On the projective normality of complete linear series on an algebraic curve.
On the projective normality of the adjunction bundles.
On the projectivity of the moduli spaces of curves.
On the real X -ranks of points of P n (R) with respect to a real variety X ⊂ P n
Let X ⊂ Pn be an integral and non-degenerate m-dimensional variety defined over R. For any P ∈ Pn(R) the real X-rank r x,R(P) is the minimal cardinality of S ⊂ X(R) such that P ∈ . Here we extend to the real case an upper bound for the X-rank due to Landsberg and Teitler.
On the secant varieties to the osculating variety of a Veronese surface
In this paper we study the k-th osculating variety of the order d Veronese embedding of P n. In particular, for k=n=2 we show that the corresponding secant varieties have the expected dimension except in one case.
On the Severi problem.
On the space curves with the same dual varietey.
On the stability of the restricted bundle of space curves contained in a quartic surface.
On the superadditivity of secant defects
On the weak non-defectivity of veronese embeddings of projective spaces
Fix integers n, x, k such that n≥3, k>0, x≥4, (n, x)≠(3, 4) and k(n+1)<(nn+x). Here we prove that the order x Veronese embedding ofP n is not weakly (k−1)-defective, i.e. for a general S⊃P n such that #(S) = k+1 the projective space | I 2S (x)| of all degree t hypersurfaces ofP n singular at each point of S has dimension (n/n+x )−1− k(n+1) (proved by Alexander and Hirschowitz) and a general F∈| I 2S (x)| has an ordinary double point at each P∈ S and Sing (F)=S.
On unions of scrolls along linear spaces
On vanishing inflection points of plane curves
We study the local behaviour of inflection points of families of plane curves in the projective plane. We develop normal forms and versal deformation concepts for holomorphic function germs which take into account the inflection points of the fibres of . We give a classification of such function- germs which is a projective analog of Arnold’s A,D,E classification. We compute the versal deformation with respect to inflections of Morse function-germs.
Osculatory behavior and second dual varieties of Del Pezzo surfaces.
Picard Numbers of Surfaces in 3-Dimensional Weighted Projective Spaces.
Picard-Borel-Räume.
Plane curves of genus p and degree 2p as projections of space curves of the same degree.
Plane projections of a smooth space curve
Let C be a smooth non-degenerate integral curve of degree d and genus g in over an algebraically closed field of characteristic zero. For each point P in let be the linear system on C induced by the hyperplanes through P. By one maps C onto a plane curve , such a map can be seen as a projection of C from P. If P is not the vertex of a cone of bisecant lines, then will have only finitely many singular points; or to put it slightly different: The secant scheme parametrizing divisors in...
Plücker conditions on plane rational curves.
Polar classes of singular varieties
Poncelet 5-gons and abelian surfaces.