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Liaison of families of subschemes in $${\mathbb{P}}^ n$$. (English) Zbl 0697.14003
Algebraic curves and projective geometry, Proc. Conf., Trento/Italy 1988, Lect. Notes Math. 1389, 128-173 (1989).
[For the entire collection see Zbl 0667.00008.]
The goal of this article is to understand better the structure of the Hilbert scheme $$H(p)=Hilb^ p({\mathbb{P}}^ n_ k)$$ of projective subschemes of $${\mathbb{P}}^ n_ k$$ with Hilbert polynomial p by considering linkage not just of individual subschemes but instead of entire flat families of them, in effect a study of linkage behavior under very general deformation.
For S locally noetherian, an S-point of D(p,q) is a sequence of closed embeddings XYP of flat S-schemes such that for any $$s\in S$$ the schemes $$X_ s$$ and $$Y_ s$$ have Hilbert polynomials p and q, respectively. Further D(p;f)$${}_{CM}$$ is the open subscheme of D(p,q) for which the fibers $$X_ s$$ are Cohen-Macaulay and equidimensional and $$Y_ s$$ are complete intersections of multidegree f$$=f_ 1,...,f_ r$$ for all $$s\in S$$. The main result is that linkage of families X and $$X'$$ with respect to a family of complete intersections Y defines an isomorphism D(p;f)$${}_{CM}\to D(p';{\mathbf{f}})_{CM}$$. If U is a subset of $$H(p)_{CM}$$, all the members of U are contained in complete intersections of the same type, and $$U'$$ is the set of linked subschemes in $$H(p')_{CM}$$, then, under various additional hypotheses, properties of U (openness, irreducibility, smoothness of H(p) along it) can be carried over to $$U'$$. As a corollary, if $$X\in H(p)_{CM}$$ is non- obstructed, linked to $$X'$$, and certain cohomological conditions hold on X and its “generizations”, then $$X'$$ is also non-obstructed. The author also gives a number of concrete examples, and methods for constructing these.
If, for example X is a (locally Cohen-Macaulay) curve in $${\mathbb{P}}^ 3$$ with $$H^ 1(N_ X)=0$$, then by linking geometrically by Y to $$X'$$ (subject to certain constraints on the degrees for Y) and then again by suitable $$Y'$$ to $$X''$$, one obtains an obstructed curve that is doubly linked to X. Further results concern invariance of the cotangent sheaf $$A^ 2_ X$$ and the obstruction space $$A^ 2(XY)$$ under geometric linkage. There is also a subgroup C(XY) of $$A^ 2(XY)$$ that is still invariant under appropriate conditions, but easier to compute and with useful applications to calculation of $$H^ 1(N_ X)$$ for generic complete intersection curves in $${\mathbb{P}}^ 3$$. For related results on invariance of cotangent modules the reader may consult R.-O. Buchweitz [Thesis, Paris VII (1981)], R.-O. Buchweitz and B. Ulrich [“Homological properties invariant under linkage” (preprint 1983)], B. Ulrich [Math. Z. 196, 463-484 (1987; Zbl 0657.13023)], and for results on linkage of curves in $${\mathbb{P}}^ 3$$, the sequence of papers by Bolondi and Migliore.
Reviewer: M.Miller

##### MSC:
 14C05 Parametrization (Chow and Hilbert schemes) 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) 14D15 Formal methods and deformations in algebraic geometry 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)