On the Homology of Two-Dimensional
Elimination
Jooyoun Hong
Department de Mathematics
University of California, Riverside
900 Big Springs Drive
Riverside, CA 92521
e-mail: jyhong@math.ucr.edu
Aron Simis
Departamento de Matemática
Universidade Federal de Pernambuco
50740-540 Recife, PE, Brazil
e-mail: aron@dmat.ufpe.br
Wolmer V. Vasconcelos
Department of Mathematics
Rutgers University
110 Frelinghuysen Rd
Piscataway, New Jersey 08854-8019
e-mail: vasconce@math.rutgers.edu
September 9, 2021
Abstract
We study birational maps with empty base locus defined by almost complete intersection
ideals. Birationality is shown to be expressed by the equality of two Chern numbers.
We provide a relatively effective method of their calculation in terms of certain Hilbert
coefficients. In dimension two the structure of the irreducible ideals leads naturally to
the calculation of Sylvester determinants via a computer-assisted method. For degree
at most 5 we produce the full set of defining equations of the base ideal. The results
answer affirmatively some questions raised by D. Cox ([9]).
AMS 2000 Mathematics Subject Classification. Primary 13H10; Secondary 13F20, 05C90.
Partially supported by CNPq, Brazil.
Partially supported by the NSF.
http://arxiv.org/abs/0704.0608v1
1 Introduction
Let R be a Noetherian ring and f = {f1, . . . , fm} a set of elements of R. Such sets are
the ingredients of rational maps between affine and other spaces. At the cost of losing
some definition, we choose to examine them in the setting of the ideal I they generate.
Specifically, we consider the presentation of the Rees algebra of I
0 → M −→ S = R[T1, . . . , Tm]
ϕ−→ R[It] → 0, Ti 7→ fit.
The context of Rees algebra theory allows for the examination of the syzygies of the fi but
also of the relations of all orders, which are carriers of analytic information.
We set R = R[It] for the Rees algebra of I. The ideal M will be referred to as the
equations of the fj, or by abuse of terminology, of the ideal I. If M is generated by forms
of degree 1, I is said to be of linear type (this is independent of the set of generators). The
Rees algebra R[It] is then the symmetric algebra S = Sym(I) of I. Such is the case when
the fi form a regular sequence, M is then generated by the Koszul forms fiTj − fjTi, i < j.
We will treat mainly almost complete intersections in a Cohen-Macaulay ring R, that is,
ideals of codimension r generated by r + 1 elements. Almost exclusively, I will be an ideal
of finite co-length in a local ring, or in a ring of polynomials over a field.
Our focus on R is shaped by the following fact. The class of ideals I to be considered
will have the property that both its symmetric algebra S and the normalization R′ of R
have amenable properties, for instance, one of them (when not both) is Cohen-Macaulay.
In such case, the diagram
S ։ R ⊂ R′
gives a convenient dual platform from which to examine R.
There are specific motivations for looking at (and for) these equations. In order to
describe our results in some detail, let us indicate their contexts.
(i) Ideals which are almost complete intersections occur in some of the more notable
birational maps and in geometric modelling ([3], [4], [5], [6], [7], [8], [9], [10], [17], [18],
[21]).
(ii) It is possible interpret questions of birationality of certain maps as an interaction
between the Rees algebra of the ideal and its special fiber. The mediation is carried
by the first Chern coefficient of the associated graded ring of I. In the case of almost
complete intersections the analysis is more tractable, including the construction of
suitable algorithms.
(iii) At a recent talk in Luminy ([9]), D. Cox raised several questions about the character of
the equations of Rees algebras in polynomial rings in two variables. They are addressed
in Section 4 as part of a general program of devising algorithms that produce all the
equations of an ideal, or at least some distinguished polynomial (e.g. the ‘elimination
equation’ in it) ([3], [13]).
We now describe our results. Section 2 is an assemblage for the ideals treated here
of basics on symmetric and Rees algebras, and on their Cohen-Macaulayness. We also
introduce the general notion of a Sylvester form in terms of contents and coefficients in a
polynomial ring over a base ring. This is concretely taken up in Section 4 when the base
ring is a polynomial ring in 2 variables over a field.
In Section 3 we examine the connection between typical algebraic invariants and the
geometric background of rational maps and their images. Here, besides the dimension and
the degree of the related algebras, we also consider the Chern number e1(I) of an ideal. In
particular we explain a criterion for a rational map to be birational in terms of an equality
of two such Chern numbers, provided the base locus of the map is empty and defined by an
almost complete intersection ideal.
In Section 4, we discuss the role of irreducible ideals in producing Sylvester forms. Of
a general nature, we describe a method to obtain an irreducible decomposition of ideals of
finite co-length. In rings such as k[s, t], due to a theorem of Serre, irreducible ideals are
complete intersections, a fact that leads to Sylvester forms of low degree.
Turning to the equations of almost complete intersections, we derive several Sylvester
forms over a polynomial ring R = k[s, t], package them into ideals and examine the incident
homological properties of these ideals and the associated algebras. It is a computer-assisted
approach whose role is to produce a set of syzygies that afford hand computation: the
required equations themselves are not generated by computation. Concretely, we model
a generic class of ideals cases to define ‘super-generic’ ideals L in rings with several new
variables
L = (f, g, h1, . . . , hm) ⊂ A.
Using Macaulay2 ([11]), we obtain the free resolutions of L. In degrees ≤ 5, the resolution
has length ≤ 3 (2 when degree = 4)
0 → F3
d3−→ F2
d2−→ F1 −→ F0 −→ L → 0.
It has the property that after specialization the ideals of maximal minors of d3 and d2 have
codimension 5 and ≥ 4, respectively. Standard arguments of the theory of free resolutions
will suffice to show that the specialization of L is a prime ideal.
For ideals in R = k[s, t] generated by forms of degrees ≤ 5, the method succeeds in
describing the full set of equations. In higher degree, in cases of special interest, it predicts
the precise form of the elimination equation.
For a technical reason–due to the character of irreducible ideals–the method is limited
to dimension two. Nevertheless, it is supple enough to apply to non-homogeneous ideals.
This may be exploited elsewhere, along with the treatment of ideals with larger numbers of
generators in a two-dimensional ring.
2 Preliminaries on symmetric and Rees algebras
We will introduce some basic material of Rees algebras ([2], [12], [22]). Since most of
the questions we will consider have a local character, we pick local rings as our setting.
Whenever required, the transition to graded rings will be direct.
Throughout we will consider a Noetherian local ring (R,m) and I an m-primary ideal
(or a graded algebra over a field k, R =
n≥0Rn = R0[R1], R0 = k, and I a homogeneous
ideal of finite colength λ(R/I) < ∞).
We assume that I admits a minimal reduction J generated by n = dimR elements.
This is always possible when k is infinite. The terminology means that for some integer r,
Ir+1 = JIr. This condition in turn means that the inclusion of Rees algebras R[Jt] ⊂ R[It]
is an integral birational extension (birational in the sense that the two algebras have the
same total ring of fractions). The smallest such integer, rJ(I), is called the reduction
number of I relative to J ; the infimum of these numbers over all minimal reductions of I
is the (absolute) reduction number r(I) of I.
