By Marvin J. Greenberg

ISBN-10: 0805335579

ISBN-13: 9780805335576

ISBN-10: 0805335587

ISBN-13: 9780805335583

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Remark: The Hilbert scheme Hilb(P2 , 2m) is empty. To see this, let Y ⊂ P2 be a subvariety with Hilbert polynomial χOY (m) = 2m, and let Y ⊂ Y be the subvariety without 0–dimension torsion. We have the exact sequence and from it 0 → H00 OY → OY → OYe → 0 χOYe (m) = 2m − length(H00 OY ) ≤ 2m. On the other hand, Y ⊂ P2 is a curve with a resolution 0 → O(−d) → O → OYe → 0 and we get χOYe (m) = dm − d(d+3) . This is a contradiction. So the first non–empty Hilbert scheme on 2 P2 with polynomial 2m + c is that of conics with c = 1.

Boundedness (S. Kleiman, SGA 6, Exp. e. a member of F is a coherent sheaf on Xs for some s ∈ S. The family F is said to be bounded if there exists a scheme T /S of finite type over S and a coherent sheaf F on XT such that each member of F is isomorphic to F t for some t ∈ T . In most cases we have S = Spec(k). Gr The following theorem is due to A. Grothendieck [13] and has been proved explicitly by S. Kl Kleiman in [18]. 1. Theorem: Let X/S be a projective scheme with relative ample line bundle OX (1) such that OXs (1) is globally generated for any s ∈ S and let F be a family of isomorphism classes of coherent sheaves on the fibres of X/S.

In most cases we have S = Spec(k). Gr The following theorem is due to A. Grothendieck [13] and has been proved explicitly by S. Kl Kleiman in [18]. 1. Theorem: Let X/S be a projective scheme with relative ample line bundle OX (1) such that OXs (1) is globally generated for any s ∈ S and let F be a family of isomorphism classes of coherent sheaves on the fibres of X/S. Then the following conditions are equivalent (i) F is bounded (ii) The set χF of Hilbert polynomials χF for [F ] ∈ F is finite and there exists an integer m such that F is m–regular for any [F ] ∈ F.

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