Proper Subvariety Research Articles

Monadic MV-algebras II: Monadic implicational subreducts

In this paper, we study the class of all monadic implicational subreducts, that is, the \({\{\rightarrow, \forall,1\}}\)-subreducts of the class of monadic MV-algebras. We prove that this class is an equational class, which we denote by \({\mathcal{ML}}\), and we give an equational basis for this variety. An algebra in \({\mathcal{ML}}\) is called a monadic Łukasiewicz implication algebra. We characterize the subdirectly irreducible members of \({\mathcal{ML}}\) and the congruences of every monadic Łukasiewicz implication algebra by monadic filters. We prove that \({\mathcal{ML}}\) is generated by its finite members. Finally, we completely describe the lattice of subvarieties, and we give an equational basis for each proper subvariety.

Monadic MV-algebras I: a study of subvarieties

Fil: Cimadamore, Cecilia Rossana. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Bahia Blanca. Instituto de Matematica Bahia Blanca. Universidad Nacional del Sur. Departamento de Matematica. Instituto de Matematica Bahia Blanca; Argentina

An almost nilpotent variety of exponent 2

We construct a non-associative algebra A over a field of characteristic zero with the following properties: if V is the variety generated by A, then V has exponential growth but any proper subvariety of V is nilpotent. Moreover, by studying the asymptotics of the sequence of codimensions of A we deduce that exp(V) = 2.

Variety of Jordan algebras var(UT 2 (F)(+)) has almost polynomial growth

It is proved that in the case of ground field of characteristic zero the variety of Jordan algebras var (UT 2 (F)(+)) has the growth with the exponent two and any its proper subvariety has a polynomial growth.

Quasiprojective varieties admitting Zariski dense entire holomorphic curves

Let X be a complex quasiprojective variety. A result of Noguchi–Winkelmann–Yamanoi shows that if X admits a Zariski dense entire curve, then its quasi-Albanese map is a fiber space. We show that the orbifold structure induced by a proper birationally equivalent map on the base is special in this case. As a consequence, if X is of log-general type with q(X)dimX, then any entire curve is contained in a proper subvariety in X.

Transversality of holomorphic mappings between real hypersurfaces in complex spaces of different dimensions

We consider holomorphic mappings $H$ between a smooth real hypersurface $M\subset \bC^{n+1}$ and another $M'\subset \bC^{N+1}$ with $N\geq n$. We provide conditions guaranteeing that $H$ is transversal to $M'$ along all of $M$. In the strictly pseudoconvex case, this is well known and follows from the classical Hopf boundary lemma. In the equidimensional case ($N=n$), transversality holds for maps of full generic rank provided that the source is of finite type in view of recent results by the authors (see also a previous paper by the first author and L. Rothschild). In the positive codimensional case ($N>n$), the situation is more delicate as examples readily show. In recent work by S. Baouendi, the first author, and L. Rothschild, conditions were given guaranteeing that the map $H$ is transversal outside a proper subvariety of $M$, and examples were given showing that transversality may fail at certain points. One of the results in this paper implies that if $N\le 2n-2$, $M'$ is Levi-nondegenerate, and $H$ has maximal rank outside a complex subvariety of codimension 2, then $H$ is transversal to $M'$ at all points of $M$. We show by examples that this conclusion fails in general if $N\geq 2n$, or if the set $W_H$ of points where $H$ is not of maximal rank has codimension one. We also show that $H$ is transversal at all points if $H$ is assumed to be a finite map (which allows $W_H$ to have codimension one) and the stronger inequality $N\leq 2n-3$ holds, provided that $M$ is of finite type.

Rigid and non-smoothable Schubert classes

A Schubert class $\sigma_\lambda$ in the Grassmannian $G(k, n)$ is rigid if the only proper subvarieties representing that class are Schubert varieties $\Sigma_\lambda$. We prove that a Schubert class $\sigma_\lambda$ is rigid if and only if it is defined by a partition $\lambda$ satisfying a simple numerical criterion. If $\lambda$ fails this criterion, then there is a corresponding hyperplane class in another Grassmannian $G(k', n')$ such that the deformations of the hyperplane in $G(k', n')$ yield non-trivial deformations of the Schubert variety $\Sigma_\lambda$. We also prove that, if a partition λ contains a sub-partition defining a rigid and singular Schubert class in another Grassmannian $G(k', n')$, then there does not exist a smooth subvariety of $G(k, n)$ representing $\sigma_\lambda$.

