Non-degenerate complete intersection singularity.

*(English)*Zbl 0930.14034
Actualités Mathématiques. Paris: Hermann. viii, 309 p. (1997).

The theory of toric varieties developed by G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat [“Toroidal embeddings. I”, Lect. Notes Math. 339 (1973; Zbl 0271.14017)] has opened methods for reducing the study of algebraic varieties to the study of polyhedra and integral points on them.

This book is an exposition of some aspects of the toric theory closely related to the singularity theory. In particular, complete intersection singularities with non-degenerate Newton boundaries are treated. Results by Bernshtein, Khovanskii, Kouchnirenko, Oka, Varchenko and others are explained. We see that a lot of properties of such a singularity can be described only by the Newton polyhedron.

Chapter I is a review of some aspects of the singularity theory. Chapter II is a brief introduction to the toric theory. In chapter III the author begins the study of non-degenerate complete intersection varieties (global objects) and non-degenerate complete intersection singularities (local objects). In particular, the case of hypersurfaces is explained.

In chapter IV aspects related to Minkowski’s mixed volume and the case of complete intersections are treated.

The beginning part of chapter \(V\) treats some problems on a family of varieties. In the middle part, it is shown that, for complete intersection varieties in a torus, a Lefschetz type theorem holds under some sufficient conditions. In the last part, a method for constructing algebraic surfaces with finite abelian fundamental group is introduced.

This book is an exposition of some aspects of the toric theory closely related to the singularity theory. In particular, complete intersection singularities with non-degenerate Newton boundaries are treated. Results by Bernshtein, Khovanskii, Kouchnirenko, Oka, Varchenko and others are explained. We see that a lot of properties of such a singularity can be described only by the Newton polyhedron.

Chapter I is a review of some aspects of the singularity theory. Chapter II is a brief introduction to the toric theory. In chapter III the author begins the study of non-degenerate complete intersection varieties (global objects) and non-degenerate complete intersection singularities (local objects). In particular, the case of hypersurfaces is explained.

In chapter IV aspects related to Minkowski’s mixed volume and the case of complete intersections are treated.

The beginning part of chapter \(V\) treats some problems on a family of varieties. In the middle part, it is shown that, for complete intersection varieties in a torus, a Lefschetz type theorem holds under some sufficient conditions. In the last part, a method for constructing algebraic surfaces with finite abelian fundamental group is introduced.

Reviewer: Tohsuke Urabe (Ibaraki)