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 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 Augmented base loci and restricted volumes on normal varieties, II: The case of real divisors

2015 ◽  
Vol 159 (3) ◽  
pp. 517-527
Author(s):  
ANGELO FELICE LOPEZ
Keyword(s):  
Open Subset ◽  
Projective Variety ◽  
Closed Field ◽  
Projective Varieties ◽  
Algebraically Closed Field ◽  
Smooth Complex ◽  
Normal Varieties ◽  
Restricted Volume ◽  
Complex Projective ◽  
Base Loci

AbstractLet X be a normal projective variety defined over an algebraically closed field and let Z be a subvariety. Let D be an ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on X. Given an expression (*) D$\sim_{\mathbb R}$t1H1 +. . .+ tsHs with ti ∈ ${\mathbb R}$ and Hi very ample, we define the (*)-restricted volume of D to Z and we show that it coincides with the usual restricted volume when Z$\not\subseteq$B+(D). Then, using some recent results of Birkar [Bir], we generalise to ${\mathbb R}$-divisors the two main results of [BCL]: The first, proved for smooth complex projective varieties by Ein, Lazarsfeld, Mustaţă, Nakamaye and Popa, is the characterisation of B+(D) as the union of subvarieties on which the (*)-restricted volume vanishes; the second is that X − B+(D) is the largest open subset on which the Kodaira map defined by large and divisible (*)-multiples of D is an isomorphism.

scholarly journals Special rays in the Mori cone of a projective variety

2002 ◽  
Vol 168 ◽  
pp. 127-137 ◽  
Author(s):  
Marco Andreatta ◽  
Gianluca Occhetta
Keyword(s):  
Projective Variety ◽  
Maximal Length ◽  
Closed Field ◽  
Algebraically Closed Field

AbstractLet X be a smooth n-dimensional projective variety over an algebraically closed field k such that KX is not nef. We give a characterization of non nef extremal rays of X of maximal length (i.e of length n – 1); in the case of Char(k) = 0 we also characterize non nef rays of length n – 2.

scholarly journals GENERAL HYPERPLANE SECTIONS OF THREEFOLDS IN POSITIVE CHARACTERISTIC

2018 ◽  
Vol 19 (2) ◽  
pp. 647-661 ◽  
Author(s):  
Kenta Sato ◽  
Shunsuke Takagi
Keyword(s):  
Projective Variety ◽  
Positive Characteristic ◽  
Hyperplane Section ◽  
Three Dimensional ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Double Points ◽  
Canonical Singularities ◽  
Hyperplane Sections

In this paper, we study the singularities of a general hyperplane section $H$ of a three-dimensional quasi-projective variety $X$ over an algebraically closed field of characteristic $p>0$. We prove that if $X$ has only canonical singularities, then $H$ has only rational double points. We also prove, under the assumption that $p>5$, that if $X$ has only klt singularities, then so does $H$.

scholarly journals Rational connectivity and analytic contractibility

2019 ◽  
Vol 2019 (747) ◽  
pp. 45-62
Author(s):  
Morgan Brown ◽  
Tyler Foster
Keyword(s):  
Laurent Series ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Closed Field ◽  
Homotopy Equivalence ◽  
Formal Laurent Series ◽  
Projective Varieties ◽  
Algebraically Closed Field ◽  
Rationally Connected ◽  
Smooth Projective Varieties

Abstract Let {{k}} be an algebraically closed field of characteristic 0, and let {f:X\to Y} be a morphism of smooth projective varieties over the ring {k((t))} of formal Laurent series. We prove that if a general geometric fiber of f is rationally connected, then the induced map {f^{\mathrm{an}}:X^{\mathrm{an}}\to Y^{\mathrm{an}}} between the Berkovich analytifications of X and Y is a homotopy equivalence. Two important consequences of this result are that the Berkovich analytification of any {\mathbb{P}^{n}} -bundle over a smooth projective {k((t))} -variety is homotopy equivalent to the Berkovich analytification of the base, and that the Berkovich analytification of a rationally connected smooth projective variety over {k((t))} is contractible.

