Approximable Algebras as Subalgebras of Section Rings

2020 ◽  
Author(s):  
Catriona Maclean
Keyword(s):  
Line Bundle ◽  
Projective Variety ◽  
Type Theorem ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Graded Ring ◽  
Partial Converse ◽  
Big Line Bundle

Abstract In [2], Huayi Chen introduced approximable graded algebras, which he uses to prove a Fujita-type theorem in the arithmetic setting, and asked if any such algebra is the graded ring of a big line bundle on a projective variety. This was proved to be false in [ 8]. Continuing the analysis started in [8], we show that while not every approximable graded algebra is a sub algebra of the section ring of a big line bundle, we can associate to any approximable graded algebra $\textbf{B}$ a projective variety $X(\textbf{B})$ and an infinite divisor $D(\textbf{B}) =\sum _{i=1}^\infty a_i D_i$ with $a_i\rightarrow 0$ such that $\textbf{B}$ is a subalgebra of $$\begin{equation*} R( D(\textbf{B}))=\oplus_n H^0(X(\textbf{B}), n D(\textbf{B})).\end{equation*}$$We also establish a partial converse to these results by showing that if an infinite divisor $D=\sum _i a_iD_i$ converges in the space of numerical classes, then any full-dimensional sub-graded algebra of $\oplus _mH^0(X, \lfloor mD \rfloor ))$ is approximable.

scholarly journals LINEAR SYSTEMS ON IRREGULAR VARIETIES

2019 ◽  
Vol 19 (6) ◽  
pp. 2087-2125 ◽  
Author(s):  
Miguel Ángel Barja ◽  
Rita Pardini ◽  
Lidia Stoppino
Keyword(s):  
Continuous Function ◽  
Line Bundle ◽  
Linear Systems ◽  
Projective Variety ◽  
Geometrical Properties ◽  
Complex Projective Variety ◽  
Wide Range ◽  
Image Position ◽  
Irregular Varieties ◽  
Complex Projective

Let $X$ be a normal complex projective variety, $T\subseteq X$ a subvariety of dimension $m$ (possibly $T=X$) and $a:X\rightarrow A$ a morphism to an abelian variety such that $\text{Pic}^{0}(A)$ injects into $\text{Pic}^{0}(T)$; let $L$ be a line bundle on $X$ and $\unicode[STIX]{x1D6FC}\in \text{Pic}^{0}(A)$ a general element.We introduce two new ingredients for the study of linear systems on $X$. First of all, we show the existence of a factorization of the map $a$, called the eventual map of $L$ on $T$, which controls the behavior of the linear systems $|L\otimes \unicode[STIX]{x1D6FC}|_{|T}$, asymptotically with respect to the pullbacks to the connected étale covers $X^{(d)}\rightarrow X$ induced by the $d$-th multiplication map of $A$.Second, we define the so-called continuous rank function$x\mapsto h_{a}^{0}(X_{|T},L+xM)$, where $M$ is the pullback of an ample divisor of $A$. This function extends to a continuous function of $x\in \mathbb{R}$, which is differentiable except possibly at countably many points; when $X=T$ we compute the left derivative explicitly.As an application, we give quick short proofs of a wide range of new Clifford–Severi inequalities, i.e., geographical bounds of the form $$\begin{eqnarray}\displaystyle \text{vol}_{X|T}(L)\geqslant C(m)h_{a}^{0}(X_{|T},L), & & \displaystyle \nonumber\end{eqnarray}$$ where $C(m)={\mathcal{O}}(m!)$ depends on several geometrical properties of $X$, $L$ or $a$.

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.

