Let k be an algebraically closed field, char k ≠ 2, 3, and let X ⊂ ℙ2 be an elliptic curve with defining polynomial f. We show that any non-trivial torsion point of order r, determines up to equivalence, a unique minimal matrix Φr of size 3r × 3r with linear polynomial entries such that det Φr = fr. We also show that the identity, thought of as the trivial torsion point of order r, determines up to equivalence, a unique minimal matrix Ψr of size (3r - 2) × (3r - 2) with linear and quadratic polynomial entries such that det Ψr = fr.
Given a smooth, projective variety Y over an algebraically closed field of characteristic zero, and a smooth, ample hyperplane section X ⊂ Y, we study the question of when a bundle E on X, extends to a bundle [Formula: see text] on a Zariski open set U ⊂ Y containing X. The main ingredients used are explicit descriptions of various obstruction classes in the deformation theory of bundles, together with Grothendieck–Lefschetz theory. As a consequence, we prove a Noether–Lefschetz theorem for higher rank bundles, which recovers and unifies the Noether–Lefschetz theorems of Joshi and Ravindra–Srinivas.
Abstract. Let X be a smooth projective hypersurface. We derive a splitting criterion for arithmetically Cohen-Macaulay bundles over X. As an application we show that any rank 3 ACM vector bundle over X splits when dim X ≥ 7. We also derive a splitting result for rank 4 arithmetically Cohen-Macaulay bundles.
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