By P.E. Newstead
Backbone name: advent to moduli difficulties and orbit areas.
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Additional info for Lectures on introduction to moduli problems and orbit spaces
4. ∩Hλn ) ⊃ Syz(γ) ∩ Hλ1 ∩ . . ∩ Hλn . Hence we obtain Y ⊃ Y ∩ V ⊃ Syz(γ) ∩ V ⊃ X ∩ V . Since X is irreducible, this implies that Y ⊃ X. Let H1 , . . , Hn be general hyperplanes. ∩Hλn ) ∩ H1 ∩ . . ∩Hn ) for all (λ1 , . . , λn ) ∈ U . Hence Y ∩ H1 ∩ . . ∩Hn ) ∩ V, and we obtain Y ∩ H1 . . ∩Hn ) since both Y ∩ H1 . . ∩Hn ) are irreducible. Hence Y is a variety of minimal degree and dimension n + 1 = dim X + 1. 4. Rank-2 bundles and Koszul classes The aim of this section is to give a more geometric approach to the problem of describing Koszul classes of low rank.
26. 3. 28. 30. If X is a ﬁnite set in a projective space P(V ∨ ), rather than considering the Koszul cohomology with values in the trivial bundle, we should work with the Koszul group Kp,1 (S(X), V ); following [Gre84a, p. 151] we shall denote this group by Kp,1 (X). 29. Let ξ ⊂ Pr be a ﬁnite set of points in general position. Then ξ is contained in a rational normal curve ⇐⇒ Kr−1,1 (ξ) = 0. If deg ξ ≥ r + 3 then Syz(γ) is a rational normal curve for every nonzero Koszul class γ ∈ Kr−1,1 (ξ). 7.
3. The method of Koh–Stillman. Voisin’s method produces syzygies of rank ≤ p + 2. As we have seen in the previous subsection, rank p + 1 syzygies are Green–Lazarsfeld syzygies of scrollar type. Rank p + 2 syzygies can be obtained in the following way. Suppose that L is a globally generated line bundle on a projective variety X, and let [γ] ∈ Kp,1 (X, L) be a nonzero class represented by p an element γ ∈ W ⊗ V with dim W = p + 2. We view γ as an element in 2 2 ∨ W ⊗V ∼ = Hom( W, V ). As before we consider the map γ : 2 (C ⊕ W ) = W ⊕ 2 W →V W → V .