Circle Actions

2020 ◽  
pp. 161-166
Author(s):  
Loring W. Tu
Keyword(s):  
Differential Form ◽  
Circle Action ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Circle Actions ◽  
Algebra Isomorphism ◽  
Weil Algebra ◽  
Differential Graded Algebras ◽  
Cartan Model

This chapter focuses on circle actions. Specifically, it specializes the Weil algebra and the Weil model to a circle action. In this case, all the formulas simplify. The chapter derives a simpler complex, called the Cartan model, which is isomorphic to the Weil model as differential graded algebras. It considers the theorem that for a circle action, there is a graded-algebra isomorphism. Under the isomorphism F, the Weil differential δ‎ corresponds to a differential called the Cartan differential. An element of the Cartan model is called an equivariant differential form or equivariant form for a circle action on the manifold M.

Vector Fields and the Cohomology Ring of Toric Varieties

2005 ◽  
Vol 48 (3) ◽  
pp. 414-427 ◽  
Author(s):  
Kiumars Kaveh
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Projective Variety ◽  
Cohomology Ring ◽  
Holomorphic Vector Field ◽  
Zero Set ◽  
Graded Algebras ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Complex Projective

AbstractLetXbe a smooth complex projective variety with a holomorphic vector field with isolated zero setZ. From the results of Carrell and Lieberman there exists a filtrationF0⊂F1⊂ · · · ofA(Z), the ring of ℂ-valued functions onZ, such thatas graded algebras. In this note, for a smooth projective toric variety and a vector field generated by the action of a 1-parameter subgroup of the torus, we work out this filtration. Our main result is an explicit connection between this filtration and the polytope algebra ofX.

scholarly journals Effective Faithful Tropicalizations Associated to Adjoint Linear Systems

2018 ◽  
Vol 2019 (19) ◽  
pp. 6089-6112
Author(s):  
Shu Kawaguchi ◽  
Kazuhiko Yamaki
Keyword(s):  
Linear System ◽  
Line Bundle ◽  
Projective Variety ◽  
Ample Line Bundle ◽  
Smooth Projective Variety ◽  
Discrete Valuation Ring ◽  
Complete Discrete Valuation Ring ◽  
System L ◽  
Semistable Model ◽  
Adjoint Bundle

Abstract Let R be a complete discrete valuation ring of equi-characteristic zero with fraction field K. Let X be a connected smooth projective variety of dimension d over K, and let L be an ample line bundle over X. We assume that there exist a regular strictly semistable model ${\mathscr {X}}$ of X over R and a relatively ample line bundle ${\mathscr {L}}$ over ${\mathscr {X}}$ with $\left .{{\mathscr {L}}}\right \vert_{{X}} \cong L$. Let $S({\mathscr {X}})$ be the skeleton associated to ${\mathscr {X}}$ in the Berkovich analytification Xan of X. In this article, we study when $S({\mathscr {X}})$ is faithfully tropicalized into tropical projective space by the adjoint linear system |L⊗m ⊗ ωX|. Roughly speaking, our results show that if m is an integer such that the adjoint bundle is basepoint free, then the adjoint linear system admits a faithful tropicalization of $S({\mathscr {X}})$.

On the Asymptotic Behavior of Complete Kähler Metrics of Positive Ricci Curvature

2010 ◽  
Vol 62 (1) ◽  
pp. 3-18
Author(s):  
Boudjemâa Anchouche
Keyword(s):  
Asymptotic Behavior ◽  
Line Bundle ◽  
Ricci Curvature ◽  
Projective Variety ◽  
Volume Form ◽  
Positive Ricci Curvature ◽  
Standard Type ◽  
Number Of Components ◽  
Holomorphic Sections ◽  
Volume Forms

AbstractLet (X, g) be a complete noncompact Kähler manifold, of dimension n ≥ 2, with positive Ricci curvature and of standard type (see the definition below). N. Mok proved that X can be compactified, i.e., X is biholomorphic to a quasi-projective variety. The aim of this paper is to prove that the L2 holomorphic sections of the line bundle K−qXand the volume form of the metric g have no essential singularities near the divisor at infinity. As a consequence we obtain a comparison between the volume forms of the Kähler metric g and of the Fubini-Study metric induced on X. In the case of dimC X = 2, we establish a relation between the number of components of the divisor D and the dimension of the.

