UNIVERSIDAD DE BUENOS AIRES UNIVERSITE PIERRE ET MARIE CURIE Facultad de Ciencias Exactas y Naturales Sciences Mathématiques de Paris Centre Departamento de Matemática Institut de Mathématiques de Jussieu
UNIVERSIDAD DE BUENOS AIRES &
UNIVERSITÉ PIERRE ET MARIE CURIE
For the grade of:
DOCTOR de la UNIVERSIDAD DE BUENOS AIRES,
en el área de Ciencias Matemáticas
DOCTEUR de l’UNIVERSITÉ PIERRE ET MARIE CURIE,
spécialité en Sciences Mathématiques.
IMPLICITIZATION OF RATIONAL MAPS
Defended the 29th September 2010 at 14h, UPMC, Paris, France.
Jury composed by:
Marc Chardin Thesis Advisor Alicia Dickenstein Thesis Advisor David Cox Rapporteurs Jean-Pierre Jouanolou Rapporteurs Monique Lejeune-Jalabert Examinateurs Joseph Oesterlé Examinateurs Laurent Busé Examinateurs
IMPLICITIZATION OF RATIONAL MAPS.
Motivated by the interest in computing explicit formulas for resultants and discriminants initiated by Bézout, Cayley and Sylvester in the eighteenth and nineteenth centuries, and emphasized in the latest years due to the increase of computing power, we focus on the implicitization of hypersurfaces in several contexts. Implicitization means, given a rational map , to compute an implicit equation of the closed image . This is a classical problem and there are numerous approaches to its solution (cf. [SC95] and [Cox01]). However, it turns out that the implicitization problem is computationally difficult.
Our approach is based on the use of linear syzygies by means of approximation complexes, following [BJ03], [BC05], and [Cha06], where they develop the theory for a rational map . Approximation complexes were first introduced by Herzog, Simis and Vasconcelos in [HSV83a] almost 30 years ago.
The main obstruction for this approximation complex-based method comes from the bad behavior of the base locus of . Thus, it is natural to try different compatifications of , that are better suited to the map , in order to avoid unwanted base points. With this purpose, in this thesis we study toric compactifications for . First, we view embedded in a projective space. Furthermore, we compactify the codomain inside , to deal with the case of different denominators in the rational functions defining . We also approach the implicitization problem considering the toric variety defined by its Cox ring, without any particular projective embedding. In all this cases, we blow-up the base locus of the map and we approximate the Rees algebra of this blow-up by the symmetric algebra . We provide resolutions for , such that gives a multiple of the implicit equation, for a graded strand . Precisely, we give specific bounds on all these settings which depend on the regularity of . We also give a geometrical interpretation of the possible other factors appearing on .
Starting from the homogeneous structure of the Cox ring of a toric variety, graded by the divisor class group of , we give a general definition of Castelnuovo-Mumford regularity for a polynomial ring over a commutative ring , graded by a finitely generated abelian group G, in terms of the support of some local cohomology modules. As in the standard case, for a G-graded -module and an homogeneous ideal of , we relate the support of with the support of .
IMPLICITISATION D’APPLICATIONS RATIONNELLES.
Motivé par la recherche de formules explicites pour les résultants et les discriminants, qui remonte au moins aux travaux de Bézout, Cayley et Sylvester au XVIIIème et XIXème siècles et a donné lieu à de nouveaux développements dans les dernières années en raison de l’augmentation de la puissance de calcul, on se concentre sur l’implicitisation des hypersurfaces dans plusieurs contextes. Implicitisation signifie calculer une équation implicite de l’image fermée , étant donnée une application rationnelle . C’est un problème classique et il y a de nombreuses approches (cf. [SC95] et [Cox01]). Toutefois, il s’avère que le problème d’implicitisation est difficile du point de vue du calcul.
Notre approche est basée sur l’utilisation des syzygies linéaires au moyen des complexes d’approximation, en suivant [BJ03], [BC05], et [Cha06], où ils développent la théorie pour une application rationnelle . Les complexes d’approximation ont d’abord été introduits par Herzog, Simis et Vasconcelos dans [HSV83a] il y a presque 30 ans.
