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By Shafarevich I.R.

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3 — The vector bundles S d (Ω(1)) have no proper invariant subsheaves. Proof. Any invariant subsheaf G must necessarily be a subbundle, since SL(V ) acts transitively on P(V ). Let x ∈ P(V ) be a closed point corresponding to a hyperplane W ⊂ V . The isotropy subgroup SL(V )x acts via the canonical surjection SL(V )x → GL(W ) on the fibre S d (Ω(1))(x) = S d W . For any invariant subbundle G the fibre G(x) ⊂ S d W is an GL(W )-subrepresentation. But S d W is an irreducible representation, so that G(x) = 0 or = S d W , which means G = 0 or G = S d (Ω(1)).

E. the restriction of OX (1) to any fibre Xs is ample. Let F be a coherent OX -module. Consider the following assertions: 1. F is S-flat 2. For all sufficiently large m the sheaves f∗ (F (m)) are locally free. 3. The Hilbert polynomial P (Fs ) is locally constant as a function of s ∈ S. 2 — There are implications 1 ⇔ 2 ⇒ 3. If S is reduced then also 3 ⇒ 1. Proof. Thm. 9 in [98] ✷ This provides an important flatness criterion. If S is not reduced, it is easy to write down counterexamples to the implication 3 ⇒ 1.

Let K be a field extension of k. According to the previous lemma, the extended socle E of K ⊗ E is a proper submodule. The extended socle is invariant under all automorphisms of K/k and satisfies the condition Hom(E , K ⊗ E/E ) = 0. Thus E is already defined over k. 11 — If k is algebraically closed and E is a stable sheaf, then E is also gemetrically stable. 12 — Consider the full subcategory C(p) of Coh(X) consisting of all semistable sheaves E with reduced Hilbert polynomial p. Then C(p) is an abelian category in which all objects are Noetherian and Artinian.

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Algebraic geometry by Shafarevich I.R.

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