Hilbert schemes of zero-dimensional subschemes of smooth varieties
Let \(X\) be a smooth projective variety defined over an algebraically closed field \(k\). Let \(X^{[n]} := \text{Hilb}(X)\) be the Hilbert scheme of zero-dimensional subschemes of \(X\) of length \(n\). The main theme of the book is to study the Hilbert schemes \(X^{[n]}\), with particular reference to understand the cohomology and the Chow ring of \(X^{[n]}\). For \(n = 1,2\), the Hilbert scheme \(X^{[n]}\) is very well understood. In fact, \(X^{[1]}\) is just \(X\) itself and \(X^{[2]}\) can be obtained by blowing up \(X \times X\) along the diagonal and taking the quotient by the natural involution that exchanges factors in \(X \times X\). The Hilbert scheme \(X^{[n]}\) is smooth if \(n \leq 3\) or \(\dim X \leq 2\). If \(X\) is a curve, then \(X^{[n]}\) is isomorphic to the \(n\)-th symmetric product, \(X^{(n)}\), of \(X\). In general, in any dimension, the natural set- theoretic map \(X^{[n]} \to X^{(n)}\) sending each 0-scheme to its support (with multiplicities) gives a desingularization of \(X^{(n)}\) if \(X^{[n]}\) is smooth. When \(\dim X = 2\), the canonical bundle of \(X^{[n]}\) is the pullback of the dualizing sheaf of \(X^{(n)}\). In chapter 1 some fundamental facts and background material are recalled. In \S1.1 the definition and the most important properties of \(X^{[n]}\) are given. In \S1.2 the author discusses the Weil conjectures in the form he needs to compute Betti numbers of Hilbert schemes. In \S1.3 the punctual Hilbert scheme, which parametrizes subschemes concentrated in a point of a smooth variety is studied. Chapter 2 is devoted to the computation of the Betti numbers of Hilbert schemes. In \S\S2.1, 2.2 the structure of the closed subscheme of \(X^{[n]}\) which parametrizes subschemes of length \(n\) on \(X\) concentrated on a variable point of \(X\), and the punctual Hilbert schemes \(\text{Hilb}(kx,y)\) are studied. \S2.3 contains the explicit computation of the Betti numbers of \(S^{[n]}\) for a smooth surface \(S\), using the Weil conjectures. The Betti numbers of all the \(S^{[n]}\) are computed as simple power series expressions in terms of the Betti numbers of \(S\). Similar results are obtained in \S2.4, where the Betti numbers of higher order Kummer varieties \(KA_ n\) are also computed. These varieties were previously defined by A. Beauville, as new examples of Calabi-Yau manifolds. \S2.5 is devoted to the computation of the Betti numbers for triangle varieties, which parametrize length 3 0-dimensional subschemes. The main powerful tool used throughout this chapter are the Weil conjectures. The second part of the book, chapters 3 and 4, is devoted to the computation of the cohomology and the Chow ring of Hilbert schemes. In chapter 3, \S\S3.1, 3.2, varieties of second and higher order data are constructed and studied. Such varieties are needed to give precise solutions to classical problems in enumerative algebraic geometry concerning contacts of families of subvarieties of projective space. As an application (see \S\S3.2, 3.3) a formula for the numbers of higher order contacts of a smooth projective variety with linear subvarieties in the ambient space is computed. The last chapter, chapter 4, is the most elementary and classical of the book. In this chapter the Chow ring of relative Hilbert schemes of projective bundles is studied. In \S4.1 the author constructs the embeddings of relative Hilbert schemes into Grassmannian bundles and studies them. The case of the relative Hilbert scheme of a \(\mathbb{P}^ 1\)- bundle over a smooth variety is studied in more detail. In \S4.2 the Chow ring of the variety \(\widetilde{\text{Hilb}}^ 3(\mathbb{P}^ 2)\), parametrizing triangles in \(\mathbb{P}^ 2\) with a marked side is computed. \S4.3 is devoted to a generalization of this result to a relative situation. In \S4.3 the author studies the relative Hilbert scheme \(\text{Hilb}^ 3(\mathbb{P}(E)/X)\) of subschemes of length 3 in the fibers of the projectivization \(\mathbb{P}(E)\) of a vector bundle \(E\). The various chapters are rather independent from each other. To read this book the reader only needs to know the basics of algebraic geometry. The book is therefore of interest not only to experts but also to graduate students and researchers in algebraic geometry not familiar with Hilbert schemes of points. -- This book is also very well organized and very nicely written. Finally, it contains a wide up-to-date bibliography on the topic.
scientific article; zbMATH DE number 556121