L’obstruction principale de la méthode des complexes d’approximation vient du mauvais comportement du lieu base de . Ainsi, il est naturel d’essayer différentes compatifications de , qui sont mieux adaptées à , afin d’éviter des points base non désirés. A cet effet, dans cette thèse on étudie des compactifications toriques de . Tout d’abord, on considère plongée dans un espace projectif. En outre, on compactifie le codomaine dans , pour faire face aux cas des dénominateurs différents dans les fonctions rationnelles qui définissent . On a également abordé le problème implicitisation lorsque la variété torique est définie par son anneau de Cox, sans un plongement projectif particulier. Dans tous ces cas, on éclate le lieu base de et on approche l’algèbre de Rees par l’algèbre symétrique . On fournit des résolutions de , telle que donne un multiple de l’équation implicite, pour . Précisément, on donne des bornes spécifiques dans tous ces cas qui dépendent de la régularité de . On donne aussi une interprétation géométrique des autres facteurs possibles qui apparaissent dans .
Motivé par la structure homogène de l’anneau Cox d’une variété torique, graduée par le groupe de classes de diviseurs de , on donne une définition générale de régularité de Castelnuovo-Mumford pour un anneau de polynômes sur un anneau commutatif , gradué par un groupe abélien de rang fini G, en termes du support de certains modules de cohomologie locale. Comme dans le cas standard, pour un -module G-gradué et un idéal homogène de , on lie le support de avec le support de .
IMPLICITACIÓN DE APLICACIONES RACIONALES.
Motivados por el interés en el cálculo de fórmulas explícitas para resultantes y discriminantes que viene desde Bézout, Cayley y Sylvester en los siglos XVIII y XIX, y enfatizado en los últimos años por el aumento del poder de cómputo, nos concentramos en la implicitación de hipersuperficies en diversos contextos. Por implicitación entendemos que, dada una aplicación racional , calculamos una ecuación implícita de la clausura de la imagen . Éste es un problema clásico con numerosas aproximaciones para su solución (cf. [SC95] y [Cox01]). A pesar de esto, el problema de implicitación es computacionalmente difícil.
Nuestro enfoque se basa en el uso de sicigias lineales mediante complejos de aproximación, siguiendo [BJ03], [BC05], y [Cha06], donde los autores desarrollan la teoría para una aplicación racional . Los complejos de aproximación fueron introducidos por primera vez por Herzog, Simis y Vasconcelos en [HSV83a] hace casi 30 años.
La principal obstrucción para este método basado en complejos de aproximación proviene del mal comportamiento del lugar base de . Luego, es natural buscar diferentes compactificaciones de , que estén mejor adaptadas a la aplicación , con el fin de evitar puntos base no deseados. Con este objetivo, en esta tesis estudiamos compactificaciones tóricas para . Primero, vemos a sumergida en un espacio proyectivo. Más aún, compactificamos el codominio en , para tratar el caso en que las funciones racionales que definen a tengan diferentes denominadores. También abordamos el problema de implicitación considerando la variedad tórica definida por su anillo de Cox, sin una inmersión proyectiva particular. En todos estos casos, explotamos el lugar base de y aproximamos al álgebra de Rees de este blow-up , mediante el álgebra simétrica . Proveemos resoluciones de tales que da un múltiplo de la ecuación implícita, para una capa graduada . Más precisamente, en todos estos casos damos cotas para que dependen de la regularidad de . También damos una interpretación geométrica para los posibles factores extras que aparecen en .
Comenzando desde la estructura homogénea del anillo de Cox de la variedad tórica, graduado por el grupo de clases de divisores de , damos una definición general de la regularidad de Castelnuovo-Mumford para anillos de polinomios sobre un anillo conmutativo , graduado por un grupo abeliano G finitamente generado, en término de los soportes de algunos módulos de cohomología local. Tal como en el caso estándar, dado un -módulo G-graduado y un ideal homogéneo de , relacionamos el soporte de con el soporte de .
like to thank
a mi directora Alicia et à mon directeur Marc, for all they have done for me, teaching, helping, suggesting. It was simultaneously a pleasure and a honor;
a mis amigos de la facultad, por acompañarme y facilitarme el trabajo durante estos años, sin su ayuda este trabajo no hubiera sido posible. Al resto de mis amigos, por su apoyo incondicional. A Ale;
à mes amis conus en France, car vous avez faites qu’être loin de chez moi soit agreable. Avec qui j’ai bien travaillé et amusé;
to all mathematicians that helped me constructing what I have done in this science;
A mi familia, a mis padres y a mi hermana. A Flor.
