We give a partial positive answer to a conjecture of Tyurin (). Indeed we prove that on a general quintic hypersurface of P 4 every arithmetically Cohen–Macaulay rank 2 vector bundle is infinitesimally rigid.
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
In this paper we show that on a general sextic hypersurface X ⊂ P 4 , a rank 2 vector bundle E splits if and only if h 1 (E(n)) = 0 for any n ∈ Z. We get thus a characterization of complete intersection curves in X.
In this paper we show that on a general hypersurface of degree r = 3, 4, 5, 6 in P 5 a rank 2 vector bundle E splits if and only if h 1 E(n) = h 2 E(n) = 0 for all n ∈ Z. Similar results for r = 1, 2 were obtained in ,  and .
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 article we apply the classical method of focal loci of families to give a lower bound for the genus of curves lying on general surfaces. First we translate and reprove Xu's result that any curve C on a general surface in P 3 of degree d ≥ 5 has geometric genus g > 1 + degC(d − 5)/2. Then we prove a similar lower bound for the curves lying on a… (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.