JOURNAL OF COMMUTATIVE ALGEBRA

Volume 6, Number 2, Summer 2014

IMPLICITIZATION OF DE JONQUI

`

ERES

PARAMETRIZATIONS

SEYED HAMID HASSANZADEH AND ARON SIMIS

ABSTRACT. One introduces a class of projective param-

eterizations that resemble generalized de Jonqui`eres maps.

Any such parametrization deﬁnes a birational map F of P

n

onto a hypersurface V (F ) ⊂ P

n+1

with a strong handle to

implicitization. From this side, the theory developed here

extends recent work of Ben´ıtez and D’Andrea on monoid

parameterizations. The paper deals with both the ideal theo-

retic and eﬀective aspects of the problem. The ring theoretic

development gives information on the Castelnuovo-Mumford

regularity of the base ideal of F. From the eﬀective side, we

give an explicit formula of deg(F ) involving data from the

inverse map of F and show how the present parametrization

relates to monoid parameterizations.

1. Intro duction and notation. Let k denote an arbitrary inﬁnite

ﬁeld which will be assumed to be algebraically closed for the geometric

purpose. A rational map F : P

n

99K P

m

is deﬁned by m + 1 forms

f = {f

0

, . . . , f

m

} ⊂ R := k[x] = k[x

0

, . . . , x

n

] of the same degree d ≥ 1,

not all null. We often write F = (f

0

: · · · : f

m

) to underscore the

projective setup.

The image of F is the projective subvariety W ⊂ P

m

whose ho-

mogeneous coordinate ring is the k-subalgebra k[f] ⊂ R after degree

renormalization. Write S := k[f ] ≃ k[y]/I(W ), where I(W ) ⊂ k[y] =

k[y

0

, . . . , y

m

] is the homogeneous deﬁning ideal of the image in the em-

bedding W ⊂ P

m

.

We say that F is birational onto the image if there is a rational map

backwards P

m

99K P

n

such that the residue classes f

′

= {f

′

0

, . . . , f

′

n

} ⊂

S of its deﬁning coordinates do not simultaneously vanish and satisfy

2010 AMS Mathematics subject classiﬁcation. Primary 13A30, 13H15, 13D02,

13D45, 14E05, 14E07.

The ﬁrst author was partially supported by a Post-Doc Fellowship (CNPq,

Brazil). The second author was partially supported by a CNPq grant.

Received by the editors on May 11, 2012, and in revised form on November 14,

2012.

DOI:10.1216/JCA-2014-6-2-149 Copyright

c

2014 Rocky Mountain Mathematics Consortium

149

150 S.H. HASSANZADEH AND A. SIMIS

the relations

(f

′

0

(f) : · · · : f

′

n

(f)) = (x

0

: · · · : x

n

),(1)

(f

0

(f

′

) : · · · : f

m

(f

′

)) ≡ (y

0

: · · · : y

m

) mod I(W ).

Let K denote the ﬁeld of fractions of S = k[f]. Note that the

set of coordinates (f

′

0

: · · · : f

′

n

) deﬁning the “inverse” map is not

uniquely deﬁned; any other set (f

′′

0

: · · · : f

′′

n

) related to f

′

by

requiring that it deﬁnes the same element of the projective space

P

n

K

= P

n

k

×

Spec(k)

Spec (K) will do as well; both tuples are called

representatives of the rational map (see [16] for details). If k is

algebraically closed, these relations translate into the usual geometric

deﬁnition in terms of invertibility of the map on a dense Zariski open

set.

A sp ecial important case is that of a Cremona map, that is, a

birational map

G = (g

0

: · · · : g

n

) : P

n

99K P

n

of P

n

onto itself. We assume, as usual, that the coordinate forms have

no proper common factor. In this setting, the common degree d ≥ 1

of these forms is called the degree of G. Having information about the

inverse map, e.g., about its degree, will be quite relevant in the sequel.