For any ideal, not necessarily m-primary, the special fiber of R[It] – or of I by abuse of
terminology – is the algebra F(I) = R[It]⊗R (R/m). The dimension of F(I) is called the
analytic spread of I, and denoted ℓ(I). When I is m-primary, ℓ(I) = dimR. A minimal
reduction J is generated by ℓ(I) elements, and F(J) is a Noether normalization of F(I).
Hilbert polynomials
The Hilbert polynomial of I by (m ≫ 0) is the function ([2]):
λ(R/Im) = e0(I)
m+ n− 1
− e1(I)
m+ n− 2
+ lower terms.
e0(I) is the multiplicity of the ideal I. If R is Cohen-Macaulay, e0(I) = λ(R/J), where J
is a minimal reduction of I (generated by a regular sequence). For such rings, e1(I) ≥ 0.
For instance, if R = k[x1, . . . , xn], m = (x1, . . . , xn) and I = m
λ(R/Im) = λ(R/mmd) =
md+ n− 1
m+ n− 1
− e1(I)
m+ n− 2
+ lower terms
where e1(I) =
(dn − dn−1).
Both coefficients will be the focus of our interest soon.
Cohen-Macaulay Rees algebras
There is broad array of criteria expressing the Cohen-Macaulayness of Rees algebra (see [1],
[14], [19], [23, Chapter 3]). Our needs will be filled by single criterion whose proof is fairly
straightforward. We briefly review its related contents.
Let (R,m) be a Cohen-Macaulay local ring of dimension ≥ 1, and let I be an m-primary
ideal with a minimal reduction J . The Rees algebra R[Jt] is Cohen-Macaulay and serves
as an anchor to derive many properties of R[It]. Here is one that we shall make use of.
Define the Sally module SJ(I) of I relative to J to be the cokernel of the natural inclusion
of finite R[Jt]-modules I R[Jt] ⊂ I R[It]. Thus,
SJ(I) =
It/IJ t−1.
It has a Hilbert function, unlike the algebra R[It], that gives information about the Hilbert
function of I (see [22, Chapter 2]). The module on the left, I ·R[Jt], is a Cohen-Macaulay
R[Jt]-module of depth dimR + 1. The Cohen-Macaulayness of I · R[It] is directly related
to that of R[It]. These considerations lead to the criterion:
Theorem 2.1 If dimR ≥ 2 and the reduction number of I is ≤ 1, that is I2 = JI, then
R[It] is Cohen-Macaulay. The converse holds if dimR = 2.
Symmetric algebras
Throughout R is a Cohen-Macaulay ring and I is an almost complete intersection. The
symmetric algebra Sym(I) will be denoted by S. Hopefully there will be no confusion
between S and the rings of polynomials S = R[T1, . . . , Tn] that we use to give a presentation
of either R or S.
What keeps symmetric algebras of almost complete intersections fairly under control is
the following:
Proposition 2.2 Let (R,m) be a Cohen-Macaulay local ring. If I is an almost complete
intersection and depth R/I ≥ dimR/I − 1, then S is Cohen-Macaulay. In particular, if I
is m-primary then S is Cohen-Macaulay.
Proof. The general assertion follows from [12, Proposition 10.3]; see also [16]. ✷
Let R be a Noetherian ring and let I be an R-ideal with a free presentation
ϕ−→ Rn −→ I → 0.
We assume that I has a regular element. If S = R[T1, . . . , Tn], the symmetric algebra S of
I is defined by the ideal M1 ⊂ S of 1-forms,
M1 = I1([T1, . . . , Tn] · ϕ).
The ideal of definition of the Rees algebra R of I is the ideal M ⊂ S obtained by elimination
(M1 : x
t) = M1 : x
where x is a regular element of I.
Sylvester forms
To get additional elements of M , evading the above calculation, we make use of general
Sylvester forms. Recall how these are obtained. Let f = {f1, . . . , fn} be a set of polynomials
in B = R[x1, . . . , xr] and let a = {a1, . . . , an} ⊂ R. If fi ∈ (a)B for all i, we can write
f = [f1 · · · fn] = [a1 · · · an] ·A = a ·A,
where A is a n×n matrix with entries in B. By an abuse of terminology, we refer to det(A)
as a Sylvester form of f relative to a, in notation
det(f)(a) = det(A).
It is not difficult to show that det(f)(a) is well-defined mod (f). The classical Sylvester forms
are defined relative to sets of monomials (see [9]). We will make use of them in Section 4.
The structure of the matrix A may give rise to finer constructions (lower order Pfaffians,
for example) in exceptional cases (see [20]).
In our approach, the fi are elements of M1, or were obtained in a previous calculation,
and the ideal (a) is derived from the matrix of syzygies ϕ.
3 Algebraic invariants in rational parametrizations
Let f1, . . . , fn+1 ∈ R = k[x1, . . . , xn] be forms of the same degree. They define a rational
Ψ : Pn−1 99K Pn
p → (f1(p) : f2(p) : · · · : fn+1(p)).
Rational maps are defined more generally with any number m of forms of the same degree,
but in this work we only deal with the case where m = n+ 1.
There are two basic ingredients to the algebraic side of rational map theory: the ideal
theoretic and the algebra aspects, both relevant for the nature of Ψ. First the ideal I =
(f1, . . . , fn+1) ⊂ R, which in this context is called the base ideal of the rational map. Then
there is the k-subalgebra k[f1, . . . , fn+1] ⊂ R, which is homogeneous, hence a standard k-
algebra up to degree renormalization. As such it gives the homogeneous coordinate ring of
the (closed) image of Ψ. Finding the irreducible defining equation of the image is known as
elimination or implicitization.
We refer to [21] and [18] (also [20] for an even earlier overview) for the interplay between
the ideal and the algebra, as well as its geometric consequences. In particular, the Rees
algebra R = R[It] plays a fundamental role in the theory. A pleasant side of it is that, since
I is generated by forms of the same degree, one has R⊗R k ≃ k[f1t, . . . , fn+1t] ⊂ R, which
retro-explains the (closed) image of Pn−1 by Ψ as the image of the projection to Pn of the
graph of Ψ. In particular, the fiber cone is reduced and irreducible.
3.1 Elimination degrees and birationality
Although a rational map Pn−1 99K Pn has a unique set of defining forms f1, . . . , fn+1 of the
same degree and unit gcd, two such maps may look “nearly” the same if they happen to
be composite with a birational map of the target Pn - a so-called Cremona transformation.
If this is the case the two maps have the same degree, in particular the final elimination
degrees are the same.
However, it may still be the case that the two maps are composite with a rational map
of the target which is not birational, so that their degrees as maps do not coincide, yet
the degrees of the respective images are the same. In such an event, one would like to
pick among all such maps one with smallest possible degree. This leads us to he notion of
improper and proper rational parametrizations.
