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.

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.) |