The $\mathit{Quot}$ functor of a quasi-coherent sheaf

2019 ◽  
Vol 23 (1) ◽  
pp. 1-20
Author(s):  
Gennaro di Brino
Keyword(s):  
Coherent Sheaf

scholarly journals Simple Helices on Fano Threefolds

2011 ◽  
Vol 54 (3) ◽  
pp. 520-526
Author(s):  
A. Polishchuk
Keyword(s):  
Braid Group ◽  
Vector Bundles ◽  
Coherent Sheaf ◽  
Derived Category ◽  
Fano Threefolds ◽  
Fano Threefold

AbstractBuilding on the work of Nogin, we prove that the braid groupB4acts transitively on full exceptional collections of vector bundles on Fano threefolds withb2= 1 andb3= 0. Equivalently, this group acts transitively on the set of simple helices (considered up to a shift in the derived category) on such a Fano threefold. We also prove that on threefolds withb2= 1 and very ample anticanonical class, every exceptional coherent sheaf is locally free.

scholarly journals A Krull-Schmdit type theorem for coherent sheaves

2014 ◽  
Vol 22 (2) ◽  
pp. 51-56
Author(s):  
A. S. Argáez
Keyword(s):  
Finite Group ◽  
Projective Variety ◽  
Type Theorem ◽  
Coherent Sheaf ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Coherent Sheaves ◽  
A Finite Group ◽  
Natural Decomposition

AbstractLet X be projective variety over an algebraically closed field k and G be a finite group with g.c.d.(char(k), |G|) = 1. We prove that any representations of G on a coherent sheaf, ρ : G → End(ℰ), has a natural decomposition ℰ ≃ ⊕ V ⊗k ℱV, where G acts trivially on ℱV and the sum run over all irreducible representations of G over k.

scholarly journals A lifting result for local cohomology of graded modules

1982 ◽  
Vol 92 (2) ◽  
pp. 221-229 ◽  
Author(s):  
M. Brodmann
Keyword(s):  
General Position ◽  
Projective Variety ◽  
Local Cohomology ◽  
Coherent Sheaf ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Graded Modules ◽  
Hyper Plane ◽  
Special Case

In this paper we prove a lifting result for local cohomology. As a special case we get the following result for the Serre-cohomology over a projective variety:Proposition (1·1). Let ℱ be a coherent sheaf over X, where X is a projective variety over an algebraically closed field k. Let i ≽ 0 and assume that there is a pencil P of hyper-plane sections which is in general position with respect to ℱ (which means that x ∉ H, ∀x ∈ Ass(ℱ), ∀H∈p), and such that for each H ∈ P Hi(X, ℱ│H(n)) = 0, ∀n ≪ 0. Then Hi + 1(X, ℱ) = 0, ∀n ≪ 0.

scholarly journals The resolution property of algebraic surfaces

2011 ◽  
Vol 148 (1) ◽  
pp. 209-226 ◽  
Author(s):  
Philipp Gross
Keyword(s):  
Toric Varieties ◽  
Coherent Sheaf ◽  
Algebraic Surfaces ◽  
Two Dimensional ◽  
Free Sheaf ◽  
Locally Free Sheaf ◽  
Excellent Ring ◽  
Resolution Property

AbstractWe prove that on separated algebraic surfaces every coherent sheaf is a quotient of a locally free sheaf. This class contains many schemes that are neither normal, reduced, quasiprojective nor embeddable into toric varieties. Our methods extend to arbitrary two-dimensional schemes that are proper over an excellent ring.

scholarly journals THE GROUP OF COVERING AUTOMORPHISMS OF A QUASI-COHERENT SHEAF ON $\bm{P}^1(K)$

2007 ◽  
Vol 50 (2) ◽  
pp. 325-341
Author(s):  
E. Enochs ◽  
S. Estrada ◽  
J. R. García Rozas ◽  
L. Oyonarte
Keyword(s):  
Commutative Ring ◽  
Wide Class ◽  
Vector Bundles ◽  
Universal Covering ◽  
Coherent Sheaf ◽  
Projective Line ◽  
Topological Properties ◽  
Coherent Sheaves ◽  
Flat Cover ◽  
Natural Way

AbstractCoGalois groups appear in a natural way in the study of covers. They generalize the well-known group of covering automorphisms associated with universal covering spaces. Recently, it has been proved that each quasi-coherent sheaf over the projective line $\bm{P}^1(R)$ ($R$ is a commutative ring) admits a flat cover and so we have the associated coGalois group of the cover. In general the problem of computing coGalois groups is difficult. We study a wide class of quasi-coherent sheaves whose associated coGalois groups admit a very accurate description in terms of topological properties. This class includes finitely generated and cogenerated sheaves and therefore, in particular, vector bundles.

scholarly journals Torelli theorem for the moduli space of framed bundles

2009 ◽  
Vol 148 (3) ◽  
pp. 409-423 ◽  
Author(s):  
I. BISWAS ◽  
T. GÓMEZ ◽  
V. MUÑOZ
Keyword(s):  
Moduli Space ◽  
Isomorphism Class ◽  
Coherent Sheaf ◽  
Projective Curve ◽  
Smooth Complex ◽  
Torelli Theorem ◽  
The One ◽  
Pointed Curve ◽  
Complex Projective ◽  
Complex Projective Curve