Definition 3.1 Let Ψ = (f1 : · · · : fn+1) : Pn−1 99K Pn be a rational map, where
gcd(f1, . . . , fn+1) = 1. We will say that Ψ (or the parametrization defined by f1, . . . , fn+1)
is improper if there exists a rational map
Ψ′ = (f ′1 : · · · : f ′n+1) : Pn−1 99K Pn,
with gcd(f ′1, . . . , f
n+1) = 1, such that:
1. There is an inclusion of k-algebras k[f1, . . . , fn+1] ⊂ k[f ′1, . . . , f ′n+1];
2. There is an isomorphism of k-algebras k[f1, . . . , fn+1] ≃ k[f ′1, . . . , f ′n+1];
3. degΨ′ < degΨ.
We note that if Ψ is improper and Ψ′ is as above then the rational map
(P1 : · · · : Pn+1) : Pn 99K Pn
is not birational, where fj = Pj(f
1, . . . , f
n+1), for 1 ≤ j ≤ n + 1. Of course, the transition
forms Pj = Pj(y1, . . . , yn+1) are not uniquely defined.
Example 3.2 The parametrization given by f1 = x
1, f2 = x
2, f3 = x
2 is improper since
it factors through the parametrization f ′1 = x
2 = x1x2, f
3 = x
2 through either one of
the rational maps (y1 : y2 : y3) 7→ (y21 : y22 : y23) or (y1 : y2 : y3) 7→ (y21 : y1t3 : y23) neither of
which is birational. Moreover, the forms x21, x1x2, x
2 define a birational map onto its image.
We say that a rational map Ψ = (f1 : · · · : fn+1) : Pn−1 99K Pn is proper if it is
not improper. The need for considering proper rational maps will become apparent in the
context. It is also a basic assumption in elimination theory when one is looking for the
elimination degrees (see [9]).
Clearly, if Ψ is birational onto its image then it is proper. The converse does not hold
and one seeks for precise conditions under which Ψ is birational onto its image. This is the
object of the following parts of this subsection.
When the ideal I = (f1, . . . , fn+1) has finite co-length – that is, I is (x1, . . . , xn)-primary
– it is natural to consider another mapping, namely, the corresponding embedding of the
Rees algebra R = R[It] into its integral closure R̃. We will explore the attached Hilbert
functions into the determinations of various degrees, including the elimination degree of the
mapping.
Thus, assume that I has finite co-length. Then we may assume (k is infinite) that
f1, . . . , fn is a regular sequence, hence the multiplicity of J = (f1, . . . , fn) is d
n, the same
as the multiplicity of md. This implies that J is a minimal reduction of I and of md. We
will set up a comparison between R and R′ = R[md], where m = (x1, . . . , xn), through two
relevant exact sequences:
0 → R −→ R′ −→ D → 0, (1)
and its reduction mod m
R̄ −→ R̄′ −→ D̄ → 0. (2)
F = R̄ is the special fiber ofR (or, of I), and since I is generated by forms of the same degree,
one has F ≃ k[f1, . . . , fn+1] as graded k-algebras. By the same token, F ′ = R̄′ ≃ k[md] –
the d-th Veronese subring of R. In particular, since dimF = dimF ′, the leftmost map in the
exact sequence (2) is injective. AlsoD is annihilated by a power of m, hence dimD = dim D̄.
These are the degrees (multiplicities) deg(F) and deg(F ′) of the special fibers. Since F ′
is an integral extension of F , one has
deg(F ′) = deg(F)[F ′ : F ], (3)
where [F ′ : F ] = dimK(F ′ ⊗F K), where K denotes the fraction field of F (see, e.g.,
[21, Proposition 6.1 (b) and Theorem 6.6] for more general formulas). Since F ′ is besides
integrally closed, the latter is also the field extension degree [ k(md) : K ]. Note that
[F ′ : F ] = 1 means that the extension F ⊂ F ′ is birational (equivalently, the rational map
Ψ maps Pn−1 birationally onto its image). As above, set L = md. We next characterize
birationality in terms of both the coefficient e1 and the dimension of the R-module D.
Proposition 3.3 The following conditions are equivalent:
(i) [F ′ : F ] = 1, that is Ψ is birational onto its image;
(ii) deg(F) = dn−1;
(iii) dim D̄ ≤ n− 1;
(iv) dimD ≤ n− 1
(v) e1(L) = e1(I).
Proof. (i) ⇐⇒ (ii) This is clear from (3) since deg(F ′) = dn−1.
(i) ⇐⇒ (iii) Since ℓ(I) = n and F ⊂ F ′ is integral, then F ⊂ F ′ is a birational extension
if and only if its conductor F :F F ′ is nonzero, equivalently, if and only if dim D̄ ≤ n− 1.
(iv) ⇐⇒ (iii) Clearly, dimD ≤ n and in the case of equality its multiplicity is e1(L) −
e1(I) > 0. Therefore, the equivalence of the two statements follows suit. ✷
There is some advantage in examining D̄ since F is a hypersurface ring,
F = k[T1, . . . , Tn+1]/(f) = R[T1, . . . , Tn+1]/(x1, . . . , xn, f)
a complete intersection. Since F ′ is also Cohen-Macaulay, with a well-known presentation,
it affords an understanding of D̄, and sometimes, of D.
3.2 Calculation of e1(I) of the base ideal of a rational map
One objective here is to apply some general formulas for the Chern number e1(I) of an ideal
I to the case of the base ideal of a rational map with source P1 = Proj(k[x1, x2]).
Here is a method put together from scattered facts in the literature of Rees algebras
(see [23, Chapter 2]).
Proposition 3.4 Let (R,m) be a Cohen-Macaulay local ring of dimension d, let I be an
m-primary ideal with a minimal reduction J = (a1, . . . , ad). Set R
′ = R/(a1, . . . , ad−1),
I ′ = IR′. Then
(i) e0(I) = e0(I
′) = λ(R/J), e1(I) = e1(I
(ii) r(I ′) < degR′ ≤ e0(I); in particular, for n ≥ r = r(I ′), one has I ′n+1 = adI ′n
(iii) λ(R′/I ′ r+1) = λ(R′/I ′ r) + λ(I ′ r/adI
′ r) = e0(I)(r + 1)− e1(I)
(iv) e1(I) = −λ(R′/I ′ r) + e0(I)r
It would be desirable to develop a direct method suitable for the ideal I = (a, b, c)
generated by forms of R = k[s, t], of degree n. We may assume that a, b for a regular
sequence (i.e. gcd(a, b) = 1). We already know that e0(I) = n
2. For regular rings, one
knows ([15]) that e1(I) ≤ d−12 e0(I), d = dimR. Nevertheless the steps above already lead
to an efficient calculation for two reasons: the multiplicity e0(I) is known at the outset and
it does not really involve the powers of I. Forms of degree up to 10 are handled well by
Maucalay2 ([11]).