LOWER BOUNDS FOR THE NORMALIZED HEIGHT AND NON-DENSE SUBSETS OF SUBVARIETIES OF ABELIAN VARIETIES

This work is the third part of a series of papers. In the first two, we considered curves and varieties in a power of an elliptic curve. Here, we deal with subvarieties of an abelian variety in general. Let V be a proper irreducible subvariety of dimension d in an abelian variety A, both defined over the algebraic numbers. We say that V is weak-transverse if V is not contained in any proper algebraic subgroup of A, and transverse if it is not contained in any translate of such a subgroup. Assume a conjectural lower bound for the normalized height of V. Then, for V transverse, we prove that the algebraic points of bounded height of V which lie in the union of all algebraic subgroups of A of codimension at least d + 1 translated by the points close to a subgroup Γ of finite rank, are non-Zariski-dense in V. If Γ has rank zero, it is sufficient to assume that V is weak-transverse. The notion of closeness is defined using a height function.

A gap principle for dynamics

AbstractLet f1,…,fg∈ℂ(z) be rational functions, let Φ=(f1,…,fg) denote their coordinate-wise action on (ℙ1)g, let V ⊂(ℙ1)g be a proper subvariety, and let P be a point in (ℙ1)g(ℂ). We show that if 𝒮={n≥0:Φn(P)∈V (ℂ)} does not contain any infinite arithmetic progressions, then 𝒮 must be a very sparse set of integers. In particular, for any k and any sufficiently large N, the number of n≤N such that Φn(P)∈V (ℂ) is less than log kN, where log k denotes the kth iterate of the log function. This result can be interpreted as an analogue of the gap principle of Davenport–Roth and Mumford.

On the Variety of m-generalized Łukasiewicz Algebras of Order n

In this paper we pursue the study of the variety \({\mathcal{L}^{m}_{n}}\) of m-generalized Łukasiewicz algebras of order n which was initiated in [1]. This variety contains the variety of Łukasiewicz algebras of order n. Given \({{\bf A}\in\mathcal{L}^{m}_{n}}\) , we establish an isomorphism from its congruence lattice to the lattice of Stone filters of a certain Łukasiewicz algebra of order n and for each congruence on A we find a description via the corresponding Stone filter. We characterize the principal congruences on A via Stone filters. In doing so, we obtain a polynomial equation which defines the principal congruences on the algebras of \({\mathcal{L}^{m}_{n}}\) . After showing that for m > 1 and n > 2, the variety of Łukasiewicz algebras of order n is a proper subvariety of \({\mathcal{L}^{m}_{n}}\) , we prove that \({\mathcal{L}^{m}_{n}}\) is a finitely generated discriminator variety and point out some consequences of this strong property, one of which is congruence permutability.

On varieties of MV-algebras with internal states

In [4,5] the authors introduced the variety SMV of MV-algebras with an internal operator, state MV-algebras. In [2,3] the authors gave a stronger version of state MV-algebras, called state-morphism MV-algebras. In this paper we continue the studies presented in [2,3] just looking at several proper subvarieties of SMV , obtained by imposing suitable conditions on the behavior of the internal operator.

The Asymptotic Schottky Problem

Let $${\mathcal{M}_g}$$ denote the moduli space of compact Riemann surfaces of genus g and let $${\mathcal{A}_g}$$ be the moduli space of principally polarized abelian varieties of dimension g. Let $${J : \mathcal{M}_g \rightarrow \mathcal{A}_g}$$ be the map which associates to a Riemann surface its Jacobian. The map J is injective, and the image $${\mathcal{J}_g := J(\mathcal{M}_g)}$$ is contained in a proper subvariety of $${\mathcal{A}_g}$$ when g ≥ 4. The classical and long-studied Schottky problem is to characterize the Jacobian locus $${\mathcal{J}_g}$$ in $${\mathcal{A}_g}$$ . In this paper we address a large scale version of this problem posed by Farb and called the coarse Schottky problem: What does $${\mathcal{J}_g}$$ look like “from far away”, or how “dense” is $${\mathcal{J}_g}$$ in the sense of coarse geometry? The large scale geometry of $${\mathcal{A}_g}$$ is encoded in its asymptotic cone, $${{\rm Cone}_\infty(\mathcal{A}_g)}$$ , which is a Euclidean simplicial cone of real dimension g. Our main result asserts that the Jacobian locus $${\mathcal{J}_g}$$ is “coarsely dense” in $${\mathcal{A}_g}$$ , which implies that the subset of $${{\rm Cone}_\infty(\mathcal{A}_g)}$$ determined by $${\mathcal{J}_g}$$ actually coincides with this cone. The proof shows that the Jacobian locus of hyperelliptic curves is coarsely dense in $${\mathcal{A}_g}$$ as well. We also study the boundary points of the Jacobian locus $${\mathcal{J}_g}$$ in $${\mathcal{A}_g}$$ and in the Baily–Borel and the Borel–Serre compactification. We show that for large genus g the set of boundary points of $${\mathcal{J}_g}$$ in these compactifications is “small”.

