Affine Lines on Affine Surfaces and the Makar–Limanov Invariant

2008 ◽  
Vol 60 (1) ◽  
pp. 109-139 ◽  
Author(s):  
R. V. Gurjar ◽  
K. Masuda ◽  
M. Miyanishi ◽  
P. Russell
Keyword(s):  
Complex Field ◽  
Picard Number ◽  
Affine Surface ◽  
Proper Maps ◽  
Affine Line ◽  
Smooth Affine

AbstractA smooth affine surface X defined over the complex field C is an ML0 surface if the Makar– Limanov invariant ML(X) is trivial. In this paper we study the topology and geometry of ML0 surfaces. Of particular interest is the question: Is every curve C in X which is isomorphic to the affine line a fiber component of an A1-fibration on X? We shall show that the answer is affirmative if the Picard number ρ(X) = 0, but negative in case ρ(X) ≥ 1. We shall also study the ascent and descent of the ML0 property under proper maps.

scholarly journals Zero cycles with modulus and zero cycles on singular varieties

2017 ◽  
Vol 154 (1) ◽  
pp. 120-187 ◽  
Author(s):  
Federico Binda ◽  
Amalendu Krishna
Keyword(s):  
Cartier Divisor ◽  
Chow Group ◽  
Canonical Decomposition ◽  
Smooth Variety ◽  
Affine Surface ◽  
Singular Varieties ◽  
Smooth Affine ◽  
Torsion Free ◽  
Cycle Class ◽  
Albanese Variety

Given a smooth variety$X$and an effective Cartier divisor$D\subset X$, we show that the cohomological Chow group of 0-cycles on the double of$X$along$D$has a canonical decomposition in terms of the Chow group of 0-cycles$\text{CH}_{0}(X)$and the Chow group of 0-cycles with modulus$\text{CH}_{0}(X|D)$on$X$. When$X$is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of$\text{CH}_{0}(X|D)$. As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that$\text{CH}_{0}(X|D)$is torsion-free and there is an injective cycle class map$\text{CH}_{0}(X|D){\hookrightarrow}K_{0}(X,D)$if$X$is affine. For a smooth affine surface$X$, this is strengthened to show that$K_{0}(X,D)$is an extension of$\text{CH}_{1}(X|D)$by$\text{CH}_{0}(X|D)$.

Identifying Patterns in Complex Field Data

2017 ◽  
Vol 225 (3) ◽  
pp. 268-284 ◽  
Author(s):  
Andrew J. White ◽  
Dieter Kleinböhl ◽  
Thomas Lang ◽  
Alfons O. Hamm ◽  
Alexander L. Gerlach ◽  
...  
Keyword(s):  
Heart Rate ◽  
Panic Disorder ◽  
Latent Class ◽  
Treatment Protocol ◽  
Situational Factors ◽  
Self Report ◽  
Complex Field ◽  
Cluster Assignment ◽  
Experimental Conditions ◽  
Conventional Analysis

Abstract. Ambulatory assessment methods are well suited to examine how patients with panic disorder and agoraphobia (PD/A) undertake situational exposure. But under complex field conditions of a complex treatment protocol, the variability of data can be so high that conventional analytic approaches based on group averages inadequately describe individual variability. To understand how fear responses change throughout exposure, we aimed to demonstrate the incremental value of sorting HR responses (an index of fear) prior to applying averaging procedures. As part of their panic treatment, 85 patients with PD/A completed a total of 233 bus exposure exercises. Heart rate (HR), global positioning system (GPS) location, and self-report data were collected. Patients were randomized to one of two active treatment conditions (standard exposure or fear-augmented exposure) and completed multiple exposures in four consecutive exposure sessions. We used latent class cluster analysis (CA) to cluster heart rate (HR) responses collected at the start of bus exposure exercises (5 min long, centered on bus boarding). Intra-individual patterns of assignment across exposure repetitions were examined to explore the relative influence of individual and situational factors on HR responses. The association between response types and panic disorder symptoms was determined by examining how clusters were related to self-reported anxiety, concordance between HR and self-report measures, and bodily symptom tolerance. These analyses were contrasted with a conventional analysis based on averages across experimental conditions. HR responses were sorted according to form and level criteria and yielded nine clusters, seven of which were interpretable. Cluster assignment was not stable across sessions or treatment condition. Clusters characterized by a low absolute HR level that slowly decayed corresponded with low self-reported anxiety and greater self-rated tolerance of bodily symptoms. Inconsistent individual factors influenced HR responses less than situational factors. Applying clustering can help to extend the conventional analysis of highly variable data collected in the field. We discuss the merits of this approach and reasons for the non-stereotypical pattern of cluster assignment across exposures.