4 Sylvester forms in dimension two
We establish the basic notation to be used throughout. R = k[s, t] is a polynomial ring over
the infinite field k, and I ⊂ R = k[s, t] is a codimension 2 ideal generated by 3 forms of the
same degree n+ 1, with free graded resolution
0 −→ R(−n−1−µ)⊕R(2(−n−1)+µ) ϕ−→ R3(−n−1) −→ I −→ 0, ϕ =
α1 β1 γ1
α2 β2 γ2
Then the symmetric algebra of I is S ≃ R[T1, T2, T3]/(f, g) with
f = α1T1 + β1T2 + γ1T3
g = α2T1 + β2T2 + γ2T3.
Starting out from these 2 forms, the defining equations of S, following [9], we obtain by elim-
ination higher degrees forms in the defining ideal of R(I). It will make use of a computer-
assisted methodology to show that these algorithmically specified sets generate the ideal
of definition M of R(I) in several cases of interest–in particular answering some questions
raised [9]. More precisely, the so-called ideal of moving forms M is given when I is gen-
erated by forms of degree at most 5. In arbitrary degree, the algorithm will provide the
elimination equation in significant cases.
4.1 Basic Sylvester forms in dimension 2
Let R = k[s, t] and let F,G ∈ B = R[s, t, T1, T2, T3]. If F,G ∈ (u, v)B, for some ideal
(u, v) ⊂ R, the form derived from
h = ad− bc = det(F,G)(u,v),
will be called a basic Sylvester form.
To explain their naturalness, even for ideals I not necessarily generated by forms, we
give an approach to irreducible decomposition of certain ideals.
Theorem 4.1 Let (R,m) be a Gorenstein local ring and let I be an m–primary ideal. Let
J ⊂ I be an ideal generated by a system of parameters and let E = (J : I)/J be the canonical
module of R/I. If E = (e1, . . . , er), ei 6= 0, and Ii = ann (ei), then Ii is an irreducible ideal
The statement and its proof will apply to ideals of rings of polynomials over a field.
Proof. The module E is the injective envelope of R/I, and therefore it is a faithful R/I–
module (see [2, Section 3.2] for relevant notions). For each ei, Re1 is a nonzero submodule of
E whose socle is contained in the socle of E (which is isomorphic to R/m) and therefore its
annihilator Ii (as an R-ideal) is irreducible. Since the intersection of the Ii is the annihilator
of E, the asserted equality follows. ✷
Corollary 4.2 Let (R,m) be a regular local ring of dimension two and let I be an m–
primary ideal with a free resolution
0 → Rn−1 ϕ−→ Rn −→ I → 0,

an−1,1 · · · an−1,n−1
an,1 · · · an,n−1

and suppose that the last two maximal minors ∆n−1,∆n of ϕ form a regular sequence. If
e1, . . . , en−1 are as above, then
(∆n−1,∆n) : I = In−2(ξ
′) = (e1, . . . , en−1)
and each ideal (∆n−1,∆n) : ei is a complete intersection of codimension 2.
Proof. The assertion that the irreducible Ii is a complete intersection is a result of Serre,
valid for all two-dimensional regular rings whose projective modules are free. ✷
Remark 4.3 In our applications, I = C(f, g), the content ideal of f, g. In some of these
cases, C(f, g) = (s, t)n, for some n, an ideal which admits the irreducible decomposition
(s, t)n =
(si, tn+1−i).
One can then process f, g through all the pairs {si, tn−i+1}, and collect the determinants for
the next round of elimination. As in the classical Sylvester forms, the inclusion C(f, g) ⊂
(s, t)n may be used anyway to start the process, although without the measure of control
of degrees afforded by the equality of ideals.
4.2 Cohen-Macaulay algebras
We pointed out in Theorem 2.1 that the basic control of Cohen-Macaulayness of a Rees
algebra of an ideal I ⊂ k[s, t] is that its reduction number be at most 1. We next give a
mean of checking this property directly off a free presentation of I.
Theorem 4.4 Let I ⊂ R be an ideal of codimension 2, minimally generated by 3 forms of
the same degree. Let
α1 α2
β1 β2
γ1 γ2
be the Hilbert-Burch presentation matrix of I. Then R is Cohen-Macaulay if and only if
the equalities of ideals of R hold
(α1, β1, γ1) = (α2, β2, γ2) = (u, v),
where u, v are forms.
Proof. Consider the presentation
0 → L −→ S = R[T1, T2, T3]/(f, g) −→ R → 0,
where f, g are the 1-forms [
T1 T2 T3
If R is Cohen-Macaulay, the reduction number of I is 1 by Theorem 2.1, so there must
be a nonzero quadratic form h with coefficients in k in the presentation ideal M of R. In
addition to h, this ideal contains f, g, hence in order to produce such terms its Hilbert-Burch
matrix must be of the form 
p1 p2
q1 q2
where u, v are forms of k[s, t], and the other entries are 1-forms of k[T1, T2, T3]. Since p1, p2
are q1, q2 are pairs of linearly independent 1-forms, the assertion about the ideals defined
by the columns of ϕ follow.
4.3 Base ideals generated in degree 4
This is the case treated by D. Cox in his Luminy lecture ([9]). We accordingly change the
notation to R = k[s, t], I = (f1, f2, f3), forms of degree 4. The field k is infinite, and we
further assume that f1, f2 form a regular sequence so that J = (f1, f2) is a reduction of I
and of (s, t)4. Let
0 → R(−4− µ)⊕R(−8 + µ) ϕ−→ R3(−4) −→ R −→ R/I → 0, ϕ =
α1 α2
β1 β2
γ1 γ2
 (4)
be the Hilbert-Burch presentation of I. We obtain the equations of f1, f2, f3 from this
matrix.
Note that µ is the degree of the first column of ϕ, 4 − µ the other degree. Let us first
consider (as in [9]) the case µ = 2.
Balanced case
We shall now give a computer-assisted treatment of the balanced case, that is when the
resolution (4) of the ideal I has µ = 2 and the content ideal of the syzygies is (s, t)2. Since
k is infinite, it is easy to show that there is a change of variables, T1, T2, T3 → x, y, z, so
that (s2, st, t2) is a syzygy of I. The forms f, g that define the symmetric algebra of I can
then be written
[f g] = [s2 st t2]
where u, v, w are linear forms in x, y, z. Finally, we will assume that the ideal I2
x y z
u v w
has codimension two. Note that this is a generic condition.
We introduce now the equations of I.
• Linear equations f and g:
[f g] = [x y z] ϕ = [x y z]
α1 α2
β1 β2
γ1 γ2
= [s2 st t2]
where u, v, w are linear forms in x, y, z.
• Biforms h1 and h2:
Write Γ1 and Γ2 such that
[f g] = [x y z] ϕ = [ s t2 ] Γ1 = [ s
2 t ] Γ2.
Then h1 = detΓ1 and h2 = detΓ2.
• Implicit equation F = detΘ, where [h1 h2] = [s t] Θ.
