Torsion-Free Sheaves and Moduli of Generalized Spin Curves

This article treats compactifications of the space of generalized spin curves. Generalized spin curves, or r-spin curves, are pairs (X,L) with X a smooth curve and L a line bundle whose rth tensor power is isomorphic to the canonical bundle of X. These are a natural generalization of 2-spin curves (algebraic curves with a theta-characteristic), which have been of interest recently, in part because of their applications to fermionic string theory. Three different compactifications over Z[1/r], all using torsion-free sheaves, are constructed. All three yield algebraic stacks, one of which is shown to have Gorenstein singularities that can be described explicitly, and one of which is smooth. All three compactifications generalize constructions of Deligne and Cornalba done for the case when r=2.     

Maximal Subsheaves of Torsion-Free Sheaves on Nodal Curves

Let Y be a reduced irreducible projective curve of arithmeticgenus g 2 with at most ordinary nodes as singularities. Fora subsheaf F of rank r', degree d' of a torsion-free sheaf Eof rank r, degree d, let s(E,F) = r'd-rd'. Define s_r'(E) = mins(E,F), where the minimum is taken over all subsheaves of Eof rank r'. For a fixed r', s_r' defines a stratification ofthe moduli space U(r,d) of stable torsion-free sheaves of rankr, degree d by locally closed subsets U_r',s. We study the nonemptinessand dimensions of the strata. We show that the general elementin each nonempty stratum is a vector bundle and it has onlyfinitely many (respectively unique) maximal subsheaves of rankr' for s r'(r-r')(g – 1) (respectively s < r'(r-r')(g– 1)). We prove that the tensor product of two generalstable vector bundles on an irreducible nodal curve Y is nonspecial.     

A Construction of the Rational Function Sheaves on Elliptic Curves

Abstract The authors introduce an effective method to construct the rational function sheaf κ on an elliptic curve E, and further study the relationship between κ and any coherent sheaf on E. Finally, it is shown that the category of all coherent sheaves of finite length on E is completely characterized by κ.     

Simple, Reducible Venn Diagrams on Five Curves and Hamiltonian Cycles

Recently, using graph theory, we developed procedures for the construction of Venn diagrams. Utilizing these procedures with some new methods introduced here, we determine the number of simple, reducible spherical Venn diagrams of five sets. In so doing, we obtain examples of Venn diagrams which yield answers to several problems and conjectures of Grünbaum. Among others, we construct a simple, reducible Venn diagram with five congruent ellipses. We show that this diagram is unique on the sphere and produces two different plane diagrams. This corrects some erroneous statements that started with John Venn more than a century ago in 1880 and have been repeated frequently by others ever since.     

Orbifolds, Sheaves and Groupoids

We characterize orbifolds in terms of their sheaves, and show that orbifolds correspond exactly to a specific class of smooth groupoids. As an application, we construct fibered products of orbifolds and prove a change-of-base formula for sheaf cohomology.     

Noncommutative Topologies,Localization, and Sheaves

In this note, we introduce and study the notion of sheaf on a noncommutative topology and construct an associated sheafification functor in this noncommutative context. It appears that abstract localization in the category of presheaves on a partially ordered set provides an elegant way to define this sheafification functor, generalizing similar constructions in the commutative case.     

New Roses: Simple Symmetric Venn Diagrams with 11 and 13 Curves

A symmetric \(n\) -Venn diagram is one that is invariant under \(n\) -fold rotation, up to a relabeling of curves. A simple \(n\) -Venn diagram is an \(n\) -Venn diagram in which at most two curves intersect at any point. In this paper, we introduce a new property of Venn diagrams called crosscut symmetry, which is related to dihedral symmetry. Utilizing a computer search restricted to diagrams with crosscut symmetry, we found many simple symmetric Venn diagrams with 11 curves. The question of the existence of a simple 11-Venn diagram has been open since the 1960s. The technique used to find the 11-Venn diagram is extended and a symmetric 13-Venn diagram is also demonstrated.     

Sheaves and duality

It has long been known in universal algebra that any distributive sublattice of congruences of an algebra which consists entirely of commuting congruences yields a sheaf representation of the algebra. In this paper we provide a generalization of this fact and prove a converse of the generalization. To be precise, we exhibit a one-to-one correspondence (up to isomorphism) between soft sheaf representations of universal algebras over stably compact spaces and frame homomorphisms from the dual frames of such spaces into subframes of pairwise commuting congruences of the congruence lattices of the universal algebras. For distributive-lattice-ordered algebras this allows us to dualize such sheaf representations.     