The interest in computing explicit formulas for resultants and discriminants goes back to Bézout, Cayley, Sylvester and many others in the eighteenth and nineteenth centuries. It has been emphasized in the latest years due to the increase of computing power. Under suitable hypotheses, resultants give the answer to many problems in elimination theory, including the implicitization of rational maps. In turn, both resultants and discriminants can be seen as the implicit equation of a suitable map (cf. [DFS07]). Lately, rational maps appeared in computer-engineering contexts, mostly applied to shape modeling using computer-aided design methods for curves and surfaces.
Rational algebraic curves and surfaces can be described in several different ways, the most common being parametric and implicit representations. Parametric representations describe the geometric object as the image of a rational map, whereas implicit representations describe it as the set of points verifying a certain algebraic condition, e.g. as the zeros of a polynomial equation. Both representations have a wide range of applications in Computer Aided Geometric Design (CAGD), and depending on the problem one needs to solve, one or the other might be better suited. It is thus interesting to be able to pass from parametric representations to implicit equations. This is a classical problem and there are numerous approaches to its solution (a good historical overview on this subject can be seen in [SC95] and [Cox01]). However, it turns out that the implicitization problem is computationally difficult.
A promising alternative suggested in [BD07] is to compute a so-called matrix representation instead, which is easier to compute but still shares some of the advantages of the implicit equation. Let be a field. For a given hypersurface , a matrix with entries in the polynomial ring is called a representation matrix of if it is generically of full rank and if the rank of evaluated in a point of drops if and only if the point lies on (see Chapter \thechapter, also cf. [BDD09]). Equivalently, a matrix represents if and only if the greatest common divisor of all its minors of maximal size is a power of the homogeneous implicit equation of .
In the case of a planar rational curve given by a parametrization of the form , , where are coprime polynomials of degree and is a field, a (linear) syzygy (or moving line) is a linear relation on the polynomials , i.e. a linear form in the variables and with polynomial coefficients such that . We denote by the set of all those linear syzygies forms and for any integer the graded part of syzygies of degree at most . To be precise, one should homogenize the with respect to a new variable and consider as a graded module here. It is obvious that is a finite-dimensional -vector space of dimension , obtained by solving a linear system. Let be a basis of . If , we define the matrix , that is, the coefficients of the with respect to a -basis of form the columns of the matrix. Note that the entries of this matrix are linear forms in the variables with coefficients in the field . Let denote the homogeneous implicit equation of the curve and the degree of the parametrization as a rational map. Intuitively, measures how many times the curve is traced. It is known that for , the matrix is a representation matrix; more precisely: if , then is a square matrix, such that . Also, if , then is a non-square matrix with more columns than rows, such that the greatest common divisor of its minors of maximal size equals . In other words, one can always represent the curve as a square matrix of linear syzygies. One could now actually calculate the implicit equation. We overview this subject more widely in Section 7.1.
For surfaces, matrix representations have been studied in [BDD09] for the case of -dimensional projective toric varieties, and we will analyze it in detail in Chapter \thechapter. Previous work had been done in this direction, with two main approaches: One allows the use of quadratic syzygies (or higher-order syzygies) in addition to the linear syzygies, in order to be able to construct square matrices, the other one only uses linear syzygies as in the curve case and obtains non-square representation matrices.
The first approach using linear and quadratic syzygies (or moving planes and quadrics) has been treated in [Cox03a] for base-point-free homogeneous parametrizations and some genericity assumptions, when . The authors of [BCD03] also treat the case of toric surfaces in the presence of base points. In [AHW05], square matrix representations of bihomogeneous parametrizations, i.e. , are constructed with linear and quadratic syzygies, whereas [KD06] gives such a construction for parametrizations over toric varieties of dimension 2. The methods using quadratic syzygies usually require additional conditions on the parametrization and the choice of the quadratic syzygies is often not canonical.