The module of derivations of certain hypersurfaces

2019 ◽  
Vol 19 (08) ◽  
pp. 2050158
Author(s):  
Arindam Dey ◽  
Vinay Wagh
Keyword(s):  
Closed Field ◽  
Generating Set ◽  
Algebraically Closed Field ◽  
Special Case ◽  
Minimal Generating Set

Let [Formula: see text] be an algebraically closed field of characteristic 0. Further, [Formula: see text], where [Formula: see text] is a polynomial in [Formula: see text] such that [Formula: see text]. We show that the module of derivations of [Formula: see text], namely [Formula: see text] is generated by [Formula: see text] elements. We also compute the generators explicitly. It is well known that, in such cases, [Formula: see text] is stably free of rank [Formula: see text]. As a special case, when [Formula: see text] and [Formula: see text] is quasi-homogeneous, we give an explicit minimal generating set for [Formula: see text], consisting of two derivations.

scholarly journals Bounds for the cohomology and the Castelnuovo regularity of certain surfaces

1993 ◽  
Vol 131 ◽  
pp. 109-126 ◽  
Author(s):  
M. Brodmann ◽  
W. Vogel
Keyword(s):  
Open Problem ◽  
Projective Variety ◽  
Closed Field ◽  
Algebraically Closed Field

Let X ⊆ Pr be a reduced, irreducible and non-degenerate projective variety over an algebraically closed field K of characteristic 0. Let reg(x) be the Castelnuovo-Mumford regularity of the sheaf of ideals associated to X.Then it is an open problem—due to D. Eisenbud (see e.g. [E-Go])—whether(0.1) reg(X) ≤ deg(x) - codim (x) + 1,where deg(x) denotes the degree of X and codim(x) denotes the codimension of X. In many cases, this inequality has been proven to hold true.

scholarly journals An Invitation to Model-Theoretic Galois Theory

2010 ◽  
Vol 16 (2) ◽  
pp. 261-269 ◽  
Author(s):  
Alice Medvedev ◽  
Ramin Takloo-Bighash
Keyword(s):  
Galois Group ◽  
Galois Theory ◽  
Order Theory ◽  
Closed Field ◽  
Finite Sets ◽  
Algebraically Closed Field ◽  
First Order ◽  
First Order Theory ◽  
Special Case ◽  
Large Model

AbstractWe carry out some of Galois' work in the setting of an arbitrary first-order theory T. We replace the ambient algebraically closed field by a large model M of T, replace fields by definably closed subsets of M, assume that T codes finite sets, and obtain the fundamental duality of Galois theory matching subgroups of the Galois group of L over F with intermediate extensions F ≤ K ≤ L. This exposition of a special case of [10] has the advantage of requiring almost no background beyond familiarity with fields, polynomials, first-order formulae, and automorphisms.

scholarly journals Remarks on Murre's conjecture on Chow groups

2013 ◽  
Vol 12 (1) ◽  
pp. 3-14 ◽  
Author(s):  
Kejian Xu ◽  
Ze Xu
Keyword(s):  
Abelian Variety ◽  
Function Field ◽  
Projective Variety ◽  
Product Variety ◽  
Smooth Projective Variety ◽  
Chow Groups ◽  
Closed Field ◽  
Irreducible Curve ◽  
Algebraically Closed Field

AbstractFor certain product varieties, Murre's conjecture on Chow groups is investigated. More precisely, let k be an algebraically closed field, X be a smooth projective variety over k and C be a smooth projective irreducible curve over k with function field K. Then we prove that if X (resp. XK) satisfies Murre's conjectures (A) and (B) for a set of Chow-Künneth projectors {, 0 ≤ i ≤ 2dim X} of X (resp. for {()K} of XK) and if for any j, , then the product variety X × C also satisfies Murre's conjectures (A) and (B). As consequences, it is proved that if C is a curve and X is an elliptic modular threefold over k (an algebraically closed field of characteristic 0) or an abelian variety of dimension 3, then Murre's conjecture (B) is true for the fourfold X × C.

On a criterion for hyperplane sections

1988 ◽  
Vol 103 (1) ◽  
pp. 59-67 ◽  
Author(s):  
Lucian Bǎdescu
Keyword(s):  
Line Bundle ◽  
Normal Bundle ◽  
Projective Variety ◽  
Ample Line Bundle ◽  
Cartier Divisor ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Hyperplane Sections

Throughout this paper we shall fix an algebraically closed field k. Consider the following:Problem. Let (Y, L) be a normal polarized variety over k, i.e. a normal projective variety Y over k together with an ample line bundle L on Y. Then one may ask under which conditions the following statement holds:(*) Every normal projective variety X containing Y as an ample Cartier divisor such that the normal bundle of Y in X is L, is isomorphic to the projective cone over Y.

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