Classification of regular dicritical foliations

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.

Quadro-quadric special birational transformations from projective spaces to smooth complete intersections

Let [Formula: see text] a birational transformation with a smooth connected base locus scheme, where [Formula: see text] is a nondegenerate prime Fano manifold covered by lines. We call [Formula: see text] a quadro-quadric special briational transformation if [Formula: see text] and [Formula: see text] are defined by linear subsystems of [Formula: see text] and [Formula: see text] respectively. In this paper, we classify quadro-quadric special birational transformations in the cases where either (i) [Formula: see text] is a complete intersection and the base locus scheme of [Formula: see text] is smooth, or (ii) [Formula: see text] is a hypersurface.

Blowing up Chern Classes

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.

Birational transformations possessing fundamental curves

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.

Birational transformations with isolated fundamental points

It is well known that the canonical system of curves on an algebraic surface is only relatively invariant under birational transformations of the surface. That is, if we have a birational transformation T between two surfaces F and F′, and if K and K′ denote curves of the unreduced canonical systems on F and F′, thenwhere E and E′ denote the sets of curves, on F and F′ respectively which are transformed into the neighbourhoods of simple points on the other surface.

On the transformation of certain singular surfaces

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.