Thus, for instance, the structural equality

(2) (g

0

(g

′

0

, . . . , g

′

n

) : · · · : g

n

(g

′

0

, . . . , g

′

n

)) = (y

0

: · · · : y

n

),

involving the inverse map gives a uniquely deﬁned form D ∈ R such

that g

i

(g

′

0

, . . . , g

′

n

) = y

i

D, for every i = 1, . . . , n. We call D ∈ k[y] the

target inversion factor of G. By symmetry, there is a source inversion

factor C ∈ k[x].

Our basic reference for the above is [16], which contains enough of

the introductory material in the form we use here (see also [6] for a

more general overview).

Now, the problem envisaged in this paper emerges from a particular

situation of rational maps, known as elimination. Namely, one takes

m = n + 1 and assumes that dim k[f] = dim R (= n + 1). Therefore,

W is a hypersurface deﬁned by an irreducible form F ∈ k[y] =

k[y

0

, . . . , y

n+1

]. We speak of F informally as the implicit equation of F.

Elimination theory in this formulation is the problem of determining

F or at least its properties, such as its degree. The set of the given

JONQUI

`

ERES PARAMETRIZATIONS 151

forms deﬁning F is called a parametrization of F . The theory has an

applicable side shown in a very active research area; we refer to some

of the related modern work on the subject in the bibliography.

Although the main interest classically focused on implicitization, i.e.,

in deriving the implicit equation F , more recently quite some literature

has appeared on the ideal theoretic structure of the parametrization and

the algebras naturally involved [1, 2, 4, 5, 10, 11]. In this regard, a

source of inspiration has been the classical Sylvester forms, a slightly

imprecise notion to refer to certain generators of the deﬁning ideal of

the Rees algebra associated to the base ideal of the rational map F (i.e.,

the ideal generated by the parameterizing forms).

Actually, we go even more special, by dealing with rational maps

which, in a sense, are allusive of the classical de Jonqui`eres plane

Cremona map. Namely, the class of parametrizations used here are

suggestive of the stellar Cremona maps by Pan [13], a bona ﬁde

generalization of the classical plane de Jonqui`eres maps, and inspired

by the results of Ben´ıtez and D’Andrea [1] on the so-called monoid

parametrizations.

Precisely, start with a Cremona map G = (g

0

: · · · : g

n

) : P

n

99K P

n

as explained above. Let f, g ∈ R be additional forms of arbitrary

degrees d ≥ 1 and d + d, respectively. We assume throughout that f

and g are relatively prime.

Deﬁnition 1.1. The rational map F = (g

0

f : · · · : g

n

f : g) : P

n

99K

P

n+1

will be called a de Jonqui`eres parametrization.

Note the easy, though important, fact that F is a birational map

onto its image W = V (F ). This follows immediately from the usual

ﬁeld extension criterion (see, e.g., [6, Proposition 1.11]. Moreover, if

the inverse of G is G

−1

= (g

′

0

: · · · : g

′

n

), with g

′

i

∈ k[y

0

, . . . , y

n

], then

(g

′

0

: · · · : g

′

n

) is a representative of the inverse F

−1

of F, where the bar

over an element of k[y] denotes its class modulo (F ); note that this

representative of F

−1

does not involve the last variable y

n+1

.

The Cremona map G may be called the underlying (or structural )

Cremona map of F.

The main results of the paper are stated in Theorem 2.6, Proposi-

tion 3.3, Proposition 4.2 and Theorem 4.5.

Let us brieﬂy describe the contents of the next sections.

152 S.H. HASSANZADEH AND A. SIMIS

Section 2 gives the main properties of the base ideal of the parametri-

zation, such as structure of syzygies, free resolution and regularity. Part

of the information of this section is crucial for introducing the concept

of syzygetic polynomials that arise as natural candidates for the implicit

equation (often with extraneous factors).

Section 3 deals with the implicit equation F . Here one introduces

the basic polynomials that play a role in the nature of F, such as

the syzygetic polynomials mentioned before. One heavily draws on

the hypothesis that the de Jonqui`eres parametrization is birational,

by having the deﬁning parametrization of the inverse map and the

inversion factor take control of the situation. This section also examines

the details of two main cases of the given de Jonqui`eres parametrization,

called the inclusion case and the non-zero-divisor case, respectively. It

is worth pointing out that the ﬁrst of these two cases covers as a very

special case the situation of a monoid parametrization.