The second approach, developed in more detail in Section 7.2, even though it does not produce square matrices, has certain advantages, in particular in the sparse setting that we present. In previous publications, this approach with linear syzygies, which relies on the use of the so-called approximation complexes has been developed in the case , see for example [BJ03], [BC05], and [Cha06], and in [BD07] for bihomogeneous parametrizations of degree . However, for a given affine parametrization , these two varieties are not necessarily the best choice of a compactification, since they do not always reflect well the combinatorial structure of the polynomials defining the parametrization. We extend the method to a much larger class of varieties, namely toric varieties of dimension (cf. [BDD09], see also 15). We show that it is possible to choose a “good” toric compactification of depending on the input polynomials, which makes the method applicable in cases where it failed over or . Also, it is significantly more efficient, leading to smaller representation matrices.
Later, in [Bot10], see Chapter \thechapter, we gave different compactifications for the domain and the codomain of an affine rational map that parametrizes a hypersurface in any dimension and we show that the closure of the image of this map (with possibly some other extra hypersurfaces) can be represented by a matrix of linear syzygies, relaxing the hypothesis on the base locus. More generally, we compactify into an -dimensional projective arithmetically Cohen-Macaulay subscheme of some . We studied one particular interesting compactification of which is the toric variety associated to the Newton polytope of the polynomials defining .
In [Bot09b] and [Bot10] we considered a different compactifications for the codomain of , as is detailed in Chapter \thechapter. We study the implicitization problem in this setting. This new perspective allow to deal with parametric rational maps with different denominators. Precisely, given , we can naturally consider a map (cf. [Bot09b]). As we have remarked before, need not be the best compactification of the domain of , thus, in [Bot10] we extended this method the setting where is any arithmetically Cohen-Macaulay closed subscheme of some . In this last context, we gave sufficient conditions, in terms of the nature of the base locus of the map, for getting a matrix representation of its closed image, without involving extra hypersurfaces (cf. Chapter \thechapter).
In order to avoid a particular embedding of in , we focused on the study of implicitization problem for toric varieties given by its Cox ring (see Section 11 or [Cox95]). This leaded to adapting the technique based on approximation complexes for more general graded rings and modules. In Chapter \thechapter we give a definition of Castelnuovo-Mumford regularity for a commutative ring graded by a finitely generated abelian group , in terms of the support of some local cohomology modules. A very interesting example is that of Cox rings of toric varieties, where the grading is given by the Chow group of the variety acting on a polynomial ring. Thus, this allows to study the implicitization problem for general arithmetically Cohen Macaulay toric varieties without the need of an embedding, as we do in Chapter \thechapter.
Ch. : Preliminaries on elimination theory and approximation complexes.
Ch. : Preliminaries on toric varieties.
Ch. : Implicitization for , by means of an embedding .
Ch. : Implicitization for , by means of an embedding .
Ch. : Castelnuovo-Mumford regularity for G-graded rings, for G abelian group.
Ch. : Implicitization for , where is defined by the Cox ring.
Ch. : Algorithm for following Chapter \thechapter.
Ch. : Algorithm for following Chapter \thechapter.
In Chapter \thechapter we give a fast overview of the original technique of computing implicit equations for projective rational maps by means of approximation complexes. Indeed, we introduce in Section 5 the notion of approximation complexes and of blow-up algebras in Section 3, and we give basic results that we will use later in this thesis. As it was mentioned, this approach with linear syzygies was first formulated for this purpose in [BJ03] an later improved in [BC05], [Cha06] and [BCJ09]. We give a more detailed outline of this method in Section 7.2.
Chapter \thechapter is mainly devoted to give an introduction to toric varieties. We recall some results that we will need later, in order to generalize the implicitization methods for toric compactifications. We develop this idea in Chapters \thechapter, \thechapter and \thechapter.
In Chapters \thechapter and \thechapter we adapt the method of approximation complexes to computing an implicit equation of a parametrized hypersurface, focusing on different compactifications of the domain and of the codomain ( and ). We will always assume that is a -dimensional closed subscheme of with graded and Cohen-Macaulay -dimensional coordinate ring .