Using generic entries for ϕ, in place of the true k-linear forms in old variables x, y, z, we
consider the ideal of k[s, t, x, y, z, u, v, w] defined by
f = s2x+ sty + t2z
g = s2u+ stv + t2w
h1 = −syu− tzu+ sxv + txw
h2 = −szu− tzv + sxw + tyw
F = −z2u2 + yzuv − xzv2 − y2uw + 2xzuw + xyvw − x2w2
Proposition 4.5 If I2
x y z
u v w
specializes to a codimension two ideal of k[x, y, z],
then L = (f, g, h1, h2, F ) ⊂ A = R[x, y, z, u, v, w] specializes to the defining ideal of R.
Proof. Macaulay2 ([11]) gives a resolution
0 → A d2−→ A5 −→ A5 −→ L → 0
where

zv − yw
zu− xw
−yu+ xv

The assumption on I2
x y z
u v w
says that the entries of d2 generate an ideal of
codimension four and thus implies that the specialization LS has projective dimension two
and that it is unmixed. Since LS 6⊂ (s, t)S, there is an element q ∈ (s, t)R that is regular
modulo S/LS. If
LS = Q1 ∩ · · · ∩Qr
is the primary decomposition of LS, the localization LSq has the corresponding decompo-
sition since q is not contained in any of the
Qi. But now Symq = Rq, so LSq = (f, g)u,
as Iq = Rq. ✷
Non-balanced case
We shall now give a similar computer-assisted treatment of the non-balanced case, that is
when the resolution (4) of the ideal I has µ = 3. This implies that the content ideal of the
syzygies is (s, t). Let us first indicate how the proposed algorithm would behave.
• Write the forms f, g as
f = as+ bt
g = cs+ dt,
where
x y z
u v w
• The next form is the Jacobian of f, g with respect to (s, t)
h1 = det(f, g)(s,t) = ad− bc = −bxs2 − byst− bzt2 + aus2 + avst+ awt2.
• The next two generators
h2 = det(f, h1)(s,t) = b
2xs+ b2yt− abzt− abus− abvt+ a2wt
and the elimination equation
h3 = det(f, h2)(s,t) = −b3x+ ab2y − a2bz + ab2u− a2bv + a3w.
Proposition 4.6 L = (f, g, h1, h2, h3) ⊂ A = k[s, t, x, y, z, u, v, w] specializes to the defin-
ing ideal of R.
Proof. Macaulay2 ([11]) gives the following resolution of L
0 → A2 ϕ−→ A6 ψ−→ A5 −→ L → 0,
x + abu −b
y + abz + abv − a
w −bsx− bty + asu + atv −btz + atw −s
x− sty − t
u− stv − t
t −s 0 0 0 0
a b t −s 0 0
0 0 a b t −s
0 0 0 0 a b
The ideal of 2 × 2 minors of ϕ has codimension 4, even after we specialize from A to
S in the natural manner. Since LS has projective dimension two, it will be unmixed. As
LS 6⊂ (s, t), there is an element u ∈ (s, t)R that is regular modulo S/LS. If
LS = Q1 ∩ · · · ∩Qr
is the primary decomposition of LS, the localization LSu has the corresponding decompo-
sition since u is not contained in any of the
Qi. But now Symu = Ru, so LSu = (f, g)u,
as Iu = Ru. ✷
4.4 Degree 5 and above
It may be worthwhile to extend this to arbitrary degree, that is assume that I is defined
by 3 forms of degree n+1 (for convenience in the notation to follow). We first consider the
case µ = 1. Using the procedure above, we would obtain the sequence of polynomials in
A = R[a, b, x1, . . . , xn, y1, . . . , yn]
• Write the forms f, g as
f = as+ bt
g = cs+ dt,
where
x1 · · · xn
y1 · · · yn

sn−2t
stn−2

• The next form is the Jacobian of f, g with respect to (s, t)
h1 = det(f, g)(s,t) = ad− bc
• Successively we would set
hi+1 = det(f, hi)(s,t), 1 < n.
• The polynomial
hn = det(f, hn−1)(s,t)
is the elimination equation.
Proposition 4.7 L = (f, g, h1, . . . , h5) ⊂ A specializes to the defining ideal of R.
In Macaulay2, we checked the degrees 5 and 6 cases. In both cases, the ideal L (which
has one more generator in degree 6) has a projective resolution of length 2 and the ideal of
maximal minors of the last map has codimension four.
Conjecture 4.8 For arbitrary n, L = (f, g, h1, . . . , hn) ⊂ A has projective dimension two
and specializes to the defining ideal of R.
In degree 5, the interesting case is when the Hilbert-Burch matrix φ has degrees 2 and
3. Let us describe the proposed generators. For simplicity, by a change of coordinates, we
assume that the coordinates of the degree 2 column of ϕ are s2, st, t2
f = s2x+ sty + t2z
g = (s3w1 + s
2tw2 + st
2w3 + t
3w4)x+ (s
3w5 + s
2tw6 + st
2w7 + t
3w8)y
+ (s3w9 + s
2tw10 + st
2w11 + t
3w12)z
Let [
x y z
sA sB + tC tD
 = φ
x ys+ zt
sA+ tB stC + t2D
xs+ yt z
s2A+ stB sC + tD
where A,B,C,D are k-linear forms in x, y, z.
h1 = det(B1)
= s2(−yA) + st(xC − yB − zA) + t2(xD − zB)
= s2(−yA) + t(xCs− yBs− zAs+ xDt− zBt)
= s(−yAs+ xCt− yBt− zAt) + t2(xD − zB),
h2 = det(B2)
= s2(xC − zA) + st(xD + yC − zB) + t2(yD)
= s2(xC − zA) + t(xDs+ yCs− zBs+ yDt)
= s(xCs− zAs + xDt+ yCt− zBt) + t2(yD).
x ys+ zt
−yA xCs− yBs− zAs + xDt− zBt
xs+ yt z
−yAs+ xCt− yBt− zAt xD − zB
x ys+ zt
xC − zA xDs+ yCs− zBs+ yDt
xs+ yt z
xCs− zAs+ xDt+ yCt− zBt yD
c1 = det(C1) = x
2(Cs+Dt) + xy(−Bs) + xz(−As−Bt) + yz(At) + y2(As)
c2 = det(C2) = x
2(Ds) + xy(Dt) + xz(−Bs−Ct) + yz(As) + z2(At)
c3 = det(C3) = x
2(Ds) + xy(Dt) + xz(−Bs−Ct) + yz(As) + z2(At)
c4 = det(C4) = xy(Ds) + xz(−Cs−Dt) + yz(−Ct) + z2(As +Bt) + y2(D)
x y z
−yA xC − yB − zA xD − zB
xC − zA xD + yC − zB yD
Then F = −x3D2+x2yCD+xy2(−BD)+x2z(2BD−C2)+xz2(2AC−B2)+xyz(BC−
3AD) + y2z(−AC)+ yz2(AB)+ y3(AD) + z3(−A2), an equation of degree 5. In particular,
the parametrization is birational.
Proposition 4.9 L = (f, g, h1, h2, c1, c2, c4, F ) specializes to the defining ideal of R.
