Group Pic Research Articles

On the parallelism between algebraic and analytic Picard groups of algebraic affine hypersurfaces

For any completeC-algebraic variety X and its associated compactC-analytic space X, it follows from the well known GAGA principle that the algebraic Picard group Pic(X) and the analytic Picard group Pic(X) are isomorphic. Our main purpose here is to investigate an analogous situation for non complete situation, namely C-algebraic affine hypersurfaces.

Workshop: Research on integrated and personalized care, sharing values and perspectives; a world café method

Introduction/Background: PhD research is one of the directions to contribute to the development of integrated care and contribute to international conferences such as ICIC in. Contributions are made towards the nine pillars of integrated care. Results are added, mostly as part of an individual PhD study, in a wide array of subjects which fits the nine pillars of integrated care. Doing PhD research brings difficulties but also successes. This also applies to PhD researchers and lecturers of the research and education group PIC’T&Re Personalized Integrated Care Teaching & Research of the University of Applied Sciences Utrecht in the Netherlands. The research group aims to contribute to integration knowledge which can facilitate changes towards the customization of integrated healthcare service delivery in the Netherlands on micro, meso and macro level of health and social care.
 Why a workshop? Using a workshop as a method to share values and perspectives in doing PhD research amongst the diversity of PhD researchers within the integrated care community. Also, to exchange experiences which are of value to all researchers in integrated care. Exchange of knowledge, expertise and experiences of all attendees is central in this workshop.
 Who is it for? The target audience for this workshop consists of (PhD) students, researchers, educators and professionals in integrated health and social care.
 Facilitators / speakers 
 Dennis van Kerkvoorden MSc, PhD candidate 
 Ferdy Pluck RN MSc, PhD candidate
 Dr Roelof Ettema, co-supervisor
 Dr Ruben van Zelm, co-supervisor
 What are we going to do? – Workshop 90’
 a.Introduction and background of the world café method and themes – (15’)
 b.World cafe: in three rounds of discussion and 5 table discussions based on the themes of the 2023 conference on Integrated care – (3 x 20’): 
 1.Change management implementation: macro, meso and micro level
 2.Person-centered care: focus on integration of goals and needs
 3.Collaboration: factor for successful implementation of integrated care
 4.Engaging everybody: caring neighborhoods and compassionate communities
 5.Population health approach: leverage for integrated care
 c.Pitches and Conclusions and general reflections (15’)
 How are you going to engage with the audience? Everybody is an expert in her or his own research, education or practice. In small groups of 5-10 persons, we will discuss one of the five themes mentioned above. We challenge everyone to share their own experiences and ideas. 
 How are you going to summarize the take home messages? All collected results will be summarized by the organizing committee of this workshop. These results will be shared with the participants and IFIC Academy with the intension of a yearly follow-up.
 Keywords: research, integrated and personalized care, world café

The cohomology of the general stable sheaf on a K3 surface

Let X be a K3 surface with Picard group Pic(X)≅ZH such that H2=2n. Let MH(v) be the moduli space of Gieseker semistable sheaves on X with Mukai vector v. We say that v satisfies weak Brill-Noether if the general sheaf in MH(v) has at most one nonzero cohomology group. We show that given any rank r≥2, there are only finitely many Mukai vectors of rank r on K3 surfaces of Picard rank one where weak Brill-Noether fails. We give an algorithm for finding the potential counterexamples and classify all such counterexamples up to rank 20 explicitly. Moreover, in each of these cases we calculate the cohomology of the general sheaf. Given r, we give sharp bounds on n, d, and a that guarantee that v satisfies weak Brill-Noether. As a corollary, we obtain another proof of the classification of Ulrich bundles on K3 surfaces of Picard rank one. In addition, we discuss the question of when the general sheaf in MH(v) is globally generated. Our techniques make crucial use of Bridgeland stability conditions.

Digital Veneering Techniques for Zirconia Implant-Supported Single Crowns-Bond Strength and Clinical Application.

