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 .