Bundles over the Fano Threefold V 5

ABSTRACT

Using an explicit resolution of the diagonal for the variety V ₅, we provide cohomological characterizations of the universal and quotient bundles. A splitting criterion for bundles over V ₅ is also proved.

The presentation of semistable aCM bundles is shown, together with a resolution–theoretic classification of low rank aCM bundles.     

Orbits of the Actions of Odd-Order Solvable Groups

Suppose that V is a finite faithful irreducible G-module where G is a finite solvable group of odd order. We prove if the action is quasi-primitive, then either F(G) is abelian or G has at least 212 regular orbits on V. As an application, we prove that when V is a finite faithful completely reducible G-module for a solvable group G of odd order, then there exists v ∈ V such that C _G (v) ? F ₂(G) (where F ₂(G) is the 2nd ascending Fitting subgroup of G). We also generalize a result of Espuelas and Navarro. Let G be a group of odd order and let H be a Hall π-subgroup of G. Let V be a faithful G-module over a finite field of characteristic 2, then there exists v ∈ V such that C _H (v) ? O _π(G).     

Constructing G-algebras

In this article we define G-algebras, that is, graded algebras on which a reductive group G, acts as gradation preserving automorphisms. Starting from a finite dimensional G-module V and the polynomial ring ?[V], it is shown how one constructs a sequence of projective varieties V_k such that each point of V_k corresponds to a graded algebra with the same decomposition up to degree k as a G-module. After some general theory, we apply this to the case that V is the n+1-dimensional permutation representation of S_n+1, the permutation group on n+1 letters.     

A new construction of anti-self-dual four-manifolds

We describe a new construction of anti-self-dual metrics on four-manifolds. These metrics are characterized by the property that their twistor spaces project as affine line bundles over surfaces. To any affine bundle with the appropriate sheaf of local translations, we associate a solution of a second-order partial differential equations system D ₂ V = 0 on a five-dimensional manifold Y{\mathbf{Y}}. The solution V and its differential completely determine an anti-self-dual conformal structure on an open set in {V = 0}. We show how our construction applies in the specific case of conformal structures for which the twistor space Z{\mathcal{Z}} has dim|-\frac12K_Z| 3 2{ \dim\left|-\frac{1}{2}K_\mathcal{Z}\right|\geq 2}, projecting thus over \mathbb C\mathbb P₂{\mathbb C\mathbb P_2} with twistor lines mapping onto plane conics.     

Elementary transformations and the rationality of the Moduli Spaces of Vector Bundles on ℙ<Superscript>2</Superscript>

In this work, using elementary transformations and prioritary sheaves, we establish birational maps between certain moduli spaces of stable vector bundles over ² with the same rank and different Chern classes. As an application we give a simple proof of the rationality of the moduli spaces M(r; c ₁, c ₂) of rank r stable vector bundles over ² with given Chern classes for a huge families of the triples (r; c ₁, c ₂).Partially supported by BFM2001-3584 Mathematics Subject Classification (2000):Primary 14D20, 14D05; Secondary 14F05     

Décomposition d"une théorie du potentiel en des sous-espaces elliptiques et paraboliques

Let V be a proper kernel on a measurable space (X, B), and E _V the cone of excessive functions generated by V. We give a necessary and sufficient condition to decompose the space (X, E _V ) in an ordered and countable family of subspaces (X _n, E _{V n} ), which are elliptic or parabolic. The X _i 's are finely open and measurable and form a partition of X. The kernel W = V _i on X _i is subordinate to V, and has a triangular matrix. Résumé. Soit V un noyau propre sur un espace mesurable (X, B), et E _V le cône des fonctions excessives engendré par V. Nous donnons une condition nécéssaire et suffisante pour décomposer l"espace (X, E _V ) en une famille dénombrable et ordonnée (X _n , E _{V n} ) de sous-espaces elliptiques ou paraboliques. Les X _i sont des ouverts fins mesurables et forment une partition de X. Le noyau W = V _i sur X _i est subordonné à V, et sa matrice est triangulaire.     

On Units of Group Algebras of 2-Groups of Maximal Class

Abstract

We determine the number of elements of order two in the group of normalized units V(𝔽₂ G) of the group algebra 𝔽₂ G of a 2-group of maximal class over the field 𝔽₂ of two elements. As a consequence for the 2-groups G and H of maximal class we have that V(𝔽₂ G) and V(𝔽₂ H) are isomorphic if and only if G and H are isomorphic.     

Group-universal unary varieties

LetV be a variety of unary algebras and letM(V) be the monoid of all unary polynomials ofV. Then every group appears as the automorphism group of an algebraA∈V if and only if the left ideals ofM(V) do not form an inclusion-ordered chain. The support of the National Research Council of Canada is gratefully acknowledged. Presented by J. Mycielski.     

A regularity theorem for deformations of a compact complex surface

Let ƒ:M →D ⊑C ⁿ be a holomorphic family of compact, complex surfaces, which is locally trivial onD∖Z, for an analytic subsetZ. Conditions are found under which ƒ extends trivially toD, if the fibers of ƒ|_D∖Z are either Hirzebruch surfaces (projective bundles overP ₁), Hopf surfaces (elliptic bundles overP ₁), hyperelliptic bundles, or any compact complex surface having one of these as minimal model under blowing-down. The results of this paper are motivated by the existence of non-Hausdorff moduli spaces in the deformation of complex structure for certain complex manifolds.     