In Chapter \thechapter, we focus on the implicitization problem for a rational map defined by polynomials of degree . We extend the method to maps defined over an -dimensional Cohen-Macaulay closed scheme , embedded in , emphasizing the case where is a toric variety. We show that we can relax the hypotheses on the base locus by admitting it to be a zero-dimensional almost locally complete intersection scheme. Implicitization in codimension one is well adapted in this case, as is shown in Section 13 and 14, following the spirit of many papers in this subject: [BJ03], [BCJ09], [BD07], [BDD09] and [Bot09b].
In order to consider more general parametrizations given by rational maps of the form with different denominators , we develop in Chapter \thechapter the study of the compactification of the codomain. With this approach, we study following [Bot09b] and [Bot10], the method of implicitization of projective hypersurfaces embedded in . As in Chapters \thechapter and \thechapter, we compute the implicit equation as the determinant of a complex which coincides with the gcd of the maximal minors of the last matrix of the complex, and we make deep analysis of the geometry of the base locus.
In Chapter \thechapter we exemplify the results of Chapters \thechapter and \thechapter, and we study in a more combinatorial fashion the size of the matrices obtained. We analyze, in both settings, how taking an homothety of the Newton polytope can modify the size of the matrices . We present several examples comparing our results with the previous ones. First, we show in a very sparse setting the advantage of not considering the homogeneous compactification of the domain when denominators are very different. We extend in the second example this idea to the case of a generic affine rational map in dimension with fixed Newton polytope. In the last example we give, for a parametrized toric hypersurface of , a detailed analysis of the relation between the nature of the base locus of a map and the extra factors appearing in the computed equation. We finish this section by giving an example of how the developed technique can be applied to the computation of sparse discriminants.
In order to avoid a particular embedding of in , we focus in Chapter \thechapter on the study of the implicitization problem for toric varieties given by its Cox ring (see Section 11 or the original source in [Cox95]). Motivated by this, in Chapter \thechapter we give a definition of Castelnuovo-Mumford regularity for a commutative ring graded by a finitely generated abelian group , in terms of the support of some local cohomology modules.
In Chapter \thechapter we give a definition of Castelnuovo-Mumford regularity for a commutative ring graded by a finitely generated abelian group , in terms of the support of some local cohomology modules. This generalizes [HW04] and [MS04]. With this purpose, we distinguish an ideal of , and we determine subsets of where the -graded modules are supported, this is, elements where . Also, we study the regularity of some particular rings, in particular, polynomial rings -graded, and we show that in these cases this notion of regularity coincides with the usual one. A very interesting example is that of Cox rings of toric varieties, where the grading is given by the Chow group of the variety acting on a polynomial ring (cf. [Cox95]).
Lately, we establish, for a -graded -module , a relation between the supports of the modules and the support of the Betti numbers of , generalizing the well-known duality for the -graded case.
In Chapter \thechapter we present a method for computing the implicit equation of a hypersurface given as the image of a rational map , where is an arithmetically Cohen-Macaulay toric variety defined by its Cox ring (see Section 11). In Chapters \thechapter and \thechapter, the approach consisted in embedding the space in a projective space. The need of this embedding comes from the necessity of a -grading in the coordinate ring of , in order to study its regularity. The aim of this chapter is to give an alternative to this approach: we study the implicitization problem directly, without an embedding in a projective space, by means of the results of Chapter \thechapter. Indeed, we deal with the multihomogeneous structure of the coordinate ring of , and we adapt the method developed in Chapters \thechapter, \thechapter and \thechapter to this setting. The main motivations for our change of perspective are that it is more natural to deal with the original grading on , and that the embedding leads to an artificial homogenization process that makes the effective computation slower, as the number of variables to eliminate increases.
Chapter \thechapter and Chapter \thechapter are devoted to the algorithmic approach of both cases studied in Chapters \thechapter and \thechapter. We show how to compute the sizes of the representation matrices obtained in both cases by means of the Hilbert functions of the coordinate ring and of its Koszul cycles.
\thechapter Preliminaries on elimination theory
- 1 Introduction
- 2 The image of a rational map as a scheme
- 3 Blow-up algebras
- 4 Rees and Symmetric algebras of a rational map
- 5 Approximation complexes
- 6 Acyclicity of approximation complexes
- 7 Implicitization
- \thechapter Preliminaries on toric varieties
\thechapter Implicit equations of Toric hypersurfaces in projective space by means of an embedding
- 12 Introduction.
