Some Properties of the Fundamental Curves of a Birational Transformation in Space

1932 ◽  
Vol 54 (2) ◽  
pp. 305
Author(s):  
Marian M. Torrey
Keyword(s):  
Birational Transformation

On the transformation of certain singular surfaces

1933 ◽  
Vol 29 (3) ◽  
pp. 319-330
Author(s):  
D. W. Babbage
Keyword(s):  
Algebraic Surfaces ◽  
Singular Surfaces ◽  
Ordinary Space ◽  
Birational Transformation ◽  
Classical Paper

Nöther in his classical paper Zur Theorie des eindeutigen Ent-sprechens algebraischer Gebilde has given a set of sixteen examples of the computation of his invariants pn, p(2), p(1) for algebraic surfaces in ordinary space. In the following I discuss, as a matter of some interest, the birational transformation of his surfaces into surfaces which are non-singular.

scholarly journals Classification of regular dicritical foliations

2016 ◽  
Vol 37 (5) ◽  
pp. 1443-1479 ◽  
Author(s):  
GABRIEL CALSAMIGLIA ◽  
YOHANN GENZMER
Keyword(s):  
Vector Space ◽  
Normal Forms ◽  
Blow Up ◽  
The Other ◽  
Holomorphic Foliations ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Finite Dimensional ◽  
Birational Transformation

In this paper we give complete analytic invariants for the set of germs of holomorphic foliations in $(\mathbb{C}^{2},0)$ that become regular after a single blow-up. Some of the invariants describe the holonomy pseudogroup of the germ and are called transverse invariants. The other invariants lie in a finite dimensional complex vector space. Such singularities admit separatrices tangentially to any direction at the origin. When enough separatrices are leaves of a radial foliaton (a condition that can always be attained if the multiplicity of the germ at the origin is at most four) we are able to describe and realize all the analytical invariants geometrically and provide analytic normal forms. As a consequence, we prove that any two such germs sharing the same transverse invariants are conjugated by a very particular type of birational transformation. We also provide explicit examples of universal equisingular unfoldings of foliations that cannot be produced by unfolding functions. With these at hand we are able to explicitly parametrize families of analytically distinct foliations that share the same transverse invariants.

Birational transformations possessing fundamental curves

1938 ◽  
Vol 34 (2) ◽  
pp. 144-155 ◽  
Author(s):  
J. A. Todd
Keyword(s):  
Algebraic Variety ◽  
General Character ◽  
Topological Methods ◽  
Canonical Systems ◽  
Birational Transformations ◽  
Birational Transformation ◽  
Relative Invariants

On any algebraic variety of d dimensions there exist certain systems of equivalence which are relative invariants under birational transformation. These systems include the canonical systems (Todd(3,4)) which can be defined for each dimension from zero to d − 1, and the systems defined by the intersections of these canonical systems among themselves. It is natural to enquire what is the precise behaviour of these invariant systems under birational transformations. Relatively few results of this kind are known. For threefolds, Segre has recently (2) investigated the transformations undergone by the invariant systems in any algebraic (not necessarily birational) transformation whose fundamental points and curves are of general character. More recently, A. Bassi (1) has obtained by topological methods the relations between the Zeuthen-Segre invariants of two Vd in an (α, α′) correspondence. I have recently (6) discussed the transformation of the invariant systems on a Vd for birational transformations on the assumption that the fundamental points are isolated and of general character.

Blowing up Chern Classes

1960 ◽  
Vol 56 (2) ◽  
pp. 118-124 ◽  
Author(s):  
I. R. Porteous
Keyword(s):  
Algebraic Variety ◽  
Topological Structure ◽  
Chern Classes ◽  
Complete Proof ◽  
Blowing Up ◽  
Birational Transformation

The behaviour of the Chern classes or of the canonical classes of an algebraic variety under a dilatation has been studied by several authors (Todd (8)–(11), Segre (5), van de Ven (12)). This problem is of interest since a dilatation is the simplest form of birational transformation which does not preserve the underlying topological structure of the algebraic variety. A relation between the Chern classes of the variety obtained by dilatation of a subvariety and the Chern classes of the original variety has been conjectured by the authors cited above but a complete proof of this relation is not in the literature.

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