Gorenstein liaison, complete intersection liaison invariants and unobstructedness.

*(English)*Zbl 1006.14018
Mem. Am. Math. Soc. 732, 116 p. (2001).

Classically, two equidimensional schemes \(V_1,V_2\subset \mathbb{P}^n\) are said to be directly linked by a complete intersection scheme \(X\) if \({\mathcal I}_{V_1}/ {\mathcal I}_X\cong {\mathcal H}om_{{\mathcal O}_{\mathbb{P}^n}} ({\mathcal O}_{V_2}, {\mathcal O}_X)\) and \({\mathcal I}_{V_2}/ {\mathcal I}_X\cong {\mathcal H}om_{{\mathcal O}_{\mathbb{P}^n}} ({\mathcal O}_{V_1}, {\mathcal O}_X)\). In this case, numerical invariants of \(V_1\) and \(V_2\) are known to be closely tied together. If one requires merely that \(X\) is an arithmetically Gorenstein scheme, one arrives at the notion of a direct G-linkage. The equivalence relation generated by direct G-linkage is called G-liaison.

One of the main points of the paper under review is to generalize many of the nice properties of complete intersection liaison (CI-liaison) theory to G-liaison theory and to show that G-liaison is a useful equivalence relation to study. For instance, in 1950, F. Gaeta proved that all projectively normal (i.e. all determinantal) curves in \(\mathbb{P}^3\) form the CI-liaison class of a complete intersection. The authors generalize this to higher codimension by showing that every standard determinantal scheme in \(\mathbb{P}^n\) is G-linked to a complete intersection.

Another view of liaison theory is to consider it as a theory of divisors on certain special schemes. The authors show that G-liaison is in fact a theory of divisors on arithmetically Cohen-Macaulay schemes, not on arithmetically Gorenstein schemes as one might expect at first. Using Hartshorne’s generalized divisor concept, they are able to state many facts which are well-known for CI-liaison, in the more general setting of G-liaison. They also generalize a result of Rao and prove that algebraic and geometric CI-linkage generate the same equivalence relation for equidimensional subschemes which are generic complete intersections.

The second part of the paper deals with groups which are invariant under CI-linkage. The Rao invariants \(H^i_* (\mathbb{P}^n,{\mathcal O}_X)\) are not always able to distinguish CI-liaison classes. Using a geometric approach, the authors demonstrate the CI-liaison invariance of the groups \(H^i_* ({\mathcal N}_X)\), where \({\mathcal N}_X\) is the normal sheaf of \(X\). Applications of this result include the relation between these invariants under basic double linkage on an arithmetically Cohen-Macaulay scheme. When one passes to G-liaison theory, the situation improves considerably, but one major open question remains: “Do arithmetically Cohen-Macaulay schemes of fixed codimension in \(\mathbb{P}^n\) form only one G-liaison class?” If true, this would be the ultimate generalization of Gaeta’s theorem. Although the authors make substantial progress toward an answer (and point out that some of the authors have since made further progress in other papers), they fall short of a complete solution.

The last part of the paper deals with the relation of liaison and the unobstructedness of certain families of subschemes of \(\mathbb{P}^n\). It is known that unobstructedness is preserved under CI-liaison. The authors show that the same holds for G-liaison under certain additional conditions. Using the theory of Hilbert flag schemes, the authors derive explicit results on unobstructedness and provide formulas for the dimension of the Hilbert schemes of several large classes of schemes, for instance certain arithmetically Cohen-Macaulay curves in \(\mathbb{P}^4\) and certain 0-dimensional schemes which are Cartier divisors on unobstructed curves in \(\mathbb{P}^n\). Another application is an upper bound on the dimension of the locus of good determinantal schemes which is sharp in a number of instances.

Although the paper is written in a clear and comprehensible style, it requires a high level of expertise in commutative algebra and algebraic geometry and cites numerous results from other research papers. The bibliography contains no less than 74 entries.

One of the main points of the paper under review is to generalize many of the nice properties of complete intersection liaison (CI-liaison) theory to G-liaison theory and to show that G-liaison is a useful equivalence relation to study. For instance, in 1950, F. Gaeta proved that all projectively normal (i.e. all determinantal) curves in \(\mathbb{P}^3\) form the CI-liaison class of a complete intersection. The authors generalize this to higher codimension by showing that every standard determinantal scheme in \(\mathbb{P}^n\) is G-linked to a complete intersection.

Another view of liaison theory is to consider it as a theory of divisors on certain special schemes. The authors show that G-liaison is in fact a theory of divisors on arithmetically Cohen-Macaulay schemes, not on arithmetically Gorenstein schemes as one might expect at first. Using Hartshorne’s generalized divisor concept, they are able to state many facts which are well-known for CI-liaison, in the more general setting of G-liaison. They also generalize a result of Rao and prove that algebraic and geometric CI-linkage generate the same equivalence relation for equidimensional subschemes which are generic complete intersections.

The second part of the paper deals with groups which are invariant under CI-linkage. The Rao invariants \(H^i_* (\mathbb{P}^n,{\mathcal O}_X)\) are not always able to distinguish CI-liaison classes. Using a geometric approach, the authors demonstrate the CI-liaison invariance of the groups \(H^i_* ({\mathcal N}_X)\), where \({\mathcal N}_X\) is the normal sheaf of \(X\). Applications of this result include the relation between these invariants under basic double linkage on an arithmetically Cohen-Macaulay scheme. When one passes to G-liaison theory, the situation improves considerably, but one major open question remains: “Do arithmetically Cohen-Macaulay schemes of fixed codimension in \(\mathbb{P}^n\) form only one G-liaison class?” If true, this would be the ultimate generalization of Gaeta’s theorem. Although the authors make substantial progress toward an answer (and point out that some of the authors have since made further progress in other papers), they fall short of a complete solution.

The last part of the paper deals with the relation of liaison and the unobstructedness of certain families of subschemes of \(\mathbb{P}^n\). It is known that unobstructedness is preserved under CI-liaison. The authors show that the same holds for G-liaison under certain additional conditions. Using the theory of Hilbert flag schemes, the authors derive explicit results on unobstructedness and provide formulas for the dimension of the Hilbert schemes of several large classes of schemes, for instance certain arithmetically Cohen-Macaulay curves in \(\mathbb{P}^4\) and certain 0-dimensional schemes which are Cartier divisors on unobstructed curves in \(\mathbb{P}^n\). Another application is an upper bound on the dimension of the locus of good determinantal schemes which is sharp in a number of instances.

Although the paper is written in a clear and comprehensible style, it requires a high level of expertise in commutative algebra and algebraic geometry and cites numerous results from other research papers. The bibliography contains no less than 74 entries.

Reviewer: Martin Kreuzer (Regensburg)

##### MSC:

14M12 | Determinantal varieties |

14C05 | Parametrization (Chow and Hilbert schemes) |

14N05 | Projective techniques in algebraic geometry |