To evaluate the shear bond strength (SBS) of different digital veneering techniques for zirconia and to critically discuss its suitability for application in single-implant prosthetics. A total of 112 square-shaped zirconia specimens were provided with four different veneering materials (n = 28 per group): a glass-ceramic (group GLA), a feldspathic ceramic (group FEL), a polymer-infiltrated ceramic network (group PIC), and a resin nanoceramic (group RNC). Discs in group GLA were sintered onto the core material, whereas all other specimens were adhesively connected. In each group, 14 specimens (GLA0, FEL0, PIC0, RNC0) were subjected to SBS testing before thermocycling, and the other 14 (GLA1, FEL1, PIC1, RNC1) were tested after thermocycling (10,000 cycles). Data were analyzed by applying SPSS software (P < .05). The surfaces and fracture patterns of the specimens were examined using scanning electron microscopy (SEM). Mean SBS values ranged from 14.09 ± 3.87 MPa (RNC1) to 40.82 ± 4.91 MPa (GLA0). Group GLA presented higher values than all other groups (P < .001). Groups FEL, PIC, and RNC showed no statistically significant differences between them. SBS decreased after thermocycling, but no significant impact was found. Every group exhibited a characteristic failure mode. All digital veneering techniques sufficed to present clinically acceptable SBS values and might be viable alternatives in implant prosthetics. However, some have yet to demonstrate their long-term clinical suitability. At present, lithium disilicate-veneered zirconia abutments and monolithic lithium disilicate hybrid abutment crowns seem to present a proven and reliable restorative option.

MP74-07 RENAL PROTECTION BY APPLYING ISCHEMIC PRECONDITIONING.

MP74-07 RENAL PROTECTION BY APPLYING ISCHEMIC PRECONDITIONING.

The Bernardi Process and Torsor Structures on Spanning Trees

Let G be a ribbon graph, i.e., a connected finite graph G together with a cyclic ordering of the edges around each vertex. By adapting a construction due to O. Bernardi, we associate to any pair (v,e) consisting of a vertex v and an edge e adjacent to v a bijection between spanning trees of G and elements of the set Pic^g(G) of degree g divisor classes on G, where g is the genus of G. Using the natural action of the Picard group Pic^0(G) on Pic^g(G), we show that the Bernardi bijection gives rise to a simply transitive action \beta_v of Pic^0(G) on the set of spanning trees which does not depend on the choice of e. A plane graph has a natural ribbon structure (coming from the counterclockwise orientation of the plane), and in this case we show that \beta_v is independent of v as well. Thus for plane graphs, the set of spanning trees is naturally a torsor for the Picard group. Conversely, we show that if \beta_v is independent of v then G together with its ribbon structure is planar. We also show that the natural action of Pic^0(G) on spanning trees of a plane graph is compatible with planar duality. These findings are formally quite similar to results of Holroyd et al. and Chan-Church-Grochow, who used rotor-routing to construct an action r_v of Pic^0(G) on the spanning trees of a ribbon graph G, which they show is independent of v if and only if G is planar. It is therefore natural to ask how the two constructions are related. We prove that \beta_v = r_v for all vertices v of G when G is a planar ribbon graph, i.e. the two torsor structures (Bernardi and rotor-routing) on the set of spanning trees coincide. In particular, it follows that the rotor-routing torsor is compatible with planar duality. We conjecture that for every non-planar ribbon graph G, there exists a vertex v with \beta_v \neq r_v.

On cohomology and vector bundles over monoid schemes

The aim of this paper is to study the cohomology theory of monoid schemes in general and apply it to vector and line bundles. We will prove that over separated monoid schemes, any vector bundle is a coproduct of line bundles and that the Pic functor respects finite products.Next we will introduce the notion of s-cancellative monoids and show that if X is locally of that type, OX⁎ can be embed injectively in a constant sheaf. We develop the theory of s-divisors, which generalise the Cartier divisors, and we prove that for s-cancellative monoid schemes, s-divisors describe the group Pic(X).We then generalise smooth monoid schemes with s-smooth monoid schemes, and prove that for them Hi(X,OX⁎)=0 for all i≥2. Furthermore we show that it is a local property and respects finite products.Finally we investigate the relationship between line bundles over a monoid scheme X and over its geometric realisation Xk, where k is a commutative ring. We prove that if k is an integral domain (resp. PID) and X is a cancellative and torsion-free (resp. seminormal and torsion-free) monoid scheme, then the induced map Pic(X)→Pic(Xk) is a monomorphism (resp. isomorphism).

