Search results for "Projective space"

On the Betti numbers of three fat points in P1 × P1

2019

In these notes we introduce a numerical function which allows us to describe explicitly (and nonrecursively) the Betti numbers, and hence, the Hilbert function of a set Z of three fat points whose support is an almost complete intersection (ACI) in P1 × P1 . A nonrecursively formula for the Betti numbers and the Hilbert function of these configurations is hard to give even for the corresponding set of five points on a special support in P2 and we did not find any kind of this result in the literature. Moreover, we also give a criterion that allows us to characterize the Hilbert functions of these special set of fat points.

Infinitesimal deformations of double covers of smooth algebraic varieties

2003

The goal of this paper is to give a method to compute the space of infinitesimal deformations of a double cover of a smooth algebraic variety. The space of all infinitesimal deformations has a representation as a direct sum of two subspaces. One is isomorphic to the space of simultaneous deformations of the branch locus and the base of the double covering. The second summand is the subspace of deformations of the double covering which induce trivial deformations of the branch divisor. The main result of the paper is a description of the effect of imposing singularities in the branch locus. As a special case we study deformations of Calabi--Yau threefolds which are non--singular models of do…

Segre and the Foundations of Geometry: From Complex Projective Geometry to Dual Numbers

2016

In 1886 Corrado Segre wrote to Felix Klein about his intention to study ‘geometrie projective pure’, completing and developing the work of von Staudt. He would continue this research project throughout the whole of his scientific life. In 1889, following a suggestion of Segre, Mario Pieri published his translation of the Geometrie der Lage, and from 1889 to 1890 Segre published four important papers, “Un nuovo campo di ricerche geometriche”, in which he completely developed complex projective geometry, considering new mathematical objects such as antiprojectivities and studying the Hermitian forms from a geometrical point of view with the related ‘hyperalgebraic varieties’. Segre developed …

Quasi-Projective Varieties

2000

We have developed the theory of affine and projective varieties separately. We now introduce the concept of a quasi-projective variety, a term that encompasses both cases. More than just a convenience, the notion of a quasi-projective variety will eventually allow us to think of an algebraic variety as an intrinsically defined geometric object, free from any particular embedding in affine or projective space.

QUANTIZATION OPERATORS ON QUADRICS

2008

Kollineationen und Schliessungssätze für Ebene Faserungen

1979

Every affine central collineation of a translation plane π induces a special collineation of the projective space π spanned by the spreadF belonging to π. Here the relations between these special collineations of π and certain incidence propositions inF are investigated; so new proofs are given for some characterisations of (A,B)-regular spreads included in [7].

A note on conjugation involutions on homotopy complex projective spaces

1986

Area minimizing projective planes on the projective space of dimension 3 with the Berger metric

2016

Abstract We show that, among the projective planes embedded into the real projective space R P 3 endowed with the Berger metric, those of least area are exactly the ones obtained by projection of the equatorial spheres of S 3 . This result generalizes a classical result for the projective spaces with the standard metric.

Baer cones in finite projective spaces

1987

Let R and V be two skew subspaces with dimensions r and v of P=PG(d,q). If q is a square, then there is a Baer subspace V* of V, i.e. a subspace of dimension v and order √q. We call the set C(R,V*)=\(\mathop \cup \limits_p \), where the union is taken over all PeV*, aBaer cone oftype (r,v).

An optimal bound for embedding linear spaces into projective planes

1988

Abstract Linear spaces with υ >n 2 − 1 2 n + 1 points, b⩽n2 + n + 1 lines and not constant point degree are classified. It turns out that there is essentially one class of such linear spaces which are not near pencils and which can not be embedded into any projective plane of order n.