Blow-up algebras in Algebra, Geometry and Combinatorics

Abstract

The primary topic of this thesis lies at the crossroads of Commutative Algebra and its interactions with Algebraic Geometry and Combinatorics. It is mainly focused around the following themes:

I Defining equations of blow-up algebras. II Study of rational maps via blow-up algebras. III Asymptotic properties of the powers of edge ideals of graphs. We are primarily interested in questions that arise in geometrical or combinatorial contexts and try to understand how their possible answers manifest in various algebraic structures or invariants. There is a particular algebraic object, the Rees algebra (or blow-up algebra), that appears in many constructions of Commutative Algebra, Algebraic Geometry, Geometric Modeling, Computer Aided Geometric Design and Combinatorics. The workhorse and main topic of this doctoral dissertation has been the study of this algebra under various situations.

The Rees algebra was introduced in the field of Commutative Algebra in the famous paper [45]. Since then, it has become a central and fundamental object with numerous applications. The study of this algebra has been so fruitful that it is difficult to single out particular results or papers, instead we refer the reader to the books [52] and [53] to wit the “landscape of blow-up algebras”.

From a geometrical point of view, the Rees algebra corresponds with the bi-homogeneous coordinate ring of two fundamental objects: the blow-up of a projective variety along a subvariety and the graph of a rational map between projective varieties (see [27, §II.7]). Therefore, the importance of finding the defining equations of the Rees algebra is probably beyond argument. This is a problem of tall order that has occupied commutative algebraists and algebraic geometers, and despite an extensive effort (see [6, 8, 15–19, 30, 37, 39–43, 51]), it remains open even in the case of polynomial rings in two variables. In [10], Chapter 2 of this dissertation, we use the theory of D-modules to describe the defining ideal of the Rees algebra in the case of a parametrization of a plane curve.

The study of rational and birational maps is classical in the literature from both an algebraic and geometric point of view, and it goes back to the work of Cremona [20], at least. A relatively new idea, probably first used in [31], is to look at the syzygies of the base ideal of a rational map to determine birationality. This algebraic method for studying rational maps has now become an active research topic (see [7, 22, 23, 28, 29, 38, 44, 46, 47]). In a joint work with Buse´ and D’Andrea [9], Chapter 3 of this dissertation, we introduce a new algebra that we call the saturated special fiber ring, which turns out to be an important tool to analyze the degree of a rational map. Later, in [11], Chapter 4 of this dissertation, we compute the multiplicity of this new algebra in the case of perfect ideals of height two, which, in particular, provides an effective method to determine the degree of a rational map having those ideals as base ideal.

Often a good tactic to approach a challenging problem is to go all the way up to a generic case and then find sufficient conditions for the specialization to keep some of the main features of the former. The procedure depends on taking a dramatic number of variables to allow modifying the given data into a generic shape, and usually receives the name of specialization. This method is seemingly due to Kronecker and was quite successful in the hands of Hurwitz ([34]) in establishing a new elegant theory of elimination and resultants. More recent instances where specialization is used are, e.g., [32], [33], [50], [48]. In a joint work with Simis [14], Chapter 5 of this dissertation, we consider the behavior of the degree of a rational map under specialization of the coefficients of the defining linear system.

The Rees algebra of the edge ideal of a graph is a well studied object (see [24, 25, 49, 54–57]), that relates combinatorial properties of a graph with algebraic invariants of the powers of its edge ideal. For the Rees algebra of 1

2 YAIRON CID RUIZ

the edge ideal of a bipartite graph, in [12], Chapter 6 of this dissertation, we compute the universal Gro¨ bner basis of its defining equations and its total Castelnuovo-Mumford regularity as a bigraded algebra.

It is a celebrated result that the regularity of the powers of a homogeneous ideal is asymptotically a linear function (see [21, 36]). Considerable efforts have been put forth to understand the form of this asymptotic linear function in the case of edge ideals (see [1–5, 26, 35]). In a joint work with Jafari, Picone and Nemati [13], Chapter 7 of this dissertation, for bicyclic graphs, i.e. graphs containing exactly two cycles, we characterize the regularity of its edge ideal in terms of the induced matching number and determine the previous asymptotic linear function in special cases.

The basic outline of this thesis is as follows. In Chapter 1, we recall some preliminary results and definitions to be used along this work. Then, the thesis is divided in three different parts. The first part corresponds with the theme “ I Defining equations of blow-up algebras” and consists of Chapter 2. The second part corresponds with the theme “ II Study of rational maps via blow-up algebras” and consists of Chapter 3, Chapter 4 and Chapter 5. The third part corresponds with the theme “ III Asymptotic properties of the powers of edge ideals of graphs” and consists of Chapter 6 and Chapter 7. The common thread and main tool in the three parts of this thesis is the use of blow-up algebras.

References

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BRIEF RESUME OF THE PHD THESIS 3

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