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Extra resources for Ample Subvarieties of Algebraic Varieties
16. Let Dt ⊂ P4 be an ACM curve deﬁned by the maximal minors of a t × (t + 2) matrix with linear entries. Dt has a linear resolution . 1 that Dt is glicci. Therefore, Hm (KX ⊗R I(X)) is not a G-liaison invariant. 3. G-liaison class of standard determinantal ideals 41 As another example about the existence of inﬁnitely many diﬀerent CI-liaison classes containing ACM curves C ⊂ P4 we have the following one. 17. , Bordiga) surface and let C ⊂ S be a rational, normal quartic. Consider an eﬀective divisor Ct ∈ |C + tH|, where H is a hyperplane section of S and 0 ≤ t ∈ Z.
E10 . For each general, smooth, rational, ACM surface, we classify the minimal ACM curves C on S (see [56, Section 8]). 12 to conclude that any other ACM curve C on S is glicci. Chapter 2 CI-liaison and G-liaison of Standard Determinantal Ideals It is a classical result, originally proved by F. Gaeta , in 1948, and re-proved in modern language by C. Peskine and L. Szpiro , that every ACM codimension 2 subscheme X ⊂ Pn can be CI-linked in a ﬁnite number of steps to a complete intersection subscheme or, equivalently, all codimension 2, ACM subschemes X ⊂ Pn are licci.
Schenzel in . To see the equivalence, the main point is that there are natural isomorphisms, [I(V1 ) : I(X)]/I(X) ∼ = HomR (R/I(V1 ), R/I(X)) and [I(V2 ) : I(X)]/I(X) ∼ = HomR (R/I(V2 ), R/I(X)). Another way of saying this is that I(V2 ) is the annihilator of I(V1 ) in R/I(X) and I(V1 ) is the annihilator of I(V2 ) in R/I(X). If V1 and V2 do not share any common component, the deﬁnition of direct linkage has a clear geometric meaning. , geometrically G-linked). 3. 3. (a) A simple example of directly geometrically CI-linked schemes is the following one: Let C1 be a twisted cubic in P3 and let C2 be a secant line to C1 .