a notion of equivariant vector bundles on schemes over semirings. We do this by considering the functor of points of a locally free sheaf. We prove that every toric vector bundle on a toric scheme X over an idempotent semifield equivariantly splits as a sum of toric line bundles. We then study the equivariant Picard group $\operatorname{Pic}_G(X)$ . Finally, we prove a version of Klyachko’s classification theorem for toric vector bundles over an idempotent semifield.

The inverse limit topology and profinite descent on Picard groups in K(n)-local homotopy theory

Integral Picard group of some stacks of polarized K3 surfaces of low degree

Abstract We show that K3 surfaces in characteristic 2 can admit sets of disjoint smooth rational curves whose sum is divisible by 2 in the Picard group, for each . More precisely, all values occur on supersingular K3 surfaces, with exceptions only at Artin invariants 1 and 10, while on K3 surfaces of finite height, only is possible.

Abstract It was shown by the first author that, over an algebraically closed field of characteristic zero, every fusion 2-category is Morita equivalent to a connected fusion 2-category, that is, one arising from a braided fusion 1-category. This result has recently allowed for a complete classification of fusion 2-categories. Here we establish that compact semisimple tensor 2-categories, which generalize fusion 2-categories to an arbitrary field of characteristic zero, also enjoy this “Morita connectedness” property. In order to do so, we generalize to an arbitrary field of characteristic zero many well-known results about braided fusion 1-categories over an algebraically closed field. Most notably, we prove that the Picard group of any braided fusion 1-category is indfinite, generalizing the classical fact that the Brauer group of a field is torsion. As an application of our main result, we derive the existence of braided fusion 1-categories indexed by the fourth Galois cohomology group of the absolute Galois group that represent interesting classes in the appropriate Witt groups.

Abstract We develop a mechanism of “isotropy separation for compact objects” that explicitly describes an invertible ‐spectrum through its collection of geometric fixed points and gluing data located in certain variants of the stable module category. As an application, we carry out a complete analysis of possible combinations of geometric fixed points of invertible ‐spectra in the case . A further application is given by showing that the Picard groups of and a category of derived Mackey functors agree.

On Picard groups and Jacobians of directed graphs

Abstract Given an associative ‐algebra , we call strongly rigid if for any pair of finite subgroups of its automorphism groups , such that , then and must be isomorphic. In this paper, we show that a large class of filtered quantizations are strongly rigid. We also solve the inverse Galois problem for a wide class of rational Cherednik algebras that includes all (simple) classical generalized Weyl algebras, and also for quantum tori. Finally, we show that the Picard group of an ‐dimensional quantum torus is isomorphic to the group of its outer automorphisms.

Diagonal property and weak point property of higher rank divisors and certain Hilbert schemes

Matrix invertible extensions over commutative rings. Part I: General theory

Abstract In this paper, we study the class of polytopes which can be obtained by taking the convex hull of some subset of the points $\{e_i-e_j \ \vert \ i \neq j\} \cup \{\pm e_i\}$ in $\mathbb {R}^n$ , where $e_1,\dots ,e_n$ is the standard basis of $\mathbb {R}^n$ . Such a polytope can be encoded by a quiver Q with vertices $V \subseteq \{{\upsilon }_1,\dots ,{\upsilon }_n\} \cup \{\star \}$ , where each edge ${\upsilon }_j\to {\upsilon }_i$ or $\star \to {\upsilon }_i$ or ${\upsilon }_i\to \star $ gives rise to the point $e_i-e_j$ or $e_i$ or $-e_i$ , respectively; we denote the corresponding polytope as $\operatorname {Root}(Q)$ . These polytopes have been studied extensively under names such as edge polytope and root polytope . We show that if the quiver Q is strongly-connected, then the root polytope $\operatorname {Root}(Q)$ is reflexive and terminal ; we moreover give a combinatorial description of the facets of $\operatorname {Root}(Q)$ . We also show that if Q is planar, then $\operatorname {Root}(Q)$ is (integrally equivalent to) the polar dual of the flow polytope of the planar dual quiver $Q^{\vee }$ . Finally, we consider the case that Q comes from the Hasse diagram of a finite ranked poset P and show in this case that $\operatorname {Root}(Q)$ is polar dual to (a translation of) a marked order polytope . We then go on to study the toric variety $Y(\mathcal {F}_Q)$ associated to the face fan $\mathcal {F}_Q$ of $\operatorname {Root}(Q)$ . If Q comes from a ranked poset P , we give a combinatorial description of the Picard group of $Y(\mathcal {F}_Q)$ , in terms of a new canonical ranked extension of P , and we show that $Y(\mathcal {F}_Q)$ is a small partial desingularisation of the Hibi projective toric variety $Y_{\mathcal {O}(P)}$ of the order polytope $\mathcal {O}(P)$ . We show that $Y(\mathcal {F}_Q)$ has a small crepant toric resolution of singularities $Y(\widehat {\mathcal {F}}_Q)$ and, as a consequence that the Hibi toric variety $Y_{\mathcal {O}(P)}$ has a small resolution of singularities for any ranked poset P . These results have applications to mirror symmetry [61].

