Canonical Resolutions Research Articles

Moduli of lattice polarized K3 surfaces via relative canonical resolutions

For a smooth canonically embedded curve C of genus 9 together with a pencil | L | of degree 6, we study the relative canonical resolution of C ⊂ X ⊂ P 8 , where X is the scroll swept out by the pencil | L | . We show that the second syzygy bundle in this resolution of C ⊂ X is unbalanced. The proof reveals a new geometric connection between the universal Brill–Noether variety W 9 , 6 1 and a moduli space F h of lattice polarized K 3 surfaces (for a certain rank 3 lattice h ). As a by-product we prove the unirationality of F h and show that W 9 , 6 1 is birational to a projective bundle over a moduli space of lattice polarized K 3 surfaces F h ′ for a certain rank 4 lattice h ′ which contains h as a sublattice .

The relative canonical resolution: Macaulay2-package, experiments and conjectures

This short note provides a quick introduction to relative canonical resolutions of curves on rational normal scrolls. We present our Macaulay2-package which computes the relative canonical resolution associated to a curve and a pencil of divisors. Most of our experimental data can be found on a dedicated webpage. We end with a list of conjectural shapes of relative canonical resolutions. In particular, for curves of genus $g=n\cdot k +1$ and pencils of degree $k$ for $n\ge 1$, we conjecture that the syzygy divisors on the Hurwitz space $\mathscr{H}_{g,k}$ constructed by Deopurkar and Patel all have the same support.

Canonical resolutions in hereditary abelian model categories

Each object of any abelian model category has a canonical resolution as described in this article. When the model structure is hereditary we show how morphism sets in the associated homotopy category may be realized as cohomology groups computed from these resolutions. We also give an alternative description of the morphism sets in terms of Yoneda Ext groups.

Autocephaly in the Light of the Preparations to the Pan-Orthodox Council

Among the topics submitted, but finally not elaborated and discussed at the Pan-Orthodox Synod, is found the manner of granting autocephaly. Calling the Great Synod signifies a revival of the institution of conciliarity’s activity at the Pan-Orthodox level in a similar manner as at the Great Ecumenical Councils. Unfortunately, the issue of autocephaly was not approved at the Fifth Pre-Conciliar Pan-Orthodox Conference. The topic of autocephaly and the means by which it is proclaimed did not receive the required unanimous voting (consensus) despite the in-depth discussion at the Inter-Orthodox Preparatory Commission and the Pre-Conciliar Pan-Orthodox Conference meetings. The resolutions of the abovementioned meetings and lively discussion provoked by the topic, gave the question of autocephaly a new dimension and produced new elements and theories. The manner of granting autocephalous status in the Orthodox Church is an extremely important question today and is awaiting a solution by establishing commonly accepted canons which guarantee the preservation of Orthodox unity. The difficulties experienced in the discussion are caused a the lack of specific and clear canonical resolutions regarding the meaning of autocephaly, the conditions of granting it, who is the responsible organ for its granting or deprivation, and also concerning the internal relations of daughter Churches with their Mother Churches and their relations with the remaining Autocephalous Churches.

On high order finite element spaces of differential forms

We show how the high order finite element spaces of differential forms due to Raviart-Thomas-Nedelec-Hiptmair fit into the framework of finite element systems, in an elaboration of the finite element exterior calculus of Arnold-Falk-Winther. Based on observations by Bossavit, we provide new low order degrees of freedom. As an alternative to existing choices of bases, we provide canonical resolutions in terms of scalar polynomials and Whitney forms.

Resolution of Singularities of Null Cones

AbstractWe give canonical resolutions of singularities of several cone varieties arising from invariant theory. We establish a connection between our resolutions and resolutions of singularities of closure of conjugacy classes in classical Lie algebras.

Resolutions of Associative and Lie Algebras

AbstractCertain canonical resolutions are described for free associative and free Lie algebras in the category of non-associative algebras. These resolutions derive in both cases from geometric objects, which in turn reflect the combinatorics of suitable collections of leaf-labeled trees.