Rotor-Routing and Spanning Trees on Planar Graphs

The sandpile group Pic^0(G) of a finite graph G is a discrete analogue of the Jacobian of a Riemann surface which was rediscovered several times in the contexts of arithmetic geometry, self-organized criticality, random walks, and algorithms. Given a ribbon graph G, Holroyd et al. used the "rotor-routing" model to define a free and transitive action of Pic^0(G) on the set of spanning trees of G. However, their construction depends a priori on a choice of basepoint vertex. Ellenberg asked whether this action does in fact depend on the choice of basepoint. We answer this question by proving that the action of Pic^0(G) is independent of the basepoint if and only if G is a planar ribbon graph.

Picard groups in p-adic Fourier theory

Let $${L\neq\mathbb{Q}_p}$$ be a proper finite field extension of $${\mathbb{Q}_p}$$ and $${o \subset L}$$ its ring of integers viewed as an abelian locally L-analytic group. Let o be the rigid L-analytic group parametrizing the locally analytic characters of o constructed by Schneider–Teitelbaum. Let K/L be a finite extension field. We show that the base change o K has a Picard group Pic(o K ) which is profinite and that the unit section in o K provides a divisor class of infinite order. In particular, the abelian group Pic(o K ) is not finitely generated and is not a torsion group. On the way we show that o K is a nontrivial etale covering of the affine line over K realized via the logarithm map of a Lubin-Tate formal group. We finally prove that rank and determinant mappings induce an isomorphism between K 0(o K ) and $${\mathbb{Z} \oplus {Pic}(\hat{o}_K)}$$ .

Equations of G-minimal conic bundles

We develop a method for obtaining equations for G-minimal conic bundles with an arbitrary number of singular fibres. When the number of singular fibres is equal to 4, 6, or 7, a detailed classification is given, which includes obtaining the equations for minimal conic bundles (S,G) and an explicit indication of the action of the group G on the Picard group Pic(S) and on the surface S itself.Bibliography: 19 titles.

Picard groups of punctured spectra of dimension three local hypersurfaces are torsion-free

AbstractLet (R,m) be a Noetherian local ring and UR=Spec(R)−{m} be the punctured spectrum of R. Gabber conjectured that if R is a complete intersection of dimension three, then the abelian group Pic(UR) is torsion-free. In this note we prove Gabber’s statement for the hypersurface case. We also point out certain connections between Gabber’s conjecture, Van den Bergh’s notion of non-commutative crepant resolutions and some well-studied questions in homological algebra over local rings.

TOPOLOGY OF THE REPRESENTATION VARIETIES WITH BOREL MOLD FOR UNSTABLE CASES

AbstractIn this paper we show that, in the stable case, when m≥2n−1, the cohomology ring H*(Repn(m)B) of the representation variety with Borel mold Repn(m)B and $H^{\ast }(F_{n}(\mathbb {C}^m)) \otimes H^{\ast }(\mathrm {Flag}(\mathbb { C}^n)) \otimes \Lambda (s_{1}, \ldots , s_{n-1})$ are isomorphic as algebras. Here the degree of si is 2m−3 when 1≤i&lt;n. In the unstable cases, when m≤2n−2, we also calculate the cohomology group H*(Repn(m)B) when n=3,4 . In the most exotic case, when m=2 , Rep n (2)B is homotopy equivalent to Fn (ℂ2)×PGL n (ℂ) , where Fn (ℂ2) is the configuration space of n distinct points in ℂ2. We regard Rep n (2)B as a scheme over ℤ, and show that the Picard group Pic (Rep n (2)B) of Rep n (2)B is isomorphic to ℤ/nℤ. We give an explicit generator of the Picard group.

Complex structures on products of circle bundles over complex manifolds

We propose, in this Note, a construction of complex structures on the product of two circle bundles associated to negative ample line bundles over flag varieties X i : = G i / P i , i = 1 , 2 , where the G i are complex semisimple linear Lie groups and the P i ⊂ G i are parabolic subgroups. The resulting manifold S is non-symplectic and hence non-Kählerian. We show that the group Pic 0 ( S ) of topologically trivial holomorphic line bundles on S is isomorphic to C .