Tight products and graph expansion

In this article, we study a new product of graphs called tight product. A graph H is said to be a tight product of two (undirected multi) graphs G₁ and G₂, if V(H) = V(G₁) × V(G₂) and both projection maps V(H)→V(G₁) and V(H)→V(G₂) are covering maps. It is not a priori clear when two given graphs have a tight product (in fact, it is NP‐hard to decide). We investigate the conditions under which this is possible. This perspective yields a new characterization of class‐1 (2k+ 1)‐regular graphs. We also obtain a new model of random d‐regular graphs whose second eigenvalue is almost surely at most O(d^3/4). This construction resembles random graph lifts, but requires fewer random bits. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Integral points on quadrics with prime coordinates

Let f (x ₁, . . . , x _s ) be a regular indefinite integral quadratic form, and t an integer. Denote by V the affine quadric {x : f (x) = t}, and by V(\mathbb P){V(\mathbb {P})} the set of x ? V{{\bf x}\in V} whose coordinates are simultaneously prime. It is proved that, under suitable conditions, V(\mathbbP){V(\mathbb{P})} is Zariski dense in V as long as s ≥ 10.     

M-solid varieties generated by lattices

Following W. Taylor, we define an identity to be hypersatisfied by a variety V iff, whenever the operation symbols of V are replaced by arbitrary terms (of appropriate arity) in the operations of V, then the resulting identity is satisfied by V in the usual sense. Whenever the identity is hypersatisfied by a variety V, we shall say that is a hyperidentity of V, or a V hyperidentity. When the terms being substituted are restricted to a submonoid M of all the possible choices, is called an M-hyperidentity, and a variety V is M-solid if each identity is an M-hyperidentity. In this paper we examine the solid varieties whose identities are lattice M-hyperidentities. The M-solid varieties generated by the variety of lattices in this way provide new insight on the construction and representation of various known classes of non-commutative lattices. Received October 8, 1999; accepted in final form March 22, 2000.     

Judicious partitions of bounded‐degree graphs

We prove results on partitioning graphs G with bounded maximum degree. In particular, we provide optimal bounds for bipartitions V(G) = V₁ ∪ V₂ in which we minimize {e(V₁), e(V₂)}. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 131–143, 2004     

A Random Set Extension of a Strong Law of Large Numbers of Cantrell and Rosalsky

Abstract

A strong law of large numbers under conditions irrespective of the joint distribution of the sequence is extended to random sets. The extension is such that the role of events of the form {||V _n || ≤ b _n } (where V _n is a random element of a separable Banach space) is played by events of the form {X _n  ? B _n } (where X _n is a random closed bounded set).     

On the errors of misclassification based on dichotomous and normal variables

Summary The distribution of the errors of misclassification in procedures based on dichotomous and normal variables is derived. The expressions forE(e ₁₂) andE(e ₂₁) are also obtained. The results in the paper extend those of Chang and Afifi (1974,J. Amer. Statist. Ass.,69, 336–339), using the earlier papers due to John (1961,Ann. Math. Statist.,32, 1125–1144), Subrahmaniam and Chinganda (1978,J. Statist. Plann. Inf.,2, 79–91).     

A Construction of Uniquely C4-free colourable Graphs

Abstract

An F-free colouring of a graph G is a partition {V₁,V₂,…,V_n} of the vertex set V(G) of G such that F is not an induced subgraph of G[V_i] for each i. A graph is uniquely F-free colourable if any two .F-free colourings induce the same partition of V(G). We give a constructive proof that uniquely C₄-free colourable graphs exist.     

Partitions of finite vector spaces into subspaces

Let V_n(q) denote a vector space of dimension n over the field with q elements. A set of subspaces of V_n(q) is a partition of V_n(q) if every nonzero element of V_n(q) is contained in exactly one element of . Suppose there exists a partition of V_n(q) into x_i subspaces of dimension n_i, 1 ≤ i ≤ k. Then x₁, …, x_k satisfy the Diophantine equation . However, not every solution of the Diophantine equation corresponds to a partition of V_n(q). In this article, we show that there exists a partition of V_n(2) into x subspaces of dimension 3 and y subspaces of dimension 2 if and only if 7x + 3y = 2ⁿ ? 1 and y ≠ 1. In doing so, we introduce techniques useful in constructing further partitions. We also show that partitions of V_n(q) induce uniformly resolvable designs on qⁿ points. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 329–341, 2008     

Pseudodifferential Operators on Prehomogeneous Vector Spaces

ABSTRACT

Let G be a connected, linear algebraic group defined over ?, acting regularly on a finite dimensional vector space V over ? with ?-structure V _?. Assume that V possesses a Zariski-dense orbit, so that (G, ?, V) becomes a prehomogeneous vector space over ?. We consider the left regular representation π of the group of ?-rational points G _? on the Banach space C₀(V _?) of continuous functions on V _? vanishing at infinity, and study the convolution operators π(f), where f is a rapidly decreasing function on the identity component of G _?. Denote the complement of the dense orbit by S, and put S _? = S ∩ V _?. It turns out that, on V _? ? S _?, π(f) is a smooth operator. If S _? = {0}, the restriction of the Schwartz kernel of π(f) to the diagonal defines a homogeneous distribution on V _? ? {0}. Its nonunique extension to V _? can then be regarded as a trace of π(f). If G is reductive, and S and S _? are irreducible hypersurfaces, π(f) corresponds, on each connected component of V _? ? S _?, to a totally characteristic pseudodifferential operator. In this case, the restriction of the Schwartz kernel of π(f) to the diagonal defines a distribution on V _? ? S _? given by some power |p(m)| ^s of a relative invariant p(m) of (G, ?, V) and, as a consequence of the Fundamental Theorem of Prehomogeneous Vector Spaces, its extension to V _?, and the complex s-plane, satisfies functional equations similar to those for local zeta functions. A trace of π(f) can then be defined by subtracting the singular contributions of the poles of the meromorphic extension.