Sklyanin algebras and Hilbert schemes of points

We construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P².The generic noncommutative plane corresponds to the Sklyanin algebra S=Skl(E,σ) constructed from an automorphism σ of infinite order on an elliptic curve E⊂P². In this case, the fine moduli space of line bundles over S with first Chern class zero and Euler characteristic 1−n provides a symplectic variety that is a deformation of the Hilbert scheme of n points on P²?E.     

Rational Cherednik algebras and Hilbert schemes

Let H_c be the rational Cherednik algebra of type A_n-1 with spherical subalgebra U_c=eH_ce. Then U_c is filtered by order of differential operators, with associated graded ring where W is the nth symmetric group. We construct a filtered Z-algebra B such that, under mild conditions on c:• the category B-qgr of graded noetherian B-modules modulo torsion is equivalent to U_c-mod;• the associated graded Z-algebra has grB-lqgr?coh Hilb(n), the category of coherent sheaves on the Hilbert scheme of points in the plane.This can be regarded as saying that U_c simultaneously gives a non-commutative deformation of h⊕h^*/W and of its resolution of singularities Hilb(n)→h⊕h^*/W. As we show elsewhere, this result is a powerful tool for studying the representation theory of H_c and its relationship to Hilb(n).     

The Hilbert stack

Let π:X→S$\pi :X\to S$ be a morphism of algebraic stacks that is locally of finite presentation with affine stabilizers. We prove that there is an algebraic S-stack—the Hilbert stack—parameterizing proper algebraic stacks mapping quasi-finitely to X. This was previously unknown, even for a morphism of schemes.     

Thue type problems for graphs, points, and numbers

A sequence S=s₁s₂…s_n is said to be nonrepetitive if no two adjacent blocks of S are the same. A celebrated 1906 theorem of Thue asserts that there are arbitrarily long nonrepetitive sequences over the set {0,1,2}. This result is the starting point of Combinatorics on Words—a wide area with many deep results, sophisticated methods, important applications and intriguing open problems.The main purpose of this survey is to present a range of new directions relating Thue sequences more closely to Graph Theory, Combinatorial Geometry, and Number Theory. For instance, one may consider graph colorings avoiding repetitions on paths, or colorings of points in the plane avoiding repetitions on straight lines. Besides presenting a variety of new challenges we also recall some older problems of this area.     

The realization of hyperelliptic curves through endomorphisms of Kronecker modules

Let K be an algebraically closed field and A the Kronecker algebra over K. A general problem is to study the endomorphism algebras of A-modules M that are extensions of finite-dimensional, torsion-free, rank-one A-modules P, by infinite-dimensional, torsion-free, rank-one A-modules N. Such endomorphism algebras can be studied by means of a quadratic polynomial f(Y) in one variable Y over the rational function field K(X). We call this f(Y) the regulator of the extension. We prove that if the regulator has non-zero discriminant, then is a Noetherian, commutative K-algebra. We also prove that, subject to a regulator with non-zero discriminant, is affine over K if and only if End N is affine, in which case is the coordinate ring of a hyperelliptic curve.     

Hilbert curves of polarized varieties

Let X be a normal Gorenstein complex projective variety. We introduce the Hilbert variety V_X associated to the Hilbert polynomial χ(x₁L₁+?+x_ρL_ρ), where L₁,…,L_ρ is a basis of , ρ being the Picard number of X, and x₁,…,x_ρ are complex variables. After studying general properties of V_X we specialize to the Hilbert curve of a polarized variety (X,L), namely the plane curve of degree dim(X) associated to χ(xK_X+yL). Special emphasis is given to the case of polarized threefolds.     

An explicit formula for eigenvalues of Bethe trees and upper bounds on the largest eigenvalue of any tree

A Bethe tree B_d,k is a rooted unweighted of k levels in which the root vertex has degree equal to d, the vertices at level j(2?j?k-1) have degree equal to (d+1) and the vertices at level k are the pendant vertices. In this paper, we first derive an explicit formula for the eigenvalues of the adjacency matrix of B_d,k. Moreover, we give the corresponding multiplicities. Next, we derive an explicit formula for the simple nonzero eigenvalues, among them the largest eigenvalue, of the Laplacian matrix of B_d,k. Finally, we obtain upper bounds on the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any tree T. These upper bounds are given in terms of the largest vertex degree and the radius of T, and they are attained if and only if T is a Bethe tree.     

Automorphisms of Hilbert schemes of points on a generic projective K3 surface

We study automorphisms of the Hilbert scheme of n points on a generic projective K3 surface S, for any . We show that is either trivial or generated by a non‐symplectic involution and we determine numerical and divisorial conditions which allow us to distinguish between the two cases. As an application of these results we prove that, for any , there exist infinitely many admissible degrees for the polarization of the K3 surface S such that admits a non‐natural involution. This provides a generalization of the results of [7] for .     

On the diameters of commuting graphs

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all non-central elements of R and two distinct vertices x and y are adjacent if and only if xy = yx. Let D be a division ring and n ? 3. In this paper we investigate the diameters of Γ(M_n(D)) and determine the diameters of some induced subgraphs of Γ(M_n(D)), such as the induced subgraphs on the set of all non-scalar non-invertible, nilpotent, idempotent, and involution matrices in M_n(D). For every field F, it is shown that if Γ(M_n(F)) is a connected graph, then diam Γ(M_n(F)) ? 6. We conjecture that if Γ(M_n(F)) is a connected graph, then diam Γ(M_n(F)) ? 5. We show that if F is an algebraically closed field or n is a prime number and Γ(M_n(F)) is a connected graph, then diam Γ(M_n(F)) = 4. Finally, we present some applications to the structure of pairs of idempotents which may prove of independent interest.     