A Bicategorical Version of Masuoka’s Theorem

We give a bicategorical version of the main result of Masuoka (Tsukuba J Math 13:353–362, 1989) which proposes a non-commutative version of the fact that for a faithfully flat extension of commutative rings \(R \subseteq S\), the relative Picard group Pic(S/R) is isomorphic to the Amitsur 1–cohomology group H1(S/R,U) with coefficients in the units functor U.

Equations of G-minimal conic bundles with degree 4

We consider the first nontrivial case of relatively G-minimal conic bundles which are G-minimal and have r = 4 singular fibers. Classification obtained gives explicit equations of minimal conic bundles (S, G) and an explicit action of the group G on the Picard group Pic(S) and on the surface S.

Small codimension smooth subvarieties in even-dimensional homogeneous spaces with Picard group [formula omitted

We investigate a method proposed by E. Arrondo and J. Caravantes to study the Picard group of a smooth low-codimensional subvariety X in a variety Y when Y is homogeneous. We prove that this method is strongly related to the signature σ Y of the Poincaré pairing on the middle cohomology of Y. We give under some topological assumptions a bound on the rank of Picard group Pic ( X ) in terms of σ Y and remove these assumptions for Grassmannians to recover the main result of E. Arrondo and J. Caravantes. To cite this article: N. Perrin, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Extended Picard complexes for algebraic groups and homogeneous spaces

For a smooth geometrically integral algebraic variety X over a field k of characteristic 0, we define the extended Picard complex UPic ( X ¯ ) . It is a complex of length 2 which combines the Picard group Pic ( X ¯ ) and the group U ( X ¯ ) : = k ¯ [ X ¯ ] × / k ¯ × , where k ¯ is a fixed algebraic closure of k and X ¯ = X × k k ¯ . For a connected linear k-group G we compute the complex UPic ( G ¯ ) (up to a quasi-isomorphism) in terms of the algebraic fundamental group π 1 ( G ¯ ) . We obtain similar results for a homogeneous space X of a connected k-group G. To cite this article: M. Borovoi, J. van Hamel, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Torsion subgroups of Brauer groups and extensions of constant local degree for global function fields

We show that, for all characteristic p global fields k and natural numbers n coprime to the order of the non- p-part of the Picard group Pic 0 ( k ) of k, there exists an abelian extension L / k whose local degree at every prime of k is equal to n. This answers in the affirmative in this context a question recently posed by Kisilevsky and Sonn. As a consequence, we show that, for all n and k as above, the n-torsion subgroup Br n ( k ) of the Brauer group Br ( k ) of k is algebraic, answering a question of Aldjaeff and Sonn in this context.

Picard Groups and Refined Discrete Logarithms

AbstractLet K denote a number field, and G a finite abelian group. The ring of algebraic integers in K is denoted in this paper by $/cal{O}_K$, and $/cal{A}$ denotes any $/cal{O}_K$-order in K[G]. The paper describes an algorithm that explicitly computes the Picard group Pic($/cal{A}$), and solves the corresponding (refined) discrete logarithm problem. A tamely ramified extension L/K of prime degree l of an imaginary quadratic number field K is used as an example; the class of $/cal{O}_L$ in Pic($/cal{O}_K[G]$) can be numerically determined.

On the variety of plane curves of degree d with a singular point of multiplicity m

We study a compactification of the variety U d , m , 1 < m < d , of plane curves of degree d with an ordinary singular point of multiplicity m as a unique singularity, given by means of a projective bundle X d , m . We show that the boundary X d , m - U d , m consists of two irreducible components of codimension 1: A d , m , the closure of the variety of curves with two singular points, one of multiplicity m and another of multiplicity 2, and B d , m , the closure of the variety of curves with a non-ordinary singular point of multiplicity m. We determine the relations that express the classes of A d , m and B d , m in terms of a basis of the group Pic ( X d , m ) . From this we describe the Picard group of the variety U d , m , obtaining that it is a finite group of order 3 ( m - 1 ) ( d - m ) [ 2 d 2 - ( m + 4 ) d - ( m 2 - 2 m - 2 ) ] .