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Introduction to liaison theory and deficiency modules. (English) Zbl 0921.14033
Progress in Mathematics (Boston, Mass.). 165. Boston, MA: Birkhäuser. xii, 215 p. (1998).
In the fall of 1992 the author gave a series of lectures at the Seoul National University. The resulting lecture note: J. C. Migliore, “An introduction to deficiency modules and liaison theory for subschemes of projective space”, Lecture Note Ser. 24 (Seoul 1994; Zbl 0842.14035), followed the author’s intention to introduce students to the notions of deficiency modules as well as to liaison. Over the last couple of years the author thought of improvements and additions in order to change the manuscript with several new perspectives, in particular in direction to liaison. The author’s result of his undertaking is now published in the present book. It makes his fresh introduction to the subject available to a wider community. Let $$C \subset \mathbb{P}^3_k$$ denote a reduced, equidimensional projective curve over an algebraically closed field $$k.$$ In order to control the deviation of being arithmetically Cohen-Macaulay, P. A. Rao [Invent. Math. 50, 205-217 (1979; Zbl 0406.14033)] (following a suggestion of R. Hartshorne) studied the behaviour of $$M(C):= \bigoplus_{n \in \mathbb{Z}} H^1(\mathbb{P}^3, \mathcal J_C(N))$$ under liaison. Note that $$C$$ is arithmetically Cohen-Macaulay if and only if $$M(C)=0$$, while in general $$M(C)$$ admits the structure of a graded $$S$$-module, $$S= k[x_0,\ldots,x_n]$$, of finite length. It was shown by P. Rao that $$M(C)$$ is – up to a shift in grading and duality – an invariant under liaison. This generalizes the result of C. Peskine and L. Szpiro [Invent. Math. 26, 271-302 (1974; Zbl 0298.14022)] where the particular case of $$M(C) = 0$$ was investigated in a more general setting. During the last two decades there arose a lot of interesting results concerning these invariants for curves $$C \subset \mathbb{P}^n_k$$ giving new insights in the geometry of $$C$$ [see e.g. M. Martin-Deschamps and D. Perrin, “Sur la classification des courbes gauches”, Astérisque 184-185 (1990; Zbl 0717.14017) and J. C. Migliore, J. Lond. Math. Soc., II. Ser. 48, No. 3, 396-414 (1993; Zbl 0790.14041) and J. Algebra 99, 548-572 (1986; Zbl 0596.14020)].
The main idea of the present lecture note is to summarize and to extend some of these results to the case of $$V \subset \mathbb{P}^n_k,$$ a subscheme of arbitrary dimension. To this end the author introduced the deficiency modules $$M^i(V):= \bigoplus_{n \in \mathbb{Z}} H^i(\mathbb{P}^n,{\mathcal J}_V(n))$$, $$1\leq i\leq r=\dim V$$, graded $$S$$-modules, $$S = k[x_0,\ldots,x_n],$$ whose non-vanishing measures the failure of $$V$$ to be arithmetically Cohen-Macaulay. In the case $$V$$ is a Cohen-Macaulay scheme these modules are $$S$$-modules of finite length. Moreover the property of being arithmetically Cohen-Macaulay is ‘lifted’ from the general hyperplane section of $$V$$ up to $$V$$ [a result initiated by C. Huneke and B. Ulrich, J. Algebr. Geom. 2, No. 3, 487-505 (1993; Zbl 0808.14041)] with applications to complete intersections. Liaison addition is used in order to construct Buchsbaum curves in $$\mathbb{P}^3_k$$. In generalizing P. Rao’s result it is shown that $$M^i(V_1)$$ and $$M^{r-i+1}(V_2)$$ are isomorphic up to a shift and duality for two linked subschemes $$V_1, V_2$$ provided $$M^i(V_j)$$ is of finite length for a $$j \in \{1,2\}$$ and all $$1 \leq i \leq r$$. The results focus in describing liaison invariants in codimension 2, in particular P. Rao’s parametrization of liaison classes, the structure of even liaison classes and applications to $$\mathbb{P}^n_k,$$ for $$n = 3, 4$$. The most important new feature in his book is the author’s idea of a consequent study of liaison with respect to a Gorenstein ideal instead of a complete intersection as in the classical case. In the case of codimension three this leads to surprising connections to codimension two arithmetically Cohen-Macaulay schemes.
The author’s style is conclusive. He introduces most of the important techniques by a clever arrangement of the material. There are motivations by instructive examples, completed by conclusive arguments in the proofs. Even for researchers the fresh style could give some challenge to continue with this interesting subject, a mixture of algebraic and geometric methods.
Reviewer: P.Schenzel (Halle)

##### MSC:
 14M06 Linkage 13C40 Linkage, complete intersections and determinantal ideals 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra 13D02 Syzygies, resolutions, complexes and commutative rings 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) 14M07 Low codimension problems in algebraic geometry