Let S be a scheme such that 2 is not a zero divisor. In this paper, we address the following question: given a quadratic algebra over S, how can we parametrize its Picard group in terms of quadratic forms? In 2011, Wood established a set-theoretical bijection between isomorphism classes of primary binary quadratic forms over S and isomorphism classes of pairs [Formula: see text] where [Formula: see text] is a quadratic algebra over S and [Formula: see text] is an invertible [Formula: see text]-module. Unexpectedly, examples suggest that a refinement of Wood’s bijection is needed in order to parametrize Picard groups. This is why we start by classifying quadratic algebras over S; this is achieved by using two invariants, the discriminant and the parity. Extending the notion of orientation of quadratic algebras to the non-free case is another key step, eventually leading us to the desired parametrization. All along the paper, we illustrate various notions and obstructions with a wide range of examples.

Abstract Given a connected reductive algebraic group G over an algebraically closed field, we investigate the Picard group of the moduli stack of principal G-bundles over an arbitrary family of smooth curves.

Given an integral domain D and a D-algebra R, we introduce the local Picard group LPic(R,D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{LPic}(R,D)$$\\end{document} as the quotient between the Picard group Pic(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Pic}(R)$$\\end{document} and the canonical image of Pic(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Pic}(D)$$\\end{document} in Pic(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Pic}(R)$$\\end{document}, and its subgroup LPicu(R,D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{LPic}_u(R,D)$$\\end{document} generated by the the integral ideals of R that are unitary with respect to D. We show that, when D⊆R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D\\subseteq R$$\\end{document} is a ring extension that satisfies certain properties (for example, when R is the ring of polynomial D[X] or the ring of integer-valued polynomials Int(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Int}(D)$$\\end{document}), it is possible to decompose LPic(R,D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{LPic}(R,D)$$\\end{document} as the direct sum ⨁LPic(RT,T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bigoplus \ extrm{LPic}(RT,T)$$\\end{document}, where T ranges in a Jaffard family of D. We also study under what hypothesis this isomorphism holds for pre-Jaffard families of D.

Abstract We study when the Picard group of smooth surfaces of degree $d\geq 5$ in $\mathbb{P}^{3}$ acquires extra classes. In particular we show that the so-called exceptional components of the Noether–Lefschetz locus are not Zariski dense. This answers a 1991 question of C. Voisin. We also obtain similar results for the Noether–Lefschetz locus for suitable $(Y,L)$, where $Y$ is a smooth projective three-fold and $L$ a very ample line bundle. Both results are applications of the Zilber–Pink viewpoint recently developed by the authors for arbitrary (polarized, integral) variations of Hodge structures.

Let Ln for a positive integer n denote the stable homotopy category of v −1 n BP -local spectra at a prime number p. Then, M. Hopkins defines the Picard group of Ln as a collection of isomorphism classes of invertible spectra, whose exotic summand Pic 0 (Ln) is studied by several authors. In this paper, we study the summand for n with n 2 ≤ 2p + 2. For n 2 ≤ 2p − 2, it consists of invertible spectra whose K(n)-localization is the K(n)-local sphere. In particular, X is an exotic invertible spectrum of Ln if and only if X ∧ M J is isomorphic to a v −1 n BP -localization of the generalized Moore spectrum M J for an invarinat regular ideal J of length n. For n with 2p−2 < n 2 ≤ 2p+2, we consider the cases for (p, n) = (5, 3) and (7, 4) . In these cases, we characterize them by the Smith-Toda spectra V (n − 1). For this sake, we show that L 3 V (2) at the prime five and L 4 V (3) at the prime seven are ring spectra.

We determine the rational class and Picard groups of the moduli space of stable logarithmic maps in genus zero, with target projective space relative a hyperplane. For the class group we exhibit an explicit basis consisting of boundary divisors. For the Picard group we exhibit a spanning set indexed by piecewise-linear functions on the tropicalisation. In both cases a complete set of boundary relations is obtained by pulling back the WDVV relations from the space of stable curves. Our proofs hinge on a controlled technique for manufacturing test curves in logarithmic mapping spaces, opening up the topology of these spaces to further study.

We prove duality theorems for the étale cohomology of split tori on smooth curves over a local field of positive characteristic. In particular, we show that the classical Brauer–Manin pairing between the Brauer and Picard groups of smooth projective curves over such a field extends to arbitrary smooth curves over the field. As another consequence, we obtain a description of the Brauer group of the function fields of curves over local fields in terms of the characters of the idele class groups.