- 13 General setting
- 14 The implicitization problem
- 15 The representation matrix for toric surfaces
- 16 The special case of biprojective surfaces
- 17 Examples
- 18 Final remarks
- \thechapter Implicit equations of toric hypersurfaces in multiprojective space
- \thechapter The algorithmic approach
- \thechapter G-graded Castelnuovo Mumford Regularity
- \thechapter Implicit equation of multigraded hypersurfaces
- \thechapter Algorithm1
- \thechapter Algorithm2
Chapter \thechapter Preliminaries on elimination theory
In this chapter we give a short summary of the articles written by Laurent Busé, Marc Chardin and Jean-Pierre Jouanolou on implicitization of projective hypersurfaces by means of approximation complexes [BJ03, BC05, Cha06, BCJ09]. There are many branches on mathematics and computer sciences where implicit equations of hypersurfaces are used and, hence, implicitization problems are involved. One of them is the interest in computer aided design (cf. [Hof89, GK03]).
In the beginning of the ’s, Hurgen Herzog, Aron Simis and Wolmer V. Vasconcelos developed the so called Approximation Complexes (cf. [HSV82, HSV83b, Vas94a]) for studying the syzygies of the conormal module (cf. [SV81]).
In elimination theory approximation complexes were used for the first time by Laurent Busé and Jean-Pierre Jouanolou in in order to propose a new alternative to the previous methods (see [BJ03]). This new tool generalized the work of Sederberg and Cheng, on “moving lines” and “moving surfaces” introduced a few years before in [SC95, CSC98, ZSCC03], giving also a theoretical framework.
The spirit behind the method based on approximation complexes consists in doing elimination theory by taking determinant of a graded strand of a complex. This idea is similar to the one used for the computation of a Macaulay resultant of homogeneous polynomials in variables, by means of taking determinant of a graded branch of a Koszul complex.
This resultant spans the annihilator of the quotient ring of by in big enough degree (bigger than its regularity). This annihilator can also be related to the MacRae invariant of the coordinate ring in the same degree . This theoretical method can become effective through the computation of the determinant of the degree--strand of the Koszul complex of (see [Nor76, Mac65, GKZ94, KM76]).
In this case, we wish to give a closed formula for the implicit equation of the image of a rational map , over a field . We will assume at first that this image defines a hypersurface in , and hence, is generically finite.
It is well known that a map between schemes gives rise to a map of rings that we will denote by . We will focus on computing the kernel of this map which is a principal prime ideal of the polynomial ring , and hence it describes the closed image of .
2 The image of a rational map as a scheme
We will describe henceforward in this chapter how to compute the implicit equation of the closed image of a rational map following the ideas of L. Busé, M. Chardin and J.-P. Jouanolou. Let be a commutative ring and a -graded -algebra. We will assume that , where the polynomials are homogeneous of the same degree for all . Let be a morphism of graded -algebras defined by
The map induces a morphism of -affine schemes
where is an open set of .
Also, given homogeneous of degree , is a graded morphism of graded algebras (where the grading is given by for all ). Hence, induces a morphism of -projective schemes
where is an open set of .
Denote by and , the sets of definition of and respectively, also and .
Before getting into the results, we give some notations.
We will denote by the polynomial ring , and let and be ideals of and an -module. Define
, the annihilator of ;
, the colon ideal of by ;
, the saturation of by , also written ;
, the -th local cohomology group of with support on .
Theorem 2.2 ([Bj03, Thm 2.1]).
Let and be the affine and projective sheafification of . We have that
and similarly with .
Lemma 2.3 ([Bj03, Rem 2.2]).
In particular, when , ; this means that is saturated with respect to in .
Recall that if and are ideals of , then , is defined as . We have that
This is due to the fact that and if , . Thus, that clearly belongs to . Hence, .
Recall that is the map induced by
Let be the open subscheme of definition of , and be the closed subscheme of where the sections vanish. We will blowup along .
We will denote by and the two natural projections,
The restriction of to coincides with .
Let be the Rees algebra of . Let be the map of -algebras defined by , in such a way that and , hence is of total degree .
