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Arithmetically Cohen-Macaulay sets of points in \(\mathbb P^1 \times \mathbb P^1\). (English) Zbl 1346.13001
SpringerBriefs in Mathematics. Cham: Springer (ISBN 978-3-319-24164-7/pbk; 978-3-319-24166-1/ebook). viii, 134 p. (2015).
The so-called interpolation problem is one of the most fundamental branches of classical algebraic geometry. Roughly speaking, the idea behind this problem is to consider finite sets of points in a give space (with some additional properties and restrictions) and describe all possible polynomials vanishing along these collections of points. A natural way to rephrase the interpolation problem is to use Hilbert functions of homogeneous ideals of points in a projective space. Suppose that \(\mathcal{P} = \{P_{1},\dots, P_{s}\}\) is a finite set of points in \(\mathbb{P}^{n}\). Now to each point one associates a positive integer \(m_{i}\), called its multiplicity, and then our aim is to determine the dimension of the vector space of homogeneous polynomials vanishing not only along \(\mathcal{P}\), but also having the additional property that all their \((m_{i}-1)\)-th partial derivatives also vanish at \(P_{i}\). More algebraically, let \(R = \mathbb{F}[x_{1},\dots, x_{n}]\), and consider the homogeneous ideal \(I(\mathcal{P}) = \cap_{i=1}^{s}I(P_{i})^{m_{i}}\). Then the interpolation problem asks about the Hilbert function of \(R/I(\mathcal{P})\), where the Hilbert function is defined to be \(H_{\mathcal{P}}(t) = \dim_{\mathbb{F}}(R/I(\mathcal{P}))_{t}\). Finally, we are ready to formulate the interpolation problem in the language of Hilbert functions.
Problem 1. Classify the numerical functions \(H: \mathbb{N} \rightarrow \mathbb{N}\) such that \(H = H_{\mathcal{P}}\) is the Hilbert function of \(R/I(\mathcal{P})\) where \(I(\mathcal{P})\) is the ideal of points \(\mathcal{P} = \{P_{1},\dots,P_{s}\}\) and each point has multiplicity \(m_{i} \geq 1\).
It is worth pointing out that the case of reduced sets of points (i.e., \(m_{i}=1\)) is solved, but the situation is totally different when one considers points with \(m_{i} > 1\), the so-called fat points. In the case of fat points the interpolation problem is widely open, even for the projective plane.
Problem \(1\) can be naturally generalized in some directions. One of the most natural ways is to consider configurations of linear subspaces, for instance lines in \(\mathbb{P}^{3}\), and then ask about an analogous question, namely to describe possible Hilbert functions. Another approach is to consider sets of points not in a single projective space \(\mathbb{P}^{n}\), but in a product of projective spaces \(\mathbb{P}^{n_{1}} \times\dots\times \mathbb{P}^{n_{r}}\). One of the main differences between points in a single projective space and a product of projective spaces is that in the former, any collection of points has a Cohen-Macaulay coordinate ring, while in the latter case, this is no longer true. Because the coordinate ring of a set of points in a product of projective spaces is multigraded, thus we can generalize the interpolation problem as follows.
Problem 2. Classify the numerical functions \(H : \mathbb{N}^{r} \rightarrow \mathbb{N}\) such that \(H = H_{\mathcal{P}}\) is the Hilbert function of a multigraded ring \(R/I(\mathcal{P})\), where \(\mathcal{P} = \{P_{1}, \dots, P_{s} \} \subset \mathbb{P}^{n_{1}} \times\dots \times \mathbb{P}^{n_{r}}\) is a finite set of points and each point has multiplicity \(m_{i} \geq 1\).
The main obstacle which appears in Problem \(2\) is that we know only a little about the coordinate rings of sets of points in products of projective spaces comparing with a single projective space. The one exception is the case when \(r=2\), \(n_{1} = n_{2} = 1\), and the coordinate rings of sets of points are Cohen-Macaulay. The present monograph is devoted to this particular case and provides a comprehensive survey on the arithmetically Cohen-Macaulay sets of points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\). One of the most important advantage in this setting is that these sets of points are very combinatorial in nature.