scholarly journals Rank two aCM bundles on the del Pezzo threefold with Picard number 3

2015 ◽  
Vol 429 ◽  
pp. 413-446 ◽  
Author(s):  
Gianfranco Casnati ◽  
Daniele Faenzi ◽  
Francesco Malaspina
Keyword(s):  
Picard Number ◽  
Del Pezzo ◽  
Acm Bundles

scholarly journals Two algebraic deformations of a K3 surface

1981 ◽  
Vol 82 ◽  
pp. 1-26
Author(s):  
Daniel Comenetz
Keyword(s):  
Hyperelliptic Curve ◽  
Double Cover ◽  
Rational Curves ◽  
K3 Surface ◽  
Quartic Surface ◽  
Quadric Cone ◽  
Birational Model ◽  
Affine Line ◽  
Characteristic 2 ◽  
General Member

Let X be a nonsingular algebraic K3 surface carrying a nonsingular hyperelliptic curve of genus 3 and no rational curves. Our purpose is to study two algebraic deformations of X, viz. one specialization and one generalization. We assume the characteristic ≠ 2. The generalization of X is a nonsingular quartic surface Q in P3 : we wish to show in § 1 that there is an irreducible algebraic family of surfaces over the affine line, in which X is a member and in which Q is a general member. The specialization of X is a surface Y having a birational model which is a ramified double cover of a quadric cone in P3.

scholarly journals K3 surfaces with Picard number three and canonical vector heights

2007 ◽  
Vol 76 (259) ◽  
pp. 1493-1499 ◽  
Author(s):  
Arthur Baragar ◽  
Ronald van Luijk
Keyword(s):  
K3 Surfaces ◽  
Picard Number ◽  
Canonical Vector

scholarly journals Seshadri constants on surfaces with Picard number 1

2016 ◽  
Vol 153 (3-4) ◽  
pp. 535-543
Author(s):  
Krishna Hanumanthu
Keyword(s):  
Picard Number ◽  
Seshadri Constants

scholarly journals Characterization of affine ruled surfaces

1997 ◽  
Vol 39 (1) ◽  
pp. 17-20 ◽  
Author(s):  
Włodzimierz Jelonek
Keyword(s):  
Shape Operator ◽  
Ruled Surfaces ◽  
Affine Sphere ◽  
Affine Surface ◽  
Codazzi Tensor ◽  
Difference Tensor ◽  
The Difference ◽  
In The Beginning ◽  
Affine Metric

The aim of this paper is to give certain conditions characterizing ruled affine surfaces in terms of the Blaschke structure (∇, h, S) induced on a surface (M, f) in ℝ3. The investigation of affine ruled surfaces was started by W. Blaschke in the beginning of our century (see [1]). The description of affine ruled surfaces can be also found in the book [11], [3] and [7]. Ruled extremal surfaces are described in [9]. We show in the present paper that a shape operator S is a Codazzi tensor with respect to the Levi-Civita connection ∇ of affine metric h if and only if (M, f) is an affine sphere or a ruled surface. Affine surfaces with ∇S = 0 are described in [2] (see also [4]). We also show that a surface which is not an affine sphere is ruled iff im(S - HI) =ker(S - HI) and ket(S - HI) ⊂ ker dH. Finally we prove that an affine surface with indefinite affine metric is a ruled affine sphere if and only if the difference tensor K is a Codazzi tensor with respect to ∇.

scholarly journals The Noether–Lefschetz Theorem Via Vanishing of Coherent Cohomology

2008 ◽  
Vol 51 (2) ◽  
pp. 283-290 ◽  
Author(s):  
G. V. Ravindra
Keyword(s):  
Picard Number ◽  
Generic Hypersurface ◽  
Lefschetz Theorem

AbstractWe prove that for a generic hypersurface in ℙ2n+1 of degree at least 2 + 2/n, the n-th Picard number is one. The proof is algebraic in nature and follows from certain coherent cohomology vanishing.

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