Canonical curves in P3

Let Hilb^6t–3(P³) be the Hilbert scheme of closed 1-dimensionalsubschemes of degree 6 and arithmetic genus 4 in P³. Let H bethe component of Hilb^6t–3(P³) whose generic point correspondsto a canonical curve, that is, a complete intersection of aquadric and a cubic surface in P³. Let F be the vector spaceof linear forms in the variables z₁, z₂, z₃, z₄. Denote by F_dthe vector space of homogeneous forms of degree d. Set X = (f₂,f₃)where f₂ P(F₂) is a quadric surface, and f₃ P(F₃/f₂ ·F) is a cubic modulo f₂. Wehave a rational map, : X ... Hdefined by (f₂,f₃) f₂ f₃. It fails to be regular along thelocus where f₂ and f₃ acquire a common linear component. Ourmain result gives an explicit resolution of the indeterminaciesof as well as of the singularities of H. 2000 Mathematical Subject Classification: 14C05, 14N05, 14N10,14N15.     

On Weighted Distortion in Conformal Mapping II

Some years ago, Blatter [1] gave a result of the form for any function f regular and univalentin D: |z| < 1, where is the hyperbolic distance betweenz₁ and z₂. Kim and Minda [5] pointed out that the multiplieron the right is incorrect. They say that Blatter's proof givesthe correct multiplier, but Blatter's formula for in termsof z₁, z₂ is wrong. Kim and Minda proved the generalized formula where D₁(f) = f'(z) (1 – |z|²),valid for p P with some P, . In each case there was an appropriate equality statement. Kimand Minda made the important and easily verified remark thatthese problems are linearly invariant in the sense that if theresult is proved for f, then it follows for , where U is a linear transformation of the planeonto itself and T is a linear transformation of D onto itself.This means that we need to prove such results only in an appropriatelynormalized context. 1991 Mathematics Subject Classification30C75, 30F30.     

Witt Groups of the Punctured Spectrum of a 3-Dimensional Regular Local Ring and a Purity Theorem

Let A be a regular local ring with quotient field K. Assumethat 2 is invertible in A. Let W(A)W(K) be the homomorphisminduced by the inclusion AK, where W( ) denotes the Witt groupof quadratic forms. If dim A4, it is known that this map isinjective [6, 7]. A natural question is to characterize theimage of W(A) in W(K). Let Spec¹(A) be the set of prime idealsof A of height 1. For PSpec¹(A), let _P be a parameter of thediscrete valuation ring A_P and k(P) = A_P/PA_P. For this choiceof a parameter _P, one has the second residue homomorphism _P:W(K)W(k(P))[9, p. 209]. Though the homomorphism _P depends on the choiceof the parameter _P, its kernel and cokernel do not. We havea homomorphism A part of the so-called Gersten conjecture is the followingquestion on ‘purity’. Is the sequence exact? This question has an affirmative answer for dim(A)2 [1;3, p. 277]. There have been speculations by Pardon and Barge-Sansuc-Vogelon the question of purity. However, in the literature, thereis no proof for purity even for dim(A) = 3. One of the consequencesof the main result of this paper is an affirmative answer tothe purity question for dim(A) = 3. We briefly outline our main result.     

A New Notion of Transitivity for Groups and Sets of Permutations

Let = {1, 2, ..., n} where n 2. The shape of an ordered setpartition P = (P₁, ..., P_k) of is the integer partition =(₁, ..., _k) defined by _i = |P_i|. Let G be a group of permutationsacting on . For a fixed partition of n, we say that G is -transitiveif G has only one orbit when acting on partitions P of shape. A corresponding definition can also be given when G is justa set. For example, if = (n – t, 1, ..., 1), then a -transitivegroup is the same as a t-transitive permutation group, and if = (n – t, t), then we recover the t-homogeneous permutationgroups. We use the character theory of the symmetric group S_n to establishsome structural results regarding -transitive groups and sets.In particular, we are able to generalize a celebrated resultof Livingstone and Wagner [Math. Z. 90 (1965) 393–403]about t-homogeneous groups. We survey the relevant examplescoming from groups. While it is known that a finite group ofpermutations can be at most 5-transitive unless it containsthe alternating group, we show that it is possible to constructa nontrivial t-transitive set of permutations for each positiveinteger t. We also show how these ideas lead to a combinatorialbasis for the Bose–Mesner algebra of the association schemeof the symmetric group and a design system attached to thisassociation scheme.     