Proof. Using Macaulay2, the ideal L has a resolution:
0 −→ S1 d3−→ S6 d2−→ S12 d1−→ S8 −→ L −→ 0.
d3 = [−z y x − t s 0]t
y z 0 0 0 0
x 0 z 0 0 0
−v 0 0 z 0 x2w4 − xzw7 + xyw8 + xzw12
u 0 0 0 z −xzw3 + xyw4 + z
2w6 − yzw7 + y
2w8 − xzw8 − z
2w11 + yzw12
0 x −y 0 0 0
0 −v 0 −y 0 xzw1 − x
2w3 + yzw5 + z
2w9 − xzw11
0 u 0 0 −y xzw2 − x
2w4 + z
2w10 − xzw12
0 0 u 0 −x xzw1 + yzw5 − xzw6 + x
2w8 + z
0 0 0 u v 0
0 0 v x 0 −xyw1 + x
2w2 − y
2w5 + xyw6 − x
2w7 − yzw9 + xzw10
0 0 0 0 0 −t
0 0 0 0 0 s
The ideals of maximal minors give codim I1(d3) = 5 and codim I5(d2) = 4 after special-
ization. As we have been arguing, this suffices to show that the specialization is a prime
ideal of codimension two. ✷
Elimination forms in higher degree
In degrees greater than 5, the methods above are not very suitable. However, in several
cases they are still supple enough to produce the elimination equation. We have already
seen this when one of the syzygies is of degree 1. Let us describe two other cases.
• Degree n = 2p, f and g both of degree p. We use the decomposition
(s, t)p =
(si, tp+1−i).
For each 1 ≤ i ≤ p, let
hi = det(f, g)(si,tp+1−i).
These are quadratic polynomials with coefficients in (s, t)p−1. We set
[h1, · · · , hp] = [sp−1, · · · , tp−1] ·A,
where A is a p × p matrix whose entries are 2-forms in k[x, y, z]. The Sylvester form of
degree n, F = det(A), is the required elimination equation.
• Degree n = 2p+ 1, f of degree p. We use the decomposition
(s, t)p =
(si, tp+1−i).
For each 1 ≤ i ≤ p, let
hi = det(f, g)(si,tp+1−i).
These are quadratic polynomials with coefficients in (s, t)p. We set
[f, h1, · · · , hp] = [sp, · · · , tp] ·B,
where A is a (p + 1) × (p + 1) matrix with one column whose entries are linear forms and
the remaining columns with entries 2-forms in k[x, y, z]. The Sylvester form F = det(B) is
the required elimination equation.
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2005.
	Introduction
	Preliminaries on symmetric and Rees algebras
	Algebraic invariants in rational parametrizations
	Elimination degrees and birationality
	Calculation of e1(I) of the base ideal of a rational map
	Sylvester forms in dimension two
	Basic Sylvester forms in dimension 2
	Cohen-Macaulay algebras
	Base ideals generated in degree 4
	Degree 5 and above

  We study birational maps with empty base locus defined by almost complete
intersection ideals. Birationality is shown to be expressed by the equality of
two Chern numbers. We provide a relatively effective method of their
calculation in terms of certain Hilbert coefficients. In dimension two the
structure of the irreducible ideals leads naturally to the calculation of
Sylvester determinants via a computer-assisted method. For degree at most 5 we
produce the full set of defining equations of the base ideal. The results
answer affirmatively some questions raised by D. Cox.

Introduction
Let R be a Noetherian ring and f = {f1, . . . , fm} a set of elements of R. Such sets are
the ingredients of rational maps between affine and other spaces. At the cost of losing
some definition, we choose to examine them in the setting of the ideal I they generate.
Specifically, we consider the presentation of the Rees algebra of I
0 → M −→ S = R[T1, . . . , Tm]
ϕ−→ R[It] → 0, Ti 7→ fit.
The context of Rees algebra theory allows for the examination of the syzygies of the fi but
also of the relations of all orders, which are carriers of analytic information.
We set R = R[It] for the Rees algebra of I. The ideal M will be referred to as the
equations of the fj, or by abuse of terminology, of the ideal I. If M is generated by forms
of degree 1, I is said to be of linear type (this is independent of the set of generators). The
Rees algebra R[It] is then the symmetric algebra S = Sym(I) of I. Such is the case when
the fi form a regular sequence, M is then generated by the Koszul forms fiTj − fjTi, i < j.
We will treat mainly almost complete intersections in a Cohen-Macaulay ring R, that is,
ideals of codimension r generated by r + 1 elements. Almost exclusively, I will be an ideal
of finite co-length in a local ring, or in a ring of polynomials over a field.
Our focus on R is shaped by the following fact. The class of ideals I to be considered
will have the property that both its symmetric algebra S and the normalization R′ of R
have amenable properties, for instance, one of them (when not both) is Cohen-Macaulay.
In such case, the diagram
S ։ R ⊂ R′
gives a convenient dual platform from which to examine R.
There are specific motivations for looking at (and for) these equations. In order to
describe our results in some detail, let us indicate their contexts.
(i) Ideals which are almost complete intersections occur in some of the more notable
birational maps and in geometric modelling ([3], [4], [5], [6], [7], [8], [9], [10], [17], [18],
[21]).
(ii) It is possible interpret questions of birationality of certain maps as an interaction
between the Rees algebra of the ideal and its special fiber. The mediation is carried
by the first Chern coefficient of the associated graded ring of I. In the case of almost
complete intersections the analysis is more tractable, including the construction of
suitable algorithms.
(iii) At a recent talk in Luminy ([9]), D. Cox raised several questions about the character of
the equations of Rees algebras in polynomial rings in two variables. They are addressed
in Section 4 as part of a general program of devising algorithms that produce all the
equations of an ideal, or at least some distinguished polynomial (e.g. the ‘elimination
equation’ in it) ([3], [13]).
We now describe our results. Section 2 is an assemblage for the ideals treated here
of basics on symmetric and Rees algebras, and on their Cohen-Macaulayness. We also
introduce the general notion of a Sylvester form in terms of contents and coefficients in a
polynomial ring over a base ring. This is concretely taken up in Section 4 when the base
ring is a polynomial ring in 2 variables over a field.
In Section 3 we examine the connection between typical algebraic invariants and the
geometric background of rational maps and their images. Here, besides the dimension and
the degree of the related algebras, we also consider the Chern number e1(I) of an ideal. In
particular we explain a criterion for a rational map to be birational in terms of an equality
of two such Chern numbers, provided the base locus of the map is empty and defined by an
almost complete intersection ideal.
In Section 4, we discuss the role of irreducible ideals in producing Sylvester forms. Of
a general nature, we describe a method to obtain an irreducible decomposition of ideals of
finite co-length. In rings such as k[s, t], due to a theorem of Serre, irreducible ideals are
complete intersections, a fact that leads to Sylvester forms of low degree.
Turning to the equations of almost complete intersections, we derive several Sylvester
forms over a polynomial ring R = k[s, t], package them into ideals and examine the incident
homological properties of these ideals and the associated algebras. It is a computer-assisted
approach whose role is to produce a set of syzygies that afford hand computation: the
required equations themselves are not generated by computation. Concretely, we model
a generic class of ideals cases to define ‘super-generic’ ideals L in rings with several new
variables
L = (f, g, h1, . . . , hm) ⊂ A.
