The Resolution of Singularities of an Algebraic Curve

1939 ◽  
Vol 61 (1) ◽  
pp. 107 ◽  
Author(s):  
H. T. Muhly ◽  
O. Zariski
Keyword(s):  
Algebraic Curve ◽  
Resolution Of Singularities

scholarly journals POWER GEOMETRY IN LOCAL RESOLUTION OF SINGULARITIES OF AN ALGEBRAIC CURVE

World Science ◽  
2020 ◽  
Vol 1 (6(58)) ◽  
pp. 21-26
Author(s):  
Akhmadjon Soleev
Keyword(s):  
Singular Point ◽  
Algebraic Curve ◽  
General Purpose ◽  
Resolution Of Singularities ◽  
Main Ideas ◽  
Power Geometry

The main goal of this work is to provide a consistent set of general-purpose algorithms for analyzing singularities applicable to all types of equations. We present the main ideas and algorithms of power geometry and give an overview of some of its applications. We also present a procedure that allows us to distinguish all branches of a spatial curve near a singular point and calculate the parametric appearance of these branches with any degree of accuracy. For a specific case, we show how this algorithm works.

On multiple curves. II

1945 ◽  
Vol 41 (2) ◽  
pp. 117-126
Author(s):  
W. V. D. Hodge
Keyword(s):  
Algebraic Curve ◽  
Abelian Integrals

In this note I consider the Abelian integrals of the first kind on an algebraic curve Γ which is a normal multiple of a curve C, as defined in Note I*.

A STRENGTHENING OF RESOLUTION OF SINGULARITIES IN CHARACTERISTIC ZERO

2003 ◽  
Vol 86 (2) ◽  
pp. 327-357 ◽  
Author(s):  
A. BRAVO ◽  
O. VILLAMAYOR U.
Keyword(s):  
Characteristic Zero ◽  
Resolution Of Singularities ◽  
Closed Subscheme ◽  
Subject Classification ◽  
Invertible Sheaf ◽  
Simple Manner ◽  
Birational Morphism ◽  
Exceptional Locus

Let $X$ be a closed subscheme embedded in a scheme $W$, smooth over a field ${\bf k}$ of characteristic zero, and let ${\mathcal I} (X)$ be the sheaf of ideals defining $X$. Assume that the set of regular points of $X$ is dense in $X$. We prove that there exists a proper, birational morphism, $\pi : W_r \longrightarrow W$, obtained as a composition of monoidal transformations, so that if $X_r \subset W_r$ denotes the strict transform of $X \subset W$ then:(1) the morphism $\pi : W_r \longrightarrow W$ is an embedded desingularization of $X$ (as in Hironaka's Theorem);(2) the total transform of ${\mathcal I} (X)$ in ${\mathcal O}_{W_r}$ factors as a product of an invertible sheaf of ideals ${\mathcal L}$ supported on the exceptional locus, and the sheaf of ideals defining the strict transform of $X$ (that is, ${\mathcal I}(X){\mathcal O}_{W_r} = {\mathcal L} \cdot {\mathcal I}(X_r)$).Thus (2) asserts that we can obtain, in a simple manner, the equations defining the desingularization of $X$.2000 Mathematical Subject Classification: 14E15.

Super Real Algebraic Curve

2004 ◽  
pp. 396-396
Author(s):  
Yolanda Lozano ◽  
Steven Duplij ◽  
Malte Henkel ◽  
Malte Henkel ◽  
Euro Spallucci ◽  
...  
Keyword(s):  
Algebraic Curve ◽  
Real Algebraic Curve

Number of points of an algebraic curve

1983 ◽  
Vol 17 (1) ◽  
pp. 53-54 ◽  
Author(s):  
S. G. Vlâdut ◽  
V. G. Drinfel'd
Keyword(s):  
Algebraic Curve

Resolution of Singularities of Embedded Algebraic Surfaces

1968 ◽  
Vol 52 (379) ◽  
pp. 88
Author(s):  
J. A. Todd ◽  
S. S. Abhyankar
Keyword(s):  
Resolution Of Singularities ◽  
Algebraic Surfaces
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