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THEORY OF INFINITELY NEAR SINGULAR POINTS
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 Title & Authors
THEORY OF INFINITELY NEAR SINGULAR POINTS
Hironaka, Heisuke;
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 Abstract
The notion of infinitely near singular points, classical in the case of plane curves, has been generalized to higher dimensions in my earlier articles ([5], [6], [7]). There, some basic techniques were developed, notably the three technical theorems which were Differentiation Theorem, Numerical Exponent Theorem and Ambient Reduction Theorem [7]. In this paper, using those results, we will prove the Finite Presentation Theorem, which the auther believes is the first of the most important milestones in the general theory of infinitely near singular points. The presentation is in terms of a finitely generated graded algebra which describes the total aggregate of the trees of infinitely near singular points. The totality is a priori very complex and intricate, including all possible successions of permissible blowing-ups toward the reduction of singularities. The theorem will be proven for singular data on an ambient algebraic shceme, regular and of finite type over any perfect field of any characteristics. Very interesting but not yet apparent connections are expected with many such works as ([1], [8]).
 Keywords
idealistic exponent;finite presentation;
 Language
English
 Cited by
1.
Resolution of singularities of threefolds in positive characteristic II, Journal of Algebra, 2009, 321, 7, 1836  crossref(new windwow)
2.
Resolution of singularities of threefolds in positive characteristic. I., Journal of Algebra, 2008, 320, 3, 1051  crossref(new windwow)
3.
Hypersurface singularities in positive characteristic, Advances in Mathematics, 2007, 213, 2, 687  crossref(new windwow)
4.
On the use of naturality in algorithmic resolution, Journal of Algebra, 2010, 324, 8, 1887  crossref(new windwow)
5.
Singularities in positive characteristic, stratification and simplification of the singular locus, Advances in Mathematics, 2010, 224, 4, 1349  crossref(new windwow)
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