Análisis de la jurisprudencia del Tribunal Constitucional sobre el reconocimiento de efectos civiles a las resoluciones canónicas en materia matrimonial (1981-1999)

In this material the Constitutional Tribunal has established two doctrinal lines. The first affects canonical resolutions that are given in procedures that took place before the A. A. J. These are accepted as civilly valid by virtue of the transitory law. The second outlines the growth of opposition in this procedure which is regulated by the D.A.2 of law 30/1981. With regard to the resolutions affected by the transitory law the Constitutional Tribunal considers that these must be subject to the same criteria that regulated them in the previous juridical regime established by the Concordat of 1953 and in the C.c. according to the reform of 1958. In accordance with these norms the civil recognition of canonical matrimonial resolutions consist in the execution of these decisions, which are widely protected by virtue of Article 24 of the Constitution. With regard to the opposition formulated in the process regulated by D.A.2 the following has been established: once the decision is contested the judge is to decree whether or not civil efficacy is to be automatically granted. When automatic civil recognition is rejected this does not indicate a flaw since juridical protection is given by permitting the contractants to present their petition by means of the ordinary process. If, on the other hand, the judge concedes the civil effects which have been requested the Constitutional Tribunal will grant or reject this request having considered the reasons put forward in opposition to the request. When the reasons given are based on those mentioned in Article 954 of the Law of Civil Process the judge should hear the request for confirmation. However, when the reasons are not those given in Article 954 the judge may grant civil effects to canonical resolutions if he considers that the opposition lacks foundation.

Resolution of missing label problems; a new perspective on K-matrix theory

K-matrix theory is an effective algebraic way of determining inner products and constructing orthonormal bases vectors for the carrier spaces of induced representations of Lie groups and Lie algebras. The theory is applied in this paper to give canonical resolutions of missing label problems.

Algebra structures on some canonical resolutions

The study of algebra structures on finite free resolutions of cyclic modules begins with Buchsbaum and Eisenbud [B-El]. The classes of cyclic modules whose minimal resolutions are known to admit an algebra structure include the residue field [Gu], complete intersections (Koszul complex), modules of homological dimension atmost 3 [B-El], Gorenstein of codimension four [K-M], and Herzog algebras [K-M23. Avramov [Al] gave examples to show that there are cyclic modules whose minimal resolutions do not admit an algebra structure. Let R be a noetherian local ring with maximal ideal m. In this paper we construct algebra structures on the minimal resolutions of two classes of cyclic modules R/I, namely, when Z is of the form Jk, where J is an ideal generated by a regular sequence, and when Z is the ideal of maximal minors of a generic n x m matrix, provided R contains the rationals. In [Al] Avramov defined certain obstructions to the existance of an algebra structure on the minimal resolution of a module and then produced modules with non-zero obstructions. The general question is whether the vanishing of these obstructions is also sufficient for the existence of a minimal algebra resolution of a cyclic module. When Z is an ideal generated by a regular sequence in R and M= R/Z” for any positive integer k, then these obstructions are all zero. Therefore, Avramov and Schlessinger asked whether the minimal resolutions of R/Z’ admit an algebra structure. Corollary 3.6 of this paper provides an affirmative answer to this question.

On normal two-dimensional double point singularities

Letq be a plane curve singularity and letp be the corresponding normal two-dimensional double point singularity. Let Γ and Γ1 be the topological types of the minimal and of the canonical resolutions ofp respectively. An algorithm is given for finding the equisingular type ofq in terms of Γ1. An algorithm is also given for finding all Γ1 corresponding to a given Γ. There is at most one such Γ1 in case Γ has no 1-cycles.

On the Permutability of Normal Operators

1. In this paper we obtain a theorem about the permutability of (in general unbounded) normal operators. This theorem is analogous to a theorem proved by the present authors in collaboration with J. von Neumann [3] concerning the permutability of self-adjoint operators. As we shall point out later it is not possible to generalize precisely the conditions given in [3] to the case of normal operators. Nevertheless, the proof of the theorem given here, with essential modifications, follows the proof given in [3]. We give our theorem and proof in Section 2. In Section 3 we apply this theorem on the permutability of normal operators to obtain the integral representation for a semi-group {N,} of normal operators (in general unbounded), where the index x ranges over a semi-group which is a cartesian product I+ X E where I+ is the set of vectors in m-dimensional euclidean space with non-negative integer components and Et is the set of vectors in n-dimensional euclidean space with non-negative components. This theorem contains as special cases theorems by E. Hille [6], B. v.Sz. Nagy [9, pp. 74-76], A. Devinatz [2] and a theorem recently published by R. Getoor [5]. It should be pointed out that our theorem represents an essential generalization of these previous results. For the case where { Nx ; x > 0 } is a one parameter semi-group of unbounded normal operators, the method of Nagy [9, pp. 74-75] may be pushed through to obtain the integral representation, and only in this case. However, with the use of our permutability theorem we are able to materially simplify the proof for the one parameter case. Moreover, in the unbounded case, even with the use of our permutability theorem, there remain some difficulties for multi-parameter semi-groups which do not allow us to use immediately the theorem for the one-parameter case. If the index x ranges over a more general Abelian topological semi-group than we consider, the representation problem is still open, although for semi-groups of normal operators which are uniformly bounded a method introduced by R. Phillips [8] may be used (cf. also A. E. Nussbaum [7]). 2. We start with the definition of permutability of two (in general unbounded) normal operators defined in a Hilbert space & (cf. Nagy [9] p. 50). DEFINITION 1. Let N1 and N2 be normal operators and {K1(z)} and {K2(Z)I their corresponding canonical resolutions of the identity. We say that N1 and N2 permute if Kl(zi)K2(z2) = K2(z2)Kl(zi) for each complex z1 and z2 It should be pointed out here that this definition is, by a theorem of B. Fuglede [4], equivalent to the usual definition of permutability in the case where N1 and N2 are bounded (N1N2 = N2N1) or in the case where N1 is bounded and N2 is unbounded (N1N2 _ N2N1). DEFINITION 2. We say that the operators N1, N2 and N have the property P if they are normal operators and if N = N1N2 = N2N1 .

