Determinantal Ideals (eBook)

Contents 9
Introduction 10
Background 16
1.1 Minimal free resolutions, ACM schemes, and AG schemes 16
1.2 Determinantal ideals 28
1.3 CI-liaison and G-liaison 36
CI-liaison and G-liaison of Standard Determinantal Ideals 44
2.1 CI-liaison class of Cohen–Macaulay codimension 2 ideals 45
2.2 CI-liaison class of standard determinantal ideals 48
2.3 G-liaison class of standard determinantal ideals 56
Multiplicity Conjecture for Standard Determinantal Ideals 59
3.1 The multiplicity conjecture for Cohen – Macaulay codimension 2 ideals 61
3.2 The multiplicity conjecture for standard determinantal ideals 64
Unobstructedness and Dimension of Families of Standard Determinantal Ideals 76
4.1 Families of codimension 2 Cohen–Macaulay algebras 78
4.2 Unobstructedness and dimension of families of determinantal schemes 80
Determinantal Ideals, Symmetric Determinantal Ideals, and Open Problems 117
5.1 Liaison class of determinantal and symmetric determinantal ideals 118
5.2 The multiplicity conjecture for determinantal and symmetric determinantal ideals 123
5.3 Unobstructedness and dimension of families of determinantal and symmetric determinantal ideals 131
Bibliography 141
Index 146

Introduction (p. xi-xii)

In this work, we will deal with standard determinantal ideals, determinantal ideals, and symmetric determinantal ideals, i.e., ideals generated by the maximal minors of a homogeneous polynomial matrix, by the minors (not necessarily maximal) of a homogeneous polynomial matrix, and by the minors of a homogeneous symmetric polynomial matrix, respectively. Some classical ideals that can be constructed in this way are the homogeneous ideal of Segre varieties, the homogeneous ideal of rational normal scrolls, and the homogeneous ideal of Veronese varieties.

Standard determinantal ideals, determinantal ideals, and symmetric determinantal ideals have been a central topic in both commutative algebra and algebraic geometry and they also have numerous connections with invariant theory, representation theory and combinatorics. Due to their important role, their study has attracted many researchers and has received considerable attention in the literature. Some of the most remarkable results are due to J.A. Eagon and M. Hochster [20] and to J.A. Eagon and D.G. Northcott [21]. J.A. Eagon and M. Hochster proved that generic determinantal ideals are Cohen–Macaulay while the Cohen– Macaulayness of symmetric determinantal ideals was proved by R. Kutz in [62, Theorem 1]. J.A. Eagon and D.G. Northcott constructed a .nite free resolution for any standard determinantal ideal and as a corollary they got that standard determinantal ideals are Cohen–Macaulay. In [85], B. Sturmfels uses the Knuth– Robinson–Schensted (KRS) correspondence for the computation of Gr¨obner bases of determinantal ideals. The application of the KRS correspondence to determinantal ideals has also been investigated by S.S. Abhyankar and D.V. Kulkarni in [1] and [2]. Furthermore, variants of the KRS correspondence can be used to study symmetric determinantal ideals (see [17]) or ideals generated by Pfa.ans of skew symmetric matrices (see [47], [5], and [18]). Many other authors have made important contributions to the study of standard determinantal ideals, determinantal ideals, and symmetric determinantal ideals without even being mentioned here and we apologize to those whose work we may have failed to cite properly.

In this book, we will mainly restrict our attention to standard determinantal ideals and we will attempt to address the following three crucial problems.

(1) CI-liaison class and G-liaison class of standard determinantal ideals, determinantal ideals, and symmetric determinantal ideals.
(2) The multiplicity conjecture for standard determinantal ideals, determinantal ideals, and symmetric determinantal ideals.
(3) Unobstructedness and dimension of families of standard determinantal schemes, determinantal schemes, and symmetric determinantal schemes.

Given the extensiveness of the subject, it is not possible to go into great detail in every proof. Still, it is hoped that the material that we choose will be beneficial and illuminating for the reader. The reader can refer [10], [54], [56], [59], [60], [70], [9], and [22] for background, history, and a list of important papers.