scholarly journals On the set of Hilbert polynomials

2001 ◽  
Vol 64 (2) ◽  
pp. 291-305 ◽  
Author(s):  
Alexander B. Levin
Keyword(s):  
Chromatic Polynomial ◽  
Hilbert Polynomial ◽  
Graded Algebra ◽  
Open Problems ◽  
Graded Algebras ◽  
Hilbert Polynomials

We characterise the set of all Hilbert polynomials of standard graded algebras over a field and give solutions of some open problems on Hilbert polynomials. In particular, we prove that a chromatic polynomial of a graph is a Hilbert polynomial of some standard graded algebra.

scholarly journals RANK FOUR SYMPLECTIC BUNDLES WITHOUT THETA DIVISORS OVER A CURVE OF GENUS TWO

2008 ◽  
Vol 19 (04) ◽  
pp. 387-420 ◽  
Author(s):  
GEORGE H. HITCHING
Keyword(s):  
Line Bundle ◽  
Moduli Space ◽  
Vector Bundles ◽  
Projective Variety ◽  
Theta Divisor ◽  
Weierstrass Points ◽  
Base Locus ◽  
Canonical Bijection ◽  
Genus Two ◽  
Irreducible Projective Variety

The moduli space [Formula: see text] of rank four semistable symplectic vector bundles over a curve X of genus two is an irreducible projective variety of dimension ten. Its Picard group is generated by the determinantal line bundle Ξ. The base locus of the linear system |Ξ| consists of precisely those bundles without theta divisors, that is, admitting nonzero maps from every line bundle of degree -1 over X. We show that this base locus consists of six distinct points, which are in canonical bijection with the Weierstrass points of the curve. We relate our construction of these bundles to another of Raynaud and Beauville using Fourier–Mukai transforms. As an application, we prove that the map sending a symplectic vector bundle to its theta divisor is a surjective map from [Formula: see text] to the space of even 4Θ divisors on the Jacobian variety of the curve.

scholarly journals MINIMAL MODEL PROGRAM WITH SCALING AND ADJUNCTION THEORY

2013 ◽  
Vol 24 (02) ◽  
pp. 1350007 ◽  
Author(s):  
MARCO ANDREATTA
Keyword(s):  
Minimal Model ◽  
Projective Variety ◽  
Model Program ◽  
Minimal Model Program ◽  
Projective Varieties ◽  
Complex Projective Variety ◽  
Adjunction Theory ◽  
Birational Equivalence ◽  
Big Line Bundle ◽  
Complex Projective

Let (X, L) be a quasi-polarized pair, i.e. X is a normal complex projective variety and L is a nef and big line bundle on it. We study, up to birational equivalence, the positivity (nefness) of the adjoint bundles KX + rL for high rational numbers r. For this we run a Minimal Model Program with scaling relative to the divisor KX + rL. We give then some applications, namely the classification up to birational equivalence of quasi-polarized pairs with sectional genus 0, 1 and of embedded projective varieties X ⊂ ℙN with degree smaller than 2 codim ℙN(X) + 2.

scholarly journals A partial converse to the Andreotti–Grauert theorem

2018 ◽  
Vol 155 (1) ◽  
pp. 89-99 ◽  
Author(s):  
Xiaokui Yang
Keyword(s):  
Line Bundle ◽  
Ricci Curvature ◽  
Positive Eigenvalue ◽  
Projective Manifold ◽  
Hermitian Metric ◽  
Partial Converse

Let $X$ be a smooth projective manifold with $\dim _{\mathbb{C}}X=n$. We show that if a line bundle $L$ is $(n-1)$-ample, then it is $(n-1)$-positive. This is a partial converse to the Andreotti–Grauert theorem. As an application, we show that a projective manifold $X$ is uniruled if and only if there exists a Hermitian metric $\unicode[STIX]{x1D714}$ on $X$ such that its Ricci curvature $\text{Ric}(\unicode[STIX]{x1D714})$ has at least one positive eigenvalue everywhere.

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