Corrigendum to "Classification of Varieties with canonical curve section via Gaussian maps on canonical curves"

2021 ◽  
Vol 143 (6) ◽  
pp. 1661-1663
Author(s):  
Ciro Ciliberto ◽  
Angelo Felice Lopez ◽  
Rick Miranda
Keyword(s):  
Curve Section ◽  
Canonical Curve ◽  
Canonical Curves ◽  
Classification Of Varieties ◽  
Gaussian Maps

CANONICAL CURVES IN ℙ3

2002 ◽  
Vol 85 (2) ◽  
pp. 333-366 ◽  
Author(s):  
JACQUELINE ROJAS ◽  
ISRAEL VAINSENCHER
Keyword(s):  
Vector Space ◽  
Hilbert Scheme ◽  
Linear Component ◽  
Arithmetic Genus ◽  
Generic Point ◽  
Rational Map ◽  
Linear Forms ◽  
Cubic Surface ◽  
Canonical Curve ◽  
Canonical Curves

Let ${\rm Hilb}^{6t-3}(\mathbb{P}^3)$ be the Hilbert scheme of closed 1-dimensional subschemes of degree 6 and arithmetic genus 4 in $\mathbb{P}^3$. Let $H$ be the component of ${\rm Hilb}^{6t-3}(\mathbb{P}^3)$ whose generic point corresponds to a canonical curve, that is, a complete intersection of a quadric and a cubic surface in $\mathbb{P}^3$. Let $F$ be the vector space of linear forms in the variables $z_1, z_2, z_3, z_4$. Denote by $F_d$ the vector space of homogeneous forms of degree $d$. Set $X = \{(f_2,f_3)\}$ where $f_2 \in \mathbb{P}(F_2)$ is a quadric surface, and $f_3 \in \mathbb{P}(F_3/f_2 \cdot F)$ is a cubic modulo $f_2$. We have a rational map, $\sigma : X \cdots \rightarrow H$ defined by $(f_2,f_3) \mapsto f_2 \cap f_3$. It fails to be regular along the locus where $f_2$ and $f_3$ acquire a common linear component. Our main result gives an explicit resolution of the indeterminacies of $\sigma$ as well as of the singularities of $H$. 2000 Mathematical Subject Classification: 14C05, 14N05, 14N10, 14N15.

scholarly journals Trajectory of marketing function ‒ from traditions to innovations

2018 ◽  
Vol 26 (3-4) ◽  
pp. 103-113 ◽  
Author(s):  
Petro Romanovych Putsenteilo ◽  
Vitalii Mykolaiovych Nyanko ◽  
Vitalii Leonidovych Karpenko
Keyword(s):  
Business Processes ◽  
Design Method ◽  
Future Research ◽  
Marketing Function ◽  
Technology Service ◽  
Theoretical Generalization ◽  
Marketing Activities ◽  
Market Needs ◽  
Classification Of Varieties

Purpose – to review modern marketing concepts, definitions of the constituent elements in the marketing system at enterprises and to substantiate the basic postulates underlying the development of innovative marketing at enterprises in order to improve competitiveness of products. Design/Method/Approach. In the course of the study we used methods of theoretical generalization, reasoning and abstraction, as well as analytical, monographic, dialectic methods. Findings. We have revealed the essence and content of the concept of marketing. It has been substantiated that the development of marketing support is a sophisticated dialectical process of interaction between factors from the external environment and the targeted influence of drivers associated with the production of competitive products that satisfy certain market needs. We have reviewed and substantiated the essence, objectives, basic tools and principles of marketing function. We have developed and scientifically substantiated organizational components in the innovative marketing at enterprises based on ensuring the coordinated interaction between their innovation and marketing activities. The essence, content, and principles of the term "innovative marketing" have been defined. We propose a classification of varieties of innovative marketing depending on their functional belonging. Practical implications. Results of the study can be used in the development of proposals related to the effective formation and application of marketing at an enterprise. Originality/Value. We have devised a classification of the marketing function at an enterprise. The basic principles of marketing function have been substantiated. We have defined the methodological provisions for studying the marketing function at an enterprise. The essence, content, and structure of innovative marketing have been determined as an important and integral component of conducting an economic activity by an enterprise, which implies the creation of a fundamentally new product (good, technology, service); the application of innovative marketing has been emphasized for improving business processes at an enterprise. Research limitations/Future research. The task is to develop an effective mechanism for the efficient functioning of marketing under different situational and market-led fluctuations under conditions of market volatility. Paper type – theoretical.

scholarly journals French Silk Varieties in Eighteenth Century

2020 ◽  
Vol 17 (1) ◽  
pp. 53
Author(s):  
Jialiang Lu ◽  
Feng Zhao
Keyword(s):  
Eighteenth Century ◽  
18Th Century ◽  
Material Objects ◽  
Teaching Materials ◽  
Political Factors ◽  
Silk Industry ◽  
Detailed Classification ◽  
Silk Fabrics ◽  
Classification Of Varieties

The design of French silk was very exquisite. Which has formed a clear specification and strict classification system even teaching materials in Eighteenth Century. Based on the existing material objects and teaching materials, this paper systematically sorts out the variety system of French silk fabrics, makes a detailed classification of varieties, and analyzes the political factors of the prosperity and development of French silk industry in the 18th century.

On fourfolds with canonical curve sections

1950 ◽  
Vol 46 (3) ◽  
pp. 419-428 ◽  
Author(s):  
L. Roth
Keyword(s):  
Complete Intersections ◽  
General Character ◽  
Linear Spaces ◽  
Canonical Curve ◽  
Canonical Curves ◽  
Generic Curve

In a recent note the writer has examined the varieties whose generic curve sections are canonical curves of genus p, of general character, and whose surface sections contain only complete intersections with primals; following Fano's classification, we call these varieties of the first species. Such varieties are all rational provided that r > 3 and p > 6. In the present paper we consider their representations on linear spaces for the case r = 4, from which, in conjunction with the previous results, we conclude that fourfolds of the first species exist if, and only if, p ≤ 10; this agrees with the conjecture made by Fano in the case r = 3. It will be seen that the representation of these varieties on [4] provides interesting illustrations of Semple's formulae for composite surfaces in higher space.

scholarly journals On hyperquadrics containing projective varieties

2020 ◽  
Vol 32 (5) ◽  
pp. 1199-1209
Author(s):  
Euisung Park
Keyword(s):  
Projective Variety ◽  
Minimal Degree ◽  
Zero Set ◽  
Projective Varieties ◽  
Quadratic Equations ◽  
Linearly Independent ◽  
Del Pezzo ◽  
Classification Of Varieties ◽  
Irreducible Projective Variety

AbstractClassical Castelnuovo Lemma shows that the number of linearly independent quadratic equations of a nondegenerate irreducible projective variety of codimension c is at most {{{c+1}\choose{2}}} and the equality is attained if and only if the variety is of minimal degree. Also G. Fano’s generalization of Castelnuovo Lemma implies that the next case occurs if and only if the variety is a del Pezzo variety. Recently, these results are extended to the next case in [E. Park, On hypersurfaces containing projective varieties, Forum Math. 27 2015, 2, 843–875]. This paper is intended to complete the classification of varieties satisfying at least {{{c+1}\choose{2}}-3} linearly independent quadratic equations. Also we investigate the zero set of those quadratic equations and apply our results to projective varieties of degree {\geq 2c+1}.

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