Seminormal Varieties,Torsionfree Sheaves,and Picard groups

We give a construction of torsionfree sheaves on a seminormal variety Y using torsionfree sheaves on the normalization X and the non-normal locus W. We use it to find a relation between Picard groups of X, Y, and W. We apply it to determine the Picard groups of the generalized Jacobian, the compactified Jacobian and some subschemes associated to the moduli spaces of torsionfree sheaves of rank 2 and odd degree on a nodal curve.     

Model-theoretic Imaginaries and Coherent Sheaves

We show the equivalence of categories of model-theoretic imaginaries (of various kinds) with categories of “small” (finitely generated, finitely presented, coherent) functors. We do this first for certain locally finitely presented categories and then, by localising, for much more general “definable categories” (categories of models of coherent theories). We also investigate the corresponding notion of interpretation.     

Explicit Models for Perverse Sheaves,II

In this paper we show that any p-perverse sheaf on an arbitrary stratified topological space (p is a perversity function) is functorially determined by a system of usual sheaves on the open sets U _r (r≥0) and certain gluing data, where U _r is the union of strata of perversity ≤r. Both authors are partially supported by BFM2001-3207 and MTM2004-07203-C02-01 and FEDER.     

Sheaves on quantaloids

We generalize the well-known notions of a singleton, completeQ-set, presheaf, and sheaf over a complete Heyting algebra or a right-sided idempotent quantale to arbitrary quantaloids. We show that every completeQ-set can be viewed as a sheaf and vice versa. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 133–171, April–June, 2000.     

Complex impure systems: Sheaves,freeways, and chains

Fundamental to the approach of Complex Impure Systems is the definition of the concept of an s‐impure set as a set of perceptual beliefs or denotative significances (relative beings) of material and/or energetic real objects (absolute beings). But any Subject not only the subject S perceives objects O as significances, and he perceives the existing relations between these significances or, alternatively, he infers them. The study of these relations, conceived not as a singular relation between singular objects, but as sheaves of relations in both directions and forming relational freeways, will be studied here. In this work, we approach the structure of the system, from a synchronous point of view, as a first approach to this class of systems. © 2016 Wiley Periodicals, Inc. Complexity 21: 387–400, 2016     

Sheaves and sheafification on Q-sites

We introduce noncommutative sites over a quantale, the so-called Q-sites, and define appropriate presheaves and sheaves over these. We show how most of the technical machinery which allows to construct sheaves associated to arbitrary presheaves in the commutative cases transposes to this setting. This allows us to define and study sheafification in this new, noncommutative context.     

Diagrams,orientations, and varieties

One of the central problems in the theory of ordered sets is to describe the orientations of the covering graph of an ordered set. We show that the particular operation called inversion, together with the classical constructions of retraction and product, provide a context for the classification of all such orientations. AMS subject classifications (1980). 06A10, 05C75, 05C20.     

Sheaves of Abelian l-groups

Keimel was first to associate a spectrum with an abelian l-group G. An alternative construction of a spectrum is proposed here. Its points are the prime l-ideals of G, i.e., they are the points of Keimel’s spectrum plus the improper prime l-ideal G. The spectrum is equipped with the inverse topology, which is different from the one used by Keimel. The new spectrum is denoted by Spec_inv(G) and is called the inverse spectrum of G. There is a natural construction of a sheaf of l-groups on Spec_inv(G), which is called the structure sheaf of G on Spec_inv(G). The inverse spectrum and its structure sheaf first appeared in Rump and Yang (Bull Lond Math Soc 40:263–273, 2008) via the Jaffard-Ohm Theorem. We present an elementary and direct construction. The relationship between an l-group H and an l-subgroup G is analyzed via the structure sheaves on the inverse spectra. The structure sheaves are used to give a sheaf-theoretic presentation of the valuation theory of a field. The group of Cartier divisors of an integral affine scheme is identified canonically with an l-group. Special attention is paid to Prüfer domains, where the ties between rings and l-groups are particularly close.     

Sheaves on Involutive Quantaloids

We extend the well-known notions of a singleton, complete -set, presheaf and sheaf over a complete Heyting algebra or a right-sided idempotent quantale to arbitrary involutive quantaloids. We show that sheaves on and complete -sets come to the same thing. This paper can be considered as a symmetric version of an earlier work of the author.     

Chord Diagrams, d-Diagrams, and Knots

A method of representing knots, links, and singular knots and links by words in a finite alphabet and so-called d-diagrams is given. A representation of the chord diagram algebra by words in a finite alphabet is considered. This method, unlike coding by Gauss diagrams, allows one to consider knots and links simultaneously. An algorithm for recognition of diagrams of classical knots in terms of d-diagrams is given. Bibliography: 9 titles.