Using Macaulay2 ([11]), we obtain the free resolutions of L. In degrees ≤ 5, the resolution
has length ≤ 3 (2 when degree = 4)
0 → F3
d3−→ F2
d2−→ F1 −→ F0 −→ L → 0.
It has the property that after specialization the ideals of maximal minors of d3 and d2 have
codimension 5 and ≥ 4, respectively. Standard arguments of the theory of free resolutions
will suffice to show that the specialization of L is a prime ideal.
For ideals in R = k[s, t] generated by forms of degrees ≤ 5, the method succeeds in
describing the full set of equations. In higher degree, in cases of special interest, it predicts
the precise form of the elimination equation.
For a technical reason–due to the character of irreducible ideals–the method is limited
to dimension two. Nevertheless, it is supple enough to apply to non-homogeneous ideals.
This may be exploited elsewhere, along with the treatment of ideals with larger numbers of
generators in a two-dimensional ring.
2 Preliminaries on symmetric and Rees algebras
We will introduce some basic material of Rees algebras ([2], [12], [22]). Since most of
the questions we will consider have a local character, we pick local rings as our setting.
Whenever required, the transition to graded rings will be direct.
Throughout we will consider a Noetherian local ring (R,m) and I an m-primary ideal
(or a graded algebra over a field k, R =
n≥0Rn = R0[R1], R0 = k, and I a homogeneous
ideal of finite colength λ(R/I) < ∞).
We assume that I admits a minimal reduction J generated by n = dimR elements.
This is always possible when k is infinite. The terminology means that for some integer r,
Ir+1 = JIr. This condition in turn means that the inclusion of Rees algebras R[Jt] ⊂ R[It]
is an integral birational extension (birational in the sense that the two algebras have the
same total ring of fractions). The smallest such integer, rJ(I), is called the reduction
number of I relative to J ; the infimum of these numbers over all minimal reductions of I
is the (absolute) reduction number r(I) of I.
For any ideal, not necessarily m-primary, the special fiber of R[It] – or of I by abuse of
terminology – is the algebra F(I) = R[It]⊗R (R/m). The dimension of F(I) is called the
analytic spread of I, and denoted ℓ(I). When I is m-primary, ℓ(I) = dimR. A minimal
reduction J is generated by ℓ(I) elements, and F(J) is a Noether normalization of F(I).
Hilbert polynomials
The Hilbert polynomial of I by (m ≫ 0) is the function ([2]):
λ(R/Im) = e0(I)
m+ n− 1
− e1(I)
m+ n− 2
+ lower terms.
e0(I) is the multiplicity of the ideal I. If R is Cohen-Macaulay, e0(I) = λ(R/J), where J
is a minimal reduction of I (generated by a regular sequence). For such rings, e1(I) ≥ 0.
For instance, if R = k[x1, . . . , xn], m = (x1, . . . , xn) and I = m
λ(R/Im) = λ(R/mmd) =
md+ n− 1
m+ n− 1
− e1(I)
m+ n− 2
+ lower terms
where e1(I) =
(dn − dn−1).
Both coefficients will be the focus of our interest soon.
Cohen-Macaulay Rees algebras
There is broad array of criteria expressing the Cohen-Macaulayness of Rees algebra (see [1],
[14], [19], [23, Chapter 3]). Our needs will be filled by single criterion whose proof is fairly
straightforward. We briefly review its related contents.
Let (R,m) be a Cohen-Macaulay local ring of dimension ≥ 1, and let I be an m-primary
ideal with a minimal reduction J . The Rees algebra R[Jt] is Cohen-Macaulay and serves
as an anchor to derive many properties of R[It]. Here is one that we shall make use of.
Define the Sally module SJ(I) of I relative to J to be the cokernel of the natural inclusion
of finite R[Jt]-modules I R[Jt] ⊂ I R[It]. Thus,
SJ(I) =
It/IJ t−1.
It has a Hilbert function, unlike the algebra R[It], that gives information about the Hilbert
function of I (see [22, Chapter 2]). The module on the left, I ·R[Jt], is a Cohen-Macaulay
R[Jt]-module of depth dimR + 1. The Cohen-Macaulayness of I · R[It] is directly related
to that of R[It]. These considerations lead to the criterion:
Theorem 2.1 If dimR ≥ 2 and the reduction number of I is ≤ 1, that is I2 = JI, then
R[It] is Cohen-Macaulay. The converse holds if dimR = 2.
Symmetric algebras
Throughout R is a Cohen-Macaulay ring and I is an almost complete intersection. The
symmetric algebra Sym(I) will be denoted by S. Hopefully there will be no confusion
between S and the rings of polynomials S = R[T1, . . . , Tn] that we use to give a presentation
of either R or S.
What keeps symmetric algebras of almost complete intersections fairly under control is
the following:
Proposition 2.2 Let (R,m) be a Cohen-Macaulay local ring. If I is an almost complete
intersection and depth R/I ≥ dimR/I − 1, then S is Cohen-Macaulay. In particular, if I
is m-primary then S is Cohen-Macaulay.
Proof. The general assertion follows from [12, Proposition 10.3]; see also [16]. ✷
Let R be a Noetherian ring and let I be an R-ideal with a free presentation
ϕ−→ Rn −→ I → 0.
We assume that I has a regular element. If S = R[T1, . . . , Tn], the symmetric algebra S of
I is defined by the ideal M1 ⊂ S of 1-forms,
M1 = I1([T1, . . . , Tn] · ϕ).
The ideal of definition of the Rees algebra R of I is the ideal M ⊂ S obtained by elimination
(M1 : x
t) = M1 : x
where x is a regular element of I.
Sylvester forms
To get additional elements of M , evading the above calculation, we make use of general
Sylvester forms. Recall how these are obtained. Let f = {f1, . . . , fn} be a set of polynomials
in B = R[x1, . . . , xr] and let a = {a1, . . . , an} ⊂ R. If fi ∈ (a)B for all i, we can write
f = [f1 · · · fn] = [a1 · · · an] ·A = a ·A,
where A is a n×n matrix with entries in B. By an abuse of terminology, we refer to det(A)
as a Sylvester form of f relative to a, in notation
det(f)(a) = det(A).
It is not difficult to show that det(f)(a) is well-defined mod (f). The classical Sylvester forms
are defined relative to sets of monomials (see [9]). We will make use of them in Section 4.
The structure of the matrix A may give rise to finer constructions (lower order Pfaffians,
for example) in exceptional cases (see [20]).
In our approach, the fi are elements of M1, or were obtained in a previous calculation,
and the ideal (a) is derived from the matrix of syzygies ϕ.
3 Algebraic invariants in rational parametrizations
Let f1, . . . , fn+1 ∈ R = k[x1, . . . , xn] be forms of the same degree. They define a rational
Ψ : Pn−1 99K Pn
p → (f1(p) : f2(p) : · · · : fn+1(p)).
Rational maps are defined more generally with any number m of forms of the same degree,
but in this work we only deal with the case where m = n+ 1.