On the Permutability of Self-Adjoint Operators

1. The permutability of two bounded self-adjoint operators T1 and T2 defined on a Hilbert space S& is defined in the most natural and obvious manner: We say that T1 commutes with T2 if T17T2 = T2T1 throughout S. If {E1(X) } and {E2(0-) } are the canonical resolutions of the identity of T7 and T2 respectively, then the following result is well known: A necessary and sufficient condition that T7 commutes with T2 is the validity of any one of the following conditions: (a) The product T1T2 or T2T1 is self-adjoint. (b) E1(X) commutes with E2(0-) for each X and o. If Ti is a bounded self-adjoint operator and T2 any (not necessarily bounded) self-adjoint operator, then one usually defines permutability as follows (cf. [2] p. 31 or [4] p. 404): T1 commutes with T2 if Ti T2 (7 T2T1 . That is, if and only if, whenever f is in the domain of T2, Tif is also in the domain of T2 and T71T2f = T2T1f. This definition is equivalent to (b). Therefore, the permutability of any two selfadjoint operators is usually defined by (cf. [2], p. 50). DEFINITION 1. Two self-adjoint operators T1 and T2 with the corresponding canonical resolutions of the identity {E1(X)} and {E2(0-) } respectively are said to be permutable (or T1 commutes with T2) if E1(X) commutes with E2(o-) for each X and a-. The objective of this note is to show that two self-adjoint operators T1 and T2 commute if there exists a self-adjoint operator T such that T ( 7 T1T2 or T a T2T71. For the standard results and techniques of general Hilbert space theory used in this paper the reader is referred to the various papers of J. von Neumann [3], [4] and the books by M. H. Stone [5] and B. v. Sz. Nagy [2]. 2. To elaborate on our statements in ?1 we first recall the definition of multiplication of unbounded operators. The product AB of two operators A and B is defined as follows: (AB)f is defined if and only if Bf and A (Bf) are both defined and is then equal to the latter. DEFINITION 2. We say that the operators S, T are C' if S and T are self-adjoint, D(T) z D(S) (D((T) denotes the domain of T) and for f, g E D((T), the inner product (Sf, Tg) is Hermitean symmetric in f and g. LEMMA 1. Suppose that S, T are C' and E is a projection operator which commutes with T and such that R(E) ( D((T) (R(E) denotes the range of E). Let Si and T1 be the restrictions of ES and T respectively to R(E). Then S1 and T1 are bounded self-adjoint operators and S1 commutes with T1 . PROOF. Let f, g E R(E). Since R(E) C D((T) C D(S), f and g belong to D(S) and (I-E)Sf and (I-E)Sg belong to S& E R(E). Therefore,

Integration in Locally Compact Spaces II

The general problem of producing concrete representations for continuous linear functionals on normed linear spaces, ie., of identifying conjugate spaces, has of course attracted the attention of many mathematicians during the last five decades and has been solved in many cases [1, pp. 59-72]. Likewise, the problem of extending a linear functional defined on a linear subspace of a normed linear space may be regarded as solved by the Hahn-Banach theorem [1, p. 28], although problems involving “natural” extensions, like that yielding the Lebesgue integral from the Riemann integral, remain. In the present paper, we shall consider two “natural’ methods of extending a certain linear functional and show that they are in fact identical. As a by-product, we obtain a concrete representation both for the original functional and for its “natural” extension. In subsequent communications, the writers will consider topologies in certain families of linear functionals, canonical resolutions of linear functionals, and other extension problems.