scholarly journals Singular loci of instanton sheaves on projective space

2016 ◽  
Vol 27 (07) ◽  
pp. 1640006 ◽  
Author(s):  
Michael Gargate ◽  
Marcos Jardim
Keyword(s):  
Projective Space ◽  
Singular Locus ◽  
Dimension Formula ◽  
Pure Dimension ◽  
Singular Loci ◽  
Higher Rank

We prove that the singular locus of a rank [Formula: see text] non-locally free instanton sheaf [Formula: see text] on [Formula: see text] has pure dimension [Formula: see text]. Moreover, we also show that the dual and double dual of [Formula: see text] are isomorphic locally free instanton sheaves, and that the sheaves [Formula: see text] and [Formula: see text] are rank [Formula: see text] instantons. We also provide explicit examples of instanton sheaves of ranks [Formula: see text] and [Formula: see text] illustrating that all of these claims are false for higher rank instanton sheaves.

Quarto-quartic birational maps of ℙ3(ℂ)

2017 ◽  
Vol 28 (05) ◽  
pp. 1750037
Author(s):  
Julie Déserti ◽  
Frédéric Han
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Birational Maps ◽  
Dimension Formula ◽  
Trigonal Curves ◽  
Irreducible Components ◽  
Genus Formula ◽  
Complex Projective

We construct a determinantal family of quarto-quartic transformations of a complex projective space of dimension [Formula: see text] from trigonal curves of degree [Formula: see text] and genus [Formula: see text]. Moreover, we show that the variety of [Formula: see text]-birational maps of [Formula: see text] has at least four irreducible components and describe three of them.

scholarly journals Higher codimensional foliations with Kupka singularities

2017 ◽  
Vol 28 (03) ◽  
pp. 1750019
Author(s):  
O. Calvo-Andrade ◽  
M. Corrêa ◽  
A. Fernández-Pérez
Keyword(s):  
Normal Form ◽  
Positive Integer ◽  
Projective Space ◽  
Complete Intersection ◽  
Holomorphic Foliations ◽  
Connected Component ◽  
Codimension One ◽  
Dimension Formula ◽  
Rational Fibration

We consider holomorphic foliations of dimension [Formula: see text] and codimension [Formula: see text] in the projective space [Formula: see text], with a compact connected component of the Kupka set. We prove that if the transversal type is linear with positive integer eigenvalues, then the foliation consists of the fibers of a rational fibration [Formula: see text]. As a corollary, if [Formula: see text] and has a transversal type diagonal with different eigenvalues, then the Kupka component [Formula: see text] is a complete intersection and the leaves of the foliation define a rational fibration. The same conclusion holds if the Kupka set has a radial transversal type. Finally, as an application, we find a normal form for non-integrable codimension-one distributions on [Formula: see text].

scholarly journals Defining integer-valued functions in rings of continuous definable functions over a topological field

2020 ◽  
Vol 20 (03) ◽  
pp. 2050014
Author(s):  
Luck Darnière ◽  
Marcus Tressl
Keyword(s):  
Definable Set ◽  
Ordered Field ◽  
Dimension Formula ◽  
Valued Field ◽  
Pure Dimension ◽  
Isolated Points ◽  
Topological Field ◽  
Structure Formula ◽  
Definable Functions ◽  
Ring Of Functions

Let [Formula: see text] be an expansion of either an ordered field [Formula: see text], or a valued field [Formula: see text]. Given a definable set [Formula: see text] let [Formula: see text] be the ring of continuous definable functions from [Formula: see text] to [Formula: see text]. Under very mild assumptions on the geometry of [Formula: see text] and on the structure [Formula: see text], in particular when [Formula: see text] is [Formula: see text]-minimal or [Formula: see text]-minimal, or an expansion of a local field, we prove that the ring of integers [Formula: see text] is interpretable in [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] is definably connected of pure dimension [Formula: see text], then [Formula: see text] defines the subring [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] has no isolated points, then there is a discrete ring [Formula: see text] contained in [Formula: see text] and naturally isomorphic to [Formula: see text], such that the ring of functions [Formula: see text] which take values in [Formula: see text] is definable in [Formula: see text].

