Learn More
INTRODUCTION In this paper we investigate to which extent the theory of Severi on nodal plane curves of a given degree d extends to a linear system on a complex projective nonsingular algebraic surface. As well known, in [S], Anhang F Severi proved that for every d ≥ 3 and 0 ≤ δ ≤ d−1
For a variety X of dimension n in P r , r n(k + 1) + k, the kth secant order of X is the number µ k (X) of (k + 1)-secant k-spaces passing through a general point of the kth secant variety. We show that, if r > n(k + 1) + k, then µ k (X) = 1 unless X is k-weakly defective. Furthermore we give a complete classification of surfaces X ⊂ P r , r > 3k + 2, for(More)
The geometry of a desingularization Y m of an arbitrary subvariety of a generic hypersurface X n in an ambient variety W (e.g. W = P n+1) has received much attention over the past decade or so. Clemens [CKM] has proved that for m = 1, n = 2, W = P 3 and X of degree d, Y has genus g ≥ 1 + d(d − 5)/2 and Xu [X] improved this to g ≥ d(d − 3)/2 − 2 for d ≥ 5(More)
In this paper we generalize Zak's theorems on tangencies and on linear normality as well as Zak's definition and classification of Severi varieties. In particular we find sharp lower bounds for the dimension of higher secant varieties of a given variety X under suitable regularity assumption on X, and we classify varieties for which the bound is attained.