There are two basic ingredients to the algebraic side of rational map theory: the ideal
theoretic and the algebra aspects, both relevant for the nature of Ψ. First the ideal I =
(f1, . . . , fn+1) ⊂ R, which in this context is called the base ideal of the rational map. Then
there is the k-subalgebra k[f1, . . . , fn+1] ⊂ R, which is homogeneous, hence a standard k-
algebra up to degree renormalization. As such it gives the homogeneous coordinate ring of
the (closed) image of Ψ. Finding the irreducible defining equation of the image is known as
elimination or implicitization.
We refer to [21] and [18] (also [20] for an even earlier overview) for the interplay between
the ideal and the algebra, as well as its geometric consequences. In particular, the Rees
algebra R = R[It] plays a fundamental role in the theory. A pleasant side of it is that, since
I is generated by forms of the same degree, one has R⊗R k ≃ k[f1t, . . . , fn+1t] ⊂ R, which
retro-explains the (closed) image of Pn−1 by Ψ as the image of the projection to Pn of the
graph of Ψ. In particular, the fiber cone is reduced and irreducible.
3.1 Elimination degrees and birationality
Although a rational map Pn−1 99K Pn has a unique set of defining forms f1, . . . , fn+1 of the
same degree and unit gcd, two such maps may look “nearly” the same if they happen to
be composite with a birational map of the target Pn - a so-called Cremona transformation.
If this is the case the two maps have the same degree, in particular the final elimination
degrees are the same.
However, it may still be the case that the two maps are composite with a rational map
of the target which is not birational, so that their degrees as maps do not coincide, yet
the degrees of the respective images are the same. In such an event, one would like to
pick among all such maps one with smallest possible degree. This leads us to he notion of
improper and proper rational parametrizations.
Definition 3.1 Let Ψ = (f1 : · · · : fn+1) : Pn−1 99K Pn be a rational map, where
gcd(f1, . . . , fn+1) = 1. We will say that Ψ (or the parametrization defined by f1, . . . , fn+1)
is improper if there exists a rational map
Ψ′ = (f ′1 : · · · : f ′n+1) : Pn−1 99K Pn,
with gcd(f ′1, . . . , f
n+1) = 1, such that:
1. There is an inclusion of k-algebras k[f1, . . . , fn+1] ⊂ k[f ′1, . . . , f ′n+1];
2. There is an isomorphism of k-algebras k[f1, . . . , fn+1] ≃ k[f ′1, . . . , f ′n+1];
3. degΨ′ < degΨ.
We note that if Ψ is improper and Ψ′ is as above then the rational map
(P1 : · · · : Pn+1) : Pn 99K Pn
is not birational, where fj = Pj(f
1, . . . , f
n+1), for 1 ≤ j ≤ n + 1. Of course, the transition
forms Pj = Pj(y1, . . . , yn+1) are not uniquely defined.
Example 3.2 The parametrization given by f1 = x
1, f2 = x
2, f3 = x
2 is improper since
it factors through the parametrization f ′1 = x
2 = x1x2, f
3 = x
2 through either one of
the rational maps (y1 : y2 : y3) 7→ (y21 : y22 : y23) or (y1 : y2 : y3) 7→ (y21 : y1t3 : y23) neither of
which is birational. Moreover, the forms x21, x1x2, x
2 define a birational map onto its image.
We say that a rational map Ψ = (f1 : · · · : fn+1) : Pn−1 99K Pn is proper if it is
not improper. The need for considering proper rational maps will become apparent in the
context. It is also a basic assumption in elimination theory when one is looking for the
elimination degrees (see [9]).
Clearly, if Ψ is birational onto its image then it is proper. The converse does not hold
and one seeks for precise conditions under which Ψ is birational onto its image. This is the
object of the following parts of this subsection.
When the ideal I = (f1, . . . , fn+1) has finite co-length – that is, I is (x1, . . . , xn)-primary
– it is natural to consider another mapping, namely, the corresponding embedding of the
Rees algebra R = R[It] into its integral closure R̃. We will explore the attached Hilbert
functions into the determinations of various degrees, including the elimination degree of the
mapping.
Thus, assume that I has finite co-length. Then we may assume (k is infinite) that
f1, . . . , fn is a regular sequence, hence the multiplicity of J = (f1, . . . , fn) is d
n, the same
as the multiplicity of md. This implies that J is a minimal reduction of I and of md. We
will set up a comparison between R and R′ = R[md], where m = (x1, . . . , xn), through two
relevant exact sequences:
0 → R −→ R′ −→ D → 0, (1)
and its reduction mod m
R̄ −→ R̄′ −→ D̄ → 0. (2)
F = R̄ is the special fiber ofR (or, of I), and since I is generated by forms of the same degree,
one has F ≃ k[f1, . . . , fn+1] as graded k-algebras. By the same token, F ′ = R̄′ ≃ k[md] –
the d-th Veronese subring of R. In particular, since dimF = dimF ′, the leftmost map in the
exact sequence (2) is injective. AlsoD is annihilated by a power of m, hence dimD = dim D̄.
These are the degrees (multiplicities) deg(F) and deg(F ′) of the special fibers. Since F ′
is an integral extension of F , one has
deg(F ′) = deg(F)[F ′ : F ], (3)
where [F ′ : F ] = dimK(F ′ ⊗F K), where K denotes the fraction field of F (see, e.g.,
[21, Proposition 6.1 (b) and Theorem 6.6] for more general formulas). Since F ′ is besides
integrally closed, the latter is also the field extension degree [ k(md) : K ]. Note that
[F ′ : F ] = 1 means that the extension F ⊂ F ′ is birational (equivalently, the rational map
Ψ maps Pn−1 birationally onto its image). As above, set L = md. We next characterize
birationality in terms of both the coefficient e1 and the dimension of the R-module D.
Proposition 3.3 The following conditions are equivalent:
(i) [F ′ : F ] = 1, that is Ψ is birational onto its image;
(ii) deg(F) = dn−1;
(iii) dim D̄ ≤ n− 1;
(iv) dimD ≤ n− 1
(v) e1(L) = e1(I).
Proof. (i) ⇐⇒ (ii) This is clear from (3) since deg(F ′) = dn−1.
(i) ⇐⇒ (iii) Since ℓ(I) = n and F ⊂ F ′ is integral, then F ⊂ F ′ is a birational extension
if and only if its conductor F :F F ′ is nonzero, equivalently, if and only if dim D̄ ≤ n− 1.
(iv) ⇐⇒ (iii) Clearly, dimD ≤ n and in the case of equality its multiplicity is e1(L) −
e1(I) > 0. Therefore, the equivalence of the two statements follows suit. ✷
There is some advantage in examining D̄ since F is a hypersurface ring,
F = k[T1, . . . , Tn+1]/(f) = R[T1, . . . , Tn+1]/(x1, . . . , xn, f)
a complete intersection. Since F ′ is also Cohen-Macaulay, with a well-known presentation,
it affords an understanding of D̄, and sometimes, of D.
3.2