The Resolution of Singularities of an Algebraic Curve

1939 ◽  
Vol 61 (1) ◽  
pp. 107 ◽  
Author(s):  
H. T. Muhly ◽  
O. Zariski
World Science ◽  
2020 ◽  
Vol 1 (6(58)) ◽  
pp. 21-26
Author(s):  
Akhmadjon Soleev

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.


1945 ◽  
Vol 41 (2) ◽  
pp. 117-126
Author(s):  
W. V. D. Hodge

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*.


2003 ◽  
Vol 86 (2) ◽  
pp. 327-357 ◽  
Author(s):  
A. BRAVO ◽  
O. VILLAMAYOR U.

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.


Author(s):  
Yolanda Lozano ◽  
Steven Duplij ◽  
Malte Henkel ◽  
Malte Henkel ◽  
Euro Spallucci ◽  
...  

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

1968 ◽  
Vol 52 (379) ◽  
pp. 88
Author(s):  
J. A. Todd ◽  
S. S. Abhyankar

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