scholarly journals Quantitative Singularity Theory for Random Polynomials

2020 ◽  
Author(s):  
Paul Breiding ◽  
Hanieh Keneshlou ◽  
Antonio Lerario
Keyword(s):  
Singularity Theory ◽  
Singular Locus ◽  
Zero Set ◽  
Change Of Variables ◽  
Differentiable Functions ◽  
Topological Features ◽  
Singular Loci ◽  
Semialgebraic Subset ◽  
Set Of Points ◽  
Derivatives Of

Abstract Motivated by Hilbert’s 16th problem we discuss the probabilities of topological features of a system of random homogeneous polynomials. The distribution for the polynomials is the Kostlan distribution. The topological features we consider are type-$W$ singular loci. This is a term that we introduce and that is defined by a list of equalities and inequalities on the derivatives of the polynomials. In technical terms a type-$W$ singular locus is the set of points where the jet of the Kostlan polynomials belongs to a semialgebraic subset $W$ of the jet space, which we require to be invariant under orthogonal change of variables. For instance, the zero set of polynomial functions or the set of critical points fall under this definition. We will show that, with overwhelming probability, the type-$W$ singular locus of a Kostlan polynomial is ambient isotopic to that of a polynomial of lower degree. As a crucial result, this implies that complicated topological configurations are rare. Our results extend earlier results from Diatta and Lerario who considered the special case of the zero set of a single polynomial. Furthermore, for a given polynomial function $p$ we provide a deterministic bound for the radius of the ball in the space of differentiable functions with center $p$, in which the $W$-singularity structure is constant.

scholarly journals Equivariant images of projective space under the action of SL (n, ℤ)

1981 ◽  
Vol 1 (4) ◽  
pp. 519-522 ◽  
Author(s):  
Robert J. Zimmer
Keyword(s):  
Normal Subgroup ◽  
Projective Space ◽  
Lie Group ◽  
Measure Space ◽  
Irreducible Lattice ◽  
Group 3 ◽  
Higher Rank ◽  
Measure Class

The point of this note is to answer in the affirmative a question of G. A. Margulis. In the course of his proof of the finiteness of either the cardinality or the index of a normal subgroup of an irreducible lattice in a higher rank semi-simple Lie group [3], [4], Margulis proves that if Γ = SL (n, ℤ),n≥3, (X, μ) is a measurable Γ-space, μ quasi-invariant, and φ: ℙn−1→Xis a measure class preserving Γ-map, then either φ is a measure space isomorphism or μ is supported on a point. Margulis then asks whether the topological analogue of this result is true. This is answered in the following.

scholarly journals ALGEBRAIC FOLIATIONS DEFINED BY QUASI-LINES

2011 ◽  
Vol 22 (10) ◽  
pp. 1501-1528
Author(s):  
LAURENT BONAVERO ◽  
ANDREAS HÖRING
Keyword(s):  
Projective Space ◽  
Singular Locus ◽  
Projective Manifold ◽  
Line Passing ◽  
Rational Fibration ◽  
Algebraic Foliations

Let X be a projective manifold containing a quasi-line l. An important difference between quasi-lines and lines in the projective space is that in general there is more than one quasi-line passing through two given general points. In this paper, we use this feature to construct an algebraic foliation associated to a family of quasi-lines. We prove that if the singular locus of this foliation is not too large, it induces a rational fibration on X that maps the general leaf of the foliation onto a quasi-line in a rational variety.

scholarly journals Moduli of symplectic instanton vector bundles of higher rank on projective space ℙ3

2012 ◽  
Vol 10 (4) ◽  
pp. 1232-1245 ◽  
Author(s):  
Ugo Bruzzo ◽  
Dimitri Markushevich ◽  
Alexander S. Tikhomirov
Keyword(s):  
Projective Space ◽  
Vector Bundles ◽  
Higher Rank

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

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