The density of prime divisors in the arithmetic dynamics of quadratic polynomials

Let f [x], and consider the recurrence given by a_n = f(a_{n –1}), with a₀ . Denote by P(f, a₀) the set of prime divisorsof this recurrence, that is, the set of primes dividing at leastone non-zero term, and denote the natural density of this setby D(P(f, a₀)). The problem of determining D(P(f, a₀)) whenf is linear has attracted significant study, although it remainsunresolved in full generality. In this paper, we consider thecase of f quadratic, where previously D(P(f, a₀)) was knownonly in a few cases. We show that D(P(f, a₀)) = 0 regardlessof a₀ for four infinite families of f, including f = x² + k,k \{–1}. The proof relies on tools from group theoryand probability theory to formulate a sufficient condition forD(P(f, a₀)) = 0 in terms of arithmetic properties of the forwardorbit of the critical point of f. This provides an analogy toresults in real and complex dynamics, where analytic propertiesof the forward orbit of the critical point have been shown todetermine many global dynamical properties of a quadratic polynomial.The article also includes apparently new work on the irreducibilityof iterates of quadratic polynomials.     

Homomorphisms from Automorphism Groups of Free Groups

The automorphism group of a finitely generated free group isthe normal closure of a single element of order 2. If m <n, then a homomorphism Aut(F_n)Aut(F_m) can have image of cardinalityat most 2. More generally, this is true of homomorphisms fromAut(F_n) to any group that does not contain an isomorphic imageof the symmetric group S_n+1. Strong restrictions are also obtainedon maps to groups that do not contain a copy of W_n = (Z/2)ⁿ S_n, or of Z^n–1. These results place constraints on howAut(F_n) can act. For example, if n 3, any action of Aut(F_n)on the circle (by homeomorphisms) factors through det : Aut(F_n)Z₂.2000 Mathematics Subject Classification 20F65, 20F28 (primary).     

On the Distribution of Generating Functions

Investigations concerning the generating function associatedwith the kth powers, originatewith Hardy and Littlewood in their famous series of papers inthe 1920s, ‘On some problems of "Partitio Numerorum"’(see [7, Chapters 2 and 4]). Classical analyses of this andsimilar functions show that when P is large the function approachesP in size only for in a subset of (0, 1) having small measure.Moreover, although it has never been proven, there is some expectationthat for ‘most’ , the generating function is about in magnitude. The main evidence in favour of this expectation comes from mean value estimatesof the form An asymptoticformula of the shape (1.2), with strong error term, is immediatefrom Parseval's identity when s = 2, and follows easily whens = 4 and k > 2 from the work of Hooley [2, 3, 4], Greaves[1], Skinner and Wooley [5] and Wooley [9]. On the other hand,(1.2) is false when s > 2k (see [7, Exercise 2.4]), and whens = 4 and k = 2. However, it is believed that when t < k,the total number of solutions of the diophantine equation with 1 x_j, y_j P (1 j t), is dominatedby the number of solutions in which the x_i are merely a permutationof the y_j, and the truth of such a belief would imply that (1.2)holds for even integers s with 0 s < 2k. The purpose of this paper is to investigate the extent to whichknowledge of the kind (1.2) for an initial segment of even integerexponents s can be used to establish information concerningthe general distribution of f_P(), and the behaviour of the momentsin (1.2) for general real s. 1991 Mathematics Subject Classification11L15.     

Determination of a Convex Body from Minkowski Sums of its Projections

For a convex body K in R^d and 1 K d – 1, let P_K (K)be the Minkowski sum (average) of all orthogonal projectionsof K onto k-dimensional subspaces of R^d. It is Known that theoperator P_k is injective if kd/2, k=3 for all d, and if k =2, d 14. It is shown that P_2k (K) determines a convex body K among allcentrally symmetric convex bodies and P_2k+1(K) determines aconvex body K among all bodies of constant width. Correspondingstability results are also given. Furthermore, it is shown thatany convex body K is determined by the two sets P_k (K) and P_k'(K) if 1 < k < k'. Concerning the range of P_k , 1 k d–2, it is shown that its closure (in the Hausdorff-metric)does not contain any polytopes other than singletons.     

POSITIVE QUATERNION-KAHLER 16-MANIFOLDS WITH b2 = 0

It is known that if the second Betti number of a 4n-dimensionalpositive quaterion-Kähler manifold M satisfies b₂(M) 1,then M is homothetic to the complex Grassmannian . In sixteen dimensions, it is also known that ifb₄(M) = 1, then M is homothetic to the quaternionic projectivespace ⁴. Here, we prove that the fourth Betti number of a positivequaternion-Kähler 16-manifold is at least 3 if b₂(M) =0. We also explore the consequences of other restrictions suggestedby the vanishing of certain indices of twisted Dirac operatorson the Wolf spaces.     