Poisson structures on Hilbert schemes of points¶of a surface and integrable systems

In this paper we prove that if S is a Poisson surface, i.e., a smooth algebraic surface with a Poisson structure, the Hilbert scheme of points of S has a natural Poisson structure, induced by the one of S. This generalizes previous results obtained by A. Beauville [B1] and S. Mukai [M2] in the symplectic case, i.e., when S is an abelian or K3 surface. Finally we apply our results to give some examples of integrable Hamiltonian systems naturally defined on these Hilbert schemes. In the simple case S=ℙ² we obtain by this construction a large class of integrable systems, which includes the ones studied by P. Vanhaecke in [V1] and, more generally, in [V2]. Received: 9 March 1998 / Revised version: 19 June 1998     

The Farahat-Higman ring of wreath products and Hilbert schemes

We study the structure constants of the class algebra of the wreath products Γ_n associated to an arbitrary finite group Γ with respect to the basis of conjugacy classes. We show that a suitable filtration on gives rise to the graded ring with non-negative integer structure constants independent of n (some of which are computed), which are then encoded in a Farahat-Higman ring . The real conjugacy classes of Γ come to play a distinguished role and are treated in detail in the case when Γ is a subgroup of . The above results provide new insight to the cohomology rings of Hilbert schemes of points on a quasi-projective surface X.     

Commuting graphs of matrices over semirings

We calculate diameters and girths of commuting graphs of the set of all nilpotent matrices over a semiring, the group of all invertible matrices over a semiring, and the full matrix semiring.     

Noncommuting graphs of matrices over semirings

We study diameters and girths of noncommuting graphs of semirings. For a noncommutative semiring that is either multiplicatively or additively cancellative, we find the diameter and the girth of its noncommuting graph and prove that it is Hamiltonian. Moreover, we find diameters and girths of noncommuting graphs of all nilpotent matrices over a semiring, all invertible matrices over a semiring, all noninvertible matrices over a semiring, and the full matrix semiring. In nearly all cases we prove that diameters are less than or equal to 2 and girths are less than or equal to 3, except in the case of 2×2 nilpotent matrices.     

A proof of the existence of Batanin's initial operad

In this paper we prove the existence of the n-globular operad used in Batanin's definition of weak n-category. This operad is initial in the category of n-globular operads equipped with two extra pieces of structure: a system of compositions and a contraction. Our approach closely follows a proof by Leinster of the existence of a similar n-globular operad used in his definition of weak n-category (itself a variant of Batanin's definition) – we show that there is a functor giving the free operad equipped with a contraction and system of compositions on an n-globular collection, and applying this functor to the initial collection gives the desired initial operad. Since there is no interaction between the contraction and operad structures we are able to treat their free constructions separately. This is not true of the system of compositions structure, which cannot exist separately from the operad structure, so we use an interleaving-style construction to describe the free operad with system of compositions.     

On the multiplicity and the Cohen-Macaulayness of fiber cones of graded algebras

Let A be a Noetherian local ring with the maximal ideal m and an m-primary ideal J. Let S=?_n≥0S_n be a finitely generated standard graded algebra over A. Set S⁺=?_n>0S_n. Denote by F_J(S)=?_n≥0→(S_n/JS_n) the fiber cone of S with respect to J. The paper characterizes the multiplicity and the Cohen-Macaulayness of F_J(S) in terms of minimal reductions of S⁺.     

On Mann iteration in Hilbert spaces

Let K$K$ be a compact convex subset of a real Hilbert space H$H$; T:K→K$T:K\to K$ a hemicontractive map. Let {_αn}$\left\{{\alpha }_n\right\}$ be a real sequence in [0,1] satisfying appropriate conditions; then for arbitrary _x0∈K$x_0\in K$, the sequence {_xn}$\left\{x_n\right\}$ defined iteratively by _xn=_αnxn−1+(1−_αn)T_xn$x_n={\alpha }_n{x}_{n-1}+(1-{\alpha }_n)T{x}_n$, n≥1$n\ge 1$ converges strongly to a fixed point of T$T$.     

Basic deformation theory of smooth formal schemes

We provide the main results of a deformation theory of smooth formal schemes as defined in [L. Alonso Tarrío, A. Jeremías López, M. Pérez Rodríguez, Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes, Comm. Algebra 35 (2007) 1341-1367]. Smoothness is defined by the local existence of infinitesimal liftings. Our first result is the existence of an obstruction in a certain Ext¹ group whose vanishing guarantees the existence of global liftings of morphisms. Next, given a smooth morphism f₀:X₀→Y₀ of noetherian formal schemes and a closed immersion Y₀?Y given by a square zero ideal I, we prove that the set of isomorphism classes of smooth formal schemes lifting X₀ over Y is classified by and that there exists an element in which vanishes if and only if there exists a smooth formal scheme lifting X₀ over Y.     

Singularities of the secant variety

We give positivity conditions on the embedding of a smooth variety which guarantee the normality of the secant variety, generalizing earlier results of the author and others. We also give classes of secant varieties satisfying the Hodge conjecture as well as a result on the singular locus of degenerate secant varieties.