A Note on Scrolls of Smallest Embedded Codimension
نویسنده
چکیده
The situation considered in this note is as follows. Let M be an algebraic submanifold of P with n = dimM . M is said to be a scroll over S if there is a surjective morphism π : M → S such that every fiber Fx = π (x) over x ∈ S is a linear subspace in P of dimension r − 1 = n − s, where s = dimS. This is equivalent to saying that M ∼= PS(E) for some vector bundle E of rank r = n− s+1 and the tautological bundle H(E) is the hyperplane section bundle of M . We have bi(M) = bi(P N ) for i ≤ 2n−N by Barth-Lefschetz Theorem, hence N ≥ 2n− 1 for scrolls, since otherwise 1 + b2(S) = b2(M) = b2(P N ) = 1, contradiction. Thus we want to study scrolls such that N = 2n− 1.
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