Generalized Ramsey Theory for Graphs V. the Ramsey Number of a Digraph

Continuing this series of papers on generalized Ramsey theoryfor graphs, we define the Ramsey number r(D_l, D₂) of two digraphsD₁ and D₂ as the minimum p such that every 2-colouring of thearcs (directed lines) of DK_P (the complete symmetric digraphof order p) contains a monochromatic D₁ or D₂. It is shown(Theorem1) that this number exists if and only if D₁ or D₂ is acyclic.Then r(D), the diagonal Ramsey number of a given acyclic digraphD, is defined as r(D, D). Notation: D' is the converse of D,GD is the underlying graph of D, DG is the symmetric digraphof G, and T_p is the transitive tournament of order p. Let r(m,n) be the traditional Ramsey number of the two complete graphsK_m and K_n. Finally, let S_n be the star with n arcs from onepoint u to n points v_i. Assuming the Ramsey numbers under discussionexist, we prove the following results: THEOREM 2. r(D₁, D₂) = r(D₁' D₂'). THEOREM 3. r(D₁, D₂) r(GD₁, GD₂). THEOREM 4. r(D₁, D₂) r(T_P1, T_P2) if both D₁ and D₂ (with p₁and p₂ points respectively) are acyclic. THEOREM 5. r(T_m, T_n) = r(m, n). THEOREM 6. r(m, ri) r(T_m, DK_n) r(2^m–1, n). THEOREM 7. r(S_m, S_n) = r(S_m, S_n') = m+n. Finally, we establish all Ramsey numbers r(D₁, D₂) for digraphswithout isolates and with less than four points, and all diagonalRamsey numbers r(D) of acyclic digraphs without isolates withless than five points.     

On the Approximation Numbers for the Finite Sections of Block Toeplitz Matrices

In this paper we discuss the asymptotic distribution of theapproximation numbers of the finite sections for a Toeplitzoperator and µ R, witha continuous matrix-valued generating function a. We prove thatthe approximation numbers of the finite sections T_n(a) = P_nT(a)P_nhave the k-splitting property, provided T(a) is a Fredholm operatoron . 2000 Mathematics SubjectClassification 47B35 (primary), 15A18, 47A58, 47A75, 65F20 (secondary).     

Gluing torsion endo-permutation modules

Let k be a field of characteristic p, and let P be a finitep-group, where p is an odd prime. In this paper, we considerthe problem of gluing compatible families of endo-permutationmodules: being given a torsion element M_Q in the Dade groupD(N_P(Q)/Q), for each non-trivial subgroup Q of P, subject toobvious compatibility conditions, we show that it is alwayspossible to find an element M in the Dade group of P such that for all Q, but that Mneed not be a torsion element of D(P). The obstruction to thisis controlled by an element in the zeroth cohomology group over₂ of the poset of elementary abelian subgroups of P of rankat least 2. We also give an example of a similar situation,when M_Q is only given for centric subgroups Q of P. Moreover,general results about biset functors and the Dade functor aregiven in two appendices.     

The Cohomology of the Sylow 2-Subgroup of J2

The Hall-Janko-Wales group J₂ is one of the twenty-six sporadicfinite simple groups. The cohomology of its Sylow 2-subgroupS_J is computed, an important step in calculating the mod 2 cohomologyof J₂. The spectral sequence corresponding to the central extensionfor S_J is described and shown to collapse at the eighth page.The group S_J contains two subgroups (the central product of a dihedral and a quaternionic group)and 2²⁺⁴ (the Sylow 2-subgroup of the matrix group PSL₃(₄))which detect the cohomology of S_J. The cohomology relationsfor the subgroup 2²⁺⁴ are computed.     

Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities

We prove a nearly optimal bound on the number of stable homotopytypes occurring in a k-parameter semi-algebraic family of setsin R, each defined in terms of m quadratic inequalities. Ourbound is exponential in k and m, but polynomial in . More precisely,we prove the following. Let R be a real closed field and let = {P₁, ... , P_m} R[Y₁, ... ,Y,X₁, ... ,X_k], with deg_Y(P_i) 2, deg_X(P_i) d, 1 i m. Let S R^+k be a semi-algebraic set,defined by a Boolean formula without negations, with atoms ofthe form P 0, P 0, P . Let : R^+k R^k be the projection onthe last k coordinates. Then the number of stable homotopy typesamongst the fibers S_x = ^–1(x) S is bounded by (2^mkd)^O(mk).     