Thus, there is a short exact sequence , where , namely, .
The following diagram is commutative
where , and corresponds to the restriction of to the open set .
One important difficulty is the deep understanding of the difference between and . We will give a short example to illustrate this relation.
Let be a commutative noetherian ring, and .
Invert and define . Let and hence we get . The element spans , defined as and . Since , and coincide, is not a zero divisor modulo in . We see that is a regular sequence in . Hence, the complex
is acyclic. Thus the first homology group of , , vanishes. Hence, if is a regular sequence, then the kernel of the map defined by and is spanned by . That is .
We conclude that if is spanned by a regular sequence (of length ), then the Rees algebra is isomorphic to the symmetric algebra , defined as
This can be generalized to a sequence of length . In the general case we get that the ideal of relations is spanned by the -minors of
We will deepen our understanding of the relationship between the symmetric algebra and the Rees algebra in the following section. We will also see that in the particular context of implicitization theory applied to rational maps defined over a projective scheme, this situation is never reached. Precisely, we cannot hope that the symmetric algebra and the Rees algebra coincide, we can at most ask when they coincide modulo their torsion at the maximal ideal .
3 Blow-up algebras
Henceforward let be an infinite integral domain with unity and let be a commutative -graded -algebra. Take an homogeneous ideal of , where is an homogeneous element of degree . We will write for the usual multiplication of elements of for , and . Denote times for , where . In this part we will study presentations for the algebras and gr, and the relation with the symmetric algebras and . All these algebras
are called blow-up algebras, because they are closely related to the blow-up of a ring along an ideal.
3.1 Rees algebras and symmetric algebras of an ideal
The first idea for giving equations to describe the Rees algebra , is by means of the linear syzygies of . Precisely, there is a presentation homogeneous ideal which represents the equations of , where is the module spanned by the syzygies of -products of .
Assume is of finite presentation , where is the module of syzygies of .
The map , induces a surjective morphism , defined in degree by . Denote . Then, there is a presentation for :
It can be shown that the ideal is generated by the linear form such that ,
Consider now the following presentation of the Rees algebra:
where the map is -linear and defined by . Clearly the ideal is an homogeneous ideal and its component of degree is , which is the -module of linear forms such that . Thus is spanned by .
Closely related to this presentation of there is one for the associated graded ring of , , coming from the -adic filtration in . Namely, since , there is an exact sequence
We describe in terms of a presentation of .
When is generated by a regular sequence , the Rees algebra coincides with the symmetric algebra, and the ideals and are spanned by the -minors of the matrix .
Let be a polynomial ring an ideal of , and take . Let be an ideal of . It is shown in [Vas94a] that
Let be homogeneous polynomials of the same degree that span . Consider . Then where and .
It is a well known fact that . Explicitly, .
The relation type of is the smallest integer such that . This number is independent of the generators chosen for (cf. [Vas94a]). When , we say that is of linear type.
Denote by , hence , and iff is of linear type, equivalently, is an isomorphism between and .
Let be a sequence of elements of a ring , let be an ideal of . We say that x is a:
regular sequence in , where is an -module, if:
for all , is not a zero divisor in .
x is a minimal system of generators of ;
for all and .
relative regular sequence if for all .
proper sequence if for all , where denote the -th module of Koszul homology associated to the sequence .
These conditions are related in the following way:
regular sequence d-sequence relative regular sequence proper sequence.
Every ideal generated by a -sequence is of linear type.
See [Vas94a]. ∎
4 Rees and Symmetric algebras of a rational map
Assume we have a rational map defined by homogeneous polynomials of degree . Let be a commutative ring and a -graded -algebra. Denote by the map that sends in . The map defines a morphism of -algebras , that maps . This map defines a morphism of affine schemes and a map of projective schemes .
We have mentioned that also defines a graded map of -algebras defined by , defining the Rees algebra as a quotient of a polynomial ring: . The ideal can be described as , using Proposition 3.1.
Consider the extended Rees algebra as a sub--algebra of . Denote , hence, is defined .
If , then .
It can be seen that the kernel of the map defined in (1) is given by
Writing with the inclusion map and by the composition, we have a description of
In [BJ03], the authors also proved that
If then . Moreover, if , and , then