A remark on the codimension of the Green–Griffiths locus of generic projective hypersurfaces of high degree

We show that for every smooth generic projective hypersurface , there exists a proper subvariety such that codim X Y ≧ 2 and for every non constant holomorphic entire map ƒ : ℂ → X one has ƒ (ℂ) ⊂ Y , provided deg X ≧ 2 n 5 . In particular, we obtain an effective confirmation of the Kobayashi conjecture for threefolds in .

FINITELY GENERATED LIMIT VARIETIES OF APERIODIC MONOIDS WITH CENTRAL IDEMPOTENTS

A non-finitely based variety of algebras is said to be a limit variety if all its proper subvarieties are finitely based. Recently, Marcel Jackson published two examples of finitely generated limit varieties of aperiodic monoids with central idempotents and questioned whether or not they are unique. The present article answers this question affirmatively.

A generalization of Fulton–MacPherson configuration spaces

We construct a wonderful compactification of the variety parameterizing n distinct labeled points in X away from D, where X is a nonsingular variety and D is a nonsingular proper subvariety. When D is empty, it coincides with the Fulton–MacPherson configuration space.

Komori’s characterization and top varieties of GMV-algebras

We generalize Komori’s characterization of the proper subvarieties of MV-algebras. Namely, within the variety of generalized MV-algebras (GMV-algebras) such that every maximal ideal is normal, we characterize the proper top varieties. In addition, we present equational bases for these top varieties. We show that there are only countably many different proper top varieties and each of them has uncountably many subvarieties. Finally, we study coproducts and we show that the amalgamation property fails for the class of n-perfect GMV-algebras, i.e., GMV-algebras that can be split into n + 1 comparable slices.

Semi-topological K-theory for certain projective varieties

In this paper we compute Lawson homology groups and semi-topological K-theory for certain threefolds and fourfolds. We consider smooth complex projective varieties whose zero cycles are supported on a proper subvariety. Rationally connected varieties are examples of such varieties. The computation makes use of different techniques of decomposition of the diagonal cycle, of the Bloch–Kato conjecture and of the spectral sequence relating morphic cohomology and semi-topological K-theory.

TWO REMARKS ON SUBVARIETIES OF MODULI SPACES

We study two natural questions about subvarieties of moduli spaces. In the first section, we study the locus of curves equipped with F-nilpotent bundles and its relationship to the p-rank zero locus of the moduli space of curves of genus g. In the second section, we study subvarieties of moduli spaces of vector bundles on curves. We prove an analogue of a result of F. Oort about proper subvarieties of moduli of abelian varieties.

Trisecant lemma for nonequidimensional varieties

Let X be an irreducible projective variety over an algebraically closed field of characteristic zero. For ≥ 3, if every (r−2)-plane \(\overline {x_1 , \ldots ,x_{r - 1} } \), where the x i are generic points, also meets X in a point x r different from x 1,..., x r−1, then X is contained in a linear subspace L such that codim L X ≥ r − 2. In this paper, our purpose is to present another derivation of this result for r = 3 and then to introduce a generalization to nonequidimensional varieties. For the sake of clarity, we shall reformulate our problem as follows. Let Z be an equidimensional variety (maybe singular and/or reducible) of dimension n, other than a linear space, embedded into ℙr, where r ≥ n + 1. The variety of trisecant lines of Z, say V 1,3(Z), has dimension strictly less than 2n, unless Z is included in an (n + 1)-dimensional linear space and has degree at least 3, in which case dim V 1,3(Z) = 2n. This also implies that if dim V 1,3(Z) = 2n, then Z can be embedded in ℙn + 1. Then we inquire the more general case, where Z is not required to be equidimensional. In that case, let Z be a possibly singular variety of dimension n, which may be neither irreducible nor equidimensional, embedded into ℙr, where r ≥ n + 1, and let Y be a proper subvariety of dimension k ≥ 1. Consider now S being a component of maximal dimension of the closure of \(\{ l \in \mathbb{G}(1,r)|\exists p \in Y, q_1 , q_2 \in Z\backslash Y, q_1 , q_2 ,p \in l\} \). We show that S has dimension strictly less than n + k, unless the union of lines in S has dimension n + 1, in which case dim S = n + k. In the latter case, if the dimension of the space is strictly greater than n + 1, then the union of lines in S cannot cover the whole space. This is the main result of our paper. We also introduce some examples showing that our bound is strict.

Weak analytic hyperbolicity of complements of generic surfaces of high degree in projective 3-space

In this article we prove that every entire curve in the complement of a generic hypersurface of degree d ≥ 586 in PC is algebraically degenerated i.e there exists a proper subvariety which contains the entire curve.