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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)