In Section 4 one focuses on the so-called “Rees equations” of the

parametrization. These are the elements of a minimal set of generators

of a presentation ideal (the “Rees ideal”) of the Rees algebra of the base

ideal of F, one of which, of course, is F itself. These have been variously

studied by several authors, some listed in the references. The idea in

this section is based on the method of downgrading that has been used

in diﬀerent sources (e.g., [2], [9], [11]). Ours is a modiﬁcation of this

method, hereby called birational downgrading, by which we use the

forms deﬁning the inverse map rather than the usual procedures in the

literature. The main result yields a set of Rees equations candidates for

a set of minimal generators, generating an ideal having as a minimal

prime component the entire Rees ideal. The sections end with a result

giving the precise relation between the Rees ideal of de Jonqui`eres

parameterizations and the one of the monoid parameterizations.

2. Syzygetic background. In this section we establish the basic

relations of degree 1 of the forms g

0

f, . . . , g

n

f, g deﬁning the rational

map F of Deﬁnition 1.1. For the next lemma and proposition (g

0

: · · · :

g

n

) deﬁnes any rational map, not necessarily Cremona.

2.1. A mapping cone. In this part, we state a very general result

regarding a certain mapping cone naturally associated to the present

data. The construction is completely general and does not require a

JONQUI

`

ERES PARAMETRIZATIONS 153

graded situation. Accordingly, we refresh our data just assuming that

I ⊂ R := k[x

0

, . . . , x

n

] is an arbitrary ideal and f, g ∈ R are given

elements.

Lemma 2.1. If gcd(f, g) = 1, then:

(a) If : (g) = (I : (g))f.

(b) Multiplication by g induces an isomorphism R/(I : (g))f ≃

(If, g)/If of R-modules.

Proof. (a) The inclusion If : (g) ⊃ (I : (g))f is obvious regardless of

any relative assumption about f, g. Conversely, let b ∈ R be such that

bg ∈ If. Then f divides bg and, since gcd(f, g) = 1, then f divides

b. Say, b = af, with a ∈ R. Then (ag)f ∈ If; hence, ag ∈ I, i.e.,

a ∈ I : (g). Therefore, b ∈ (I : (g))f.

(b) One has (If, g)/If ≃ (g)/(g) ∩ If = (g)/(If : (g))g ≃ R/If :

(g), where the last isomorphism is multiplication by g

−1

. Now apply

(a).

Quite generally, a surjective R-module homomorphism π : R

q

I :

(g) induces a content map c(g) : R

q

→ R

n+1

. In explicit coordinates,

let π b e induced by choosing a set of generators {c

1

, . . . , c

q

} of I : (g),

so that π(v

j

) = c

j

, where {v

1

, . . . , v

q

} is the canonical basis of R

q

.

Given a set {g

0

, . . . , g

p

} of generators of I, let {e

0

, . . . , e

p

} denote the

canonical basis of R

p+1

. Write c

j

g =

p

i

=

0

h

ij

g

i

, with h

ij

∈ R. Then

c(g)(v

j

) =

p

i

=

0

h

ij

e

i

, for j = 1, . . . , q.

This simple construction will be used in the following result.

Lemma 2.2. Let R and S denote ﬁnite free resolutions of R/I and

R/(I : (g))f, respectively. Then multiplication by g lifts to a map

S → R whose associated mapping cone is a free resolution of R/(If, g).

In particular, a syzygy matrix of the generators of (If, g) has the form

Ψ =

φ c(g)

0 −fπ

,

where φ denotes a syzygy matrix of a given set of generators of I.

Proof. As in Lemma 2.1(b), multiplication by g induces an injective

R-module homomorphism R/(I : (g))f ↩→ R/If with image (If, g)/If .