Complex Algebraic Surfaces Class 11
نویسنده
چکیده
Last time, we began analyzing geometrically ruled surfaces by studying projectivizations of rank 2 locally free sheaves. We proved: Proposition. Every geometrically ruled surface over C is C-isomorphic to PC(E) for some rank 2 locally free sheaf (vector bundle) over C. The bundles PC(E) and PC(E ′) are isomorphic (over C) iff there is an invertible sheaf (line bundle) L on C such that E ′ ∼= E ⊗ L. I then stated the first part of the following lemma. Lemma: All rank 2 locally free sheaves are filtered nicely by invertible sheaves. Suppose E is a rank 2 locally free sheaf on a curve C. (i) There exists an exact sequence 0 → L → E → M → 0 with L,M ∈ PicC. Terminology: E is an extension ofM by L. (ii) If h(E) ≥ 1, we can take L = OC(D), with D the divisor of zeros of a section of E. (Hence D is effective, i.e. D ≥ 0.) (iii) If h(E) ≥ 2 and degE > 0, we can assume D > 0.
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