Let us here summarize shortly the content of the present monograph. In Chapter \(2\), the authors review basic properties of Cohen-Macaulay rings, primarily focusing on the bigraded case. In Chapter \(3\), the authors focus on (reduced) finite sets of points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\), especially they study their combinatorics. It turns out that using the number of points lying on horizontal or vertical rulings one can define certain sequences of integers (ultimately corresponding to the so-called partitions) which provide information about the algebraic invariants of the associated coordinate rings. Next, the authors introduce the bigraded Hilbert functions of finite sets of points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) (their investigations start with Hilbert functions of points of \(\mathbb{P}^{1}\) and gradually pass to \(\mathbb{P}^{1} \times \mathbb{P}^{1}\)) and at the end of the chapter they introduce the notion of separators. The whole Chapter \(4\) is devoted to arithmetically Cohen-Macaulay (aCM) sets of points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\). aCM sets of points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) are sets of points whose associated bigraded coordinate rings are Cohen-Macaulay. It is known that sets of points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) may or may not be aCM, for instance even really simple configurations of two points might not be aCM. One of the main results of this monograph is a complete classification of aCM sets of points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) (see Theorem 4.11). For instance, this result tells us that the property of being an aCM set of points can be read off directly from the combinatorial description of this set, namely it is enough to compare the so-called partitions (and their conjugations) which decode the number of points lying on horizontal and vertical rulings. Another nice result tells us how to compute the bigraded minimal free resolution of an aCM set using its combinatorial description. In particular, it allows to describe the bigraded Hilbert function combinatorially, which is a great asset of this approach. In Chapter \(5\), the authors provide an answer to the interpolation problem for reduced finite aCM sets of points (see Theorem 5.16 and Corollary 5.17). In Chapter \(6\), the authors focus on fat points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\). Using the combinatorial description of a given set of fat points (i.e., the number of points on each vertical or horizontal rulings and multiplicities of points) one can construct sequences (ultimately partitions) which decode algebraic invariants. The main result of this chapter provides a complete classification of aCM fat points sets (see Theorem 6.21). In a similar vein, the property of being an aCM set can be read off from the combinatorial description, or just using its Hilbert function. Again, just as in the case of reduced sets of points, one can compute for ideals of aCM sets of fat points the bigraded Betti numbers in the bigraded minimal free resolutions using their combinatorics, and consequently, their Hilbert functions. However, the interpolation problem for aCM fat points sets remains an open problem. In Chapter \(7\), the authors consider sets of double points. It is known that sets of double points are rarely aCM sets. However, it is still possible to compute the bigraded minimal free resolutions for these double points. The key idea is based on the so-called complements (i.e., adding in a special way some reduced points to the given set of double points). The main result of this chapter is Algorithm 7.24 which allows to compute the minimal free resolutions for double points. In Chapter \(8\), the authors show how to apply results about double points in order to deal with the so-called Römer conjecture about the shifts that appear in graded free resolutions of ideals, and in the context of relations between symbolic and ordinary powers of ideals. From my point of view, Theorem 8.12 which tells us that if \(\mathcal{P}\) is an aCM set of points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) and \(I(\mathcal{P})\) is the associated ideal, then the second symbolic power of \(I(\mathcal{P})\) coincides with the second ordinary power of \(I(\mathcal{P})\), is one of the most interesting results of the monograph.
The present monograph is nicely written, contains some interesting examples, and almost self-contained. It provides a nice introduction to the interpolation problem for products of projective spaces. In my view, this monograph is adequate for advanced undergraduate students. There are only two small flaws, namely the authors did not provide a series of exercises for the reader, and open problems provided at the end of some chapters are quite general. It would be nice to advertise some partial problems which could be useful in order to understand the global picture. This does not change the fact that I can fully recommend this monograph.

MSC:
13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
13M10 Polynomials and finite commutative rings
13D02 Syzygies, resolutions, complexes and commutative rings
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
32S22 Relations with arrangements of hyperplanes
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