AbstractLet X be an irreducible smooth complex projective curve of genus g ≥ 2, and let x ∈ X be a fixed point. Fix r > 1, and assume that g > 2 if r = 2. A framed bundle is a pair (E, φ), where E is coherent sheaf on X of rank r and fixed determinant ξ, and φ: Ex → r is a non–zero homomorphism. There is a notion of (semi)stability for framed bundles depending on a parameter τ > 0, which gives rise to the moduli space of τ–semistable framed bundles τ. We prove a Torelli theorem for τ, for τ > 0 small enough, meaning, the isomorphism class of the one–pointed curve (X, x), and also the integer r, are uniquely determined by the isomorphism class of the variety τ.

scholarly journals -algebras associated with curves and rational functions on . I

2014 ◽  
Vol 150 (4) ◽  
pp. 621-667 ◽  
Author(s):  
Robert Fisette ◽  
Alexander Polishchuk
Keyword(s):  
Homotopy Class ◽  
Rational Functions ◽  
Coherent Sheaf ◽  
Linear Series ◽  
Projective Curve ◽  
Canonical Embedding ◽  
Rational Map ◽  
Projective Embedding ◽  
Smooth Projective Curve ◽  
Weierstrass Points

AbstractWe consider the natural$A_{\infty }$-structure on the$\mathrm{Ext}$-algebra$\mathrm{Ext}^*(G,G)$associated with the coherent sheaf$G=\mathcal{O}_C\oplus \mathcal{O}_{p_1}\oplus \cdots \oplus \mathcal{O}_{p_n}$on a smooth projective curve$C$, where$p_1,\ldots,p_n\in C$are distinct points. We study the homotopy class of the product$m_3$. Assuming that$h^0(p_1+\cdots +p_n)=1$, we prove that$m_3$is homotopic to zero if and only if$C$is hyperelliptic and the points$p_i$are Weierstrass points. In the latter case we show that$m_4$is not homotopic to zero, provided the genus of$C$is greater than$1$. In the case$n=g$we prove that the$A_{\infty }$-structure is determined uniquely (up to homotopy) by the products$m_i$with$i\le 6$. Also, in this case we study the rational map$\mathcal{M}_{g,g}\to \mathbb{A}^{g^2-2g}$associated with the homotopy class of$m_3$. We prove that for$g\ge 6$it is birational onto its image, while for$g\le 5$it is dominant. We also give an interpretation of this map in terms of tangents to$C$in the canonical embedding and in the projective embedding given by the linear series$|2(p_1+\cdots +p_g)|$.

scholarly journals Exact functors on perverse coherent sheaves

2015 ◽  
Vol 151 (9) ◽  
pp. 1688-1696
Author(s):  
Clemens Koppensteiner
Keyword(s):  
Symplectic Geometry ◽  
Coherent Sheaf ◽  
Alternative Definition ◽  
Coherent Sheaves ◽  
Constructible Sheaves ◽  
Definition Of ◽  
Symplectic Case

Inspired by symplectic geometry and a microlocal characterizations of perverse (constructible) sheaves we consider an alternative definition of perverse coherent sheaves. We show that a coherent sheaf is perverse if and only if $R{\rm\Gamma}_{Z}{\mathcal{F}}$ is concentrated in degree $0$ for special subvarieties $Z$ of $X$. These subvarieties $Z$ are analogs of Lagrangians in the symplectic case.

scholarly journals On the classification of the real vector subspaces of a quaternionic vector space

2013 ◽  
Vol 56 (2) ◽  
pp. 615-622 ◽  
Author(s):  
Radu Pantilie
Keyword(s):  
Vector Space ◽  
Coherent Sheaf ◽  
Covariant Functor ◽  
Real Vector ◽  
The Real ◽  
Real Subspace ◽  
Vector Subspaces

AbstractWe prove the classification of the real vector subspaces of a quaternionic vector space by using a covariant functor which associates, to any pair formed of a quaternionic vector space and a real subspace, a coherent sheaf over the sphere.

scholarly journals Bounded sets of sheaves on Kähler manifolds

2016 ◽  
Vol 2016 (710) ◽  
Author(s):  
Matei Toma
Keyword(s):  
Kähler Manifold ◽  
Coherent Sheaf ◽  
Connected Components ◽  
Hilbert Polynomial ◽  
Chern Classes ◽  
Boundedness Criterion ◽  
Kahler Manifold ◽  
Set Up ◽  
Compact Kähler Manifold ◽  
Bounded Sets

AbstractWe show that any set of quotients with fixed Chern classes of a given coherent sheaf on a compact Kähler manifold is bounded in a sense which we define. The result is proved by adapting Grothendieck's boundedness criterion expressed via the Hilbert polynomial to the Kähler set-up. As a consequence we obtain the compactness of the connected components of the Douady space of a compact Kähler manifold.

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