On the Greatest Prime Divisor of the Sum of Two Squares of Primes

One of the most famous theorems in number theory states thatthere are infinitely many positive prime numbers (namely p =2 and the primes p 1 mod4) that can be represented in the formx²₁+x²₂, where x₁ and x₂ are positive integers. In a recentpaper, Fouvry and Iwaniec [2] have shown that this statementremains valid even if one of the variables, say x₂, is restrictedto prime values only. In the sequel, the letter p, possiblywith an index, is reserved to denote a positive prime number.As p²₁=p²₂ = p is even for p₁, p₂ > 2, it is reasonable toconjecture that the equation p²₁=p²₂ = 2p has an infinity ofsolutions. However, a proof of this statement currently seemsfar beyond reach. As an intermediate step in this direction,one may quantify the problem by asking what can be said aboutlower bounds for the greatest prime divisor, say P(N), of thenumbers p²₁=p²₂, where p₁, p₂ N, as a function of the realparameter N 1. The well-known Chebychev–Hooley methodcombined with the Barban–Davenport–Halberstam theoremalmost immediately leads to the bound P(N) N^1–, if N N_o(); here, denotes some arbitrarily small fixed positivereal number. The first estimate going beyond the exponent 1has been achieved recently by Dartyge [1, Théorème1], who showed that P(N) N^10/9–. Note that Dartyge'sproof provides the more general result that for any irreduciblebinary form f of degree d 2 with integer coefficients the greatestprime divisor of the numbers |f(p₁, p₂)|, p₁, p₂ N, exceedsN^_d–, where _d = 2 – 8/(d = 7). We in particular wantto point out that Dartyge does not make use of the specificfeatures provided by the form x²₁+x²₂. By taking advantage ofsome special properties of this binary form, we are able toimprove upon the exponent ₂ = 10/9 considerably.     

On the Efficiency of Coxeter Groups

If G is a finitely presented group and K is any (G,2)-complex(that is, a finite 2-complex with fundamental group G), thenit is well known that X(K) (G), where (G) = 1–rk H₁G+ dH₂G. We define (G) to be min{(K): K a (G, 2)-complex}, andwe say that G is efficient if (G)=(G). In this paper we givesufficient conditions for a Coxeter group to be efficient (Theorem4.2). We also give examples of inefficient Coxeter groups (Theorem5.1). In fact, we give an infinite family G_n(n = 2, 3, 4, ...)of Coxeter groups such that (G_n)–(G_n) as n . 1991 MathematicsSubject Classification 20F05, 20F55.     

COHOMOLOGICAL DIMENSION OF MACKEY FUNCTORS FOR INFINITE GROUPS

We consider the cohomology of Mackey functors for infinite groupsand define the Mackey-cohomological dimension cdG of a groupG. We will relate this dimension to other cohomological dimensionssuch as the Bredon-cohomological dimension cdG and the relativecohomological dimension -cdG. In particular, we show that forvirtually torsion free groups the Mackey-cohomological dimensionis equal to both -cdG and the virtual cohomological dimension.     

On the Genus of a Finite Classical Group

Let G be a finite group acting faithfully and transitively ona set of size m, and let E = {x₁, ..., x_k} be a generatingset for G with x₁x₂...x_k = 1. If x G has cycles of length r₁,..., r_l in its action on , define . Then the genus g = g(G, , E) is defined by 1991 Mathematics Subject Classification 20B25, 20G40,30F99.     

Ramanujan's Eisenstein series and powers of Dedekind's eta-function

In this article, we use the theory of elliptic functions toconstruct theta function identities which are equivalent toMacdonald's identities for A₂, B₂ and G₂. Using these identities,we express, for d = 8, 10 or 14, certain theta functions inthe form ^d()F(P, Q, R), where () is Dedekind's eta-function,and F(P, Q, R) is a polynomial in Ramanujan's Eisenstein seriesP, Q and R. We also derive identities in the case when d = 26.These lead to a new expression for ²⁶(). This work generalizesthe results for d = 1 and d = 3 which were given by Ramanujanon page 369 of ‘The Lost Notebook’.     

On a Parabolic Symmetry of Finite Coxeter Groups

Let (W, S be a finite Coxeter system, and let JS. Any wW hasa unique factorization w = w^J w_J, where w_j belongs to the parabolicsubgroup W_J generated by J, and w^J is of minimal length in thecoset wW_J. It is shown here that w_I = w^J if and only if w^I =w_J, for all I, J S. Furthermore, a similar symmetry propertyin arbitrary (W_I, W_J-double cosets is conjectured, which linksthis result to the Solomon descent algebra of W. 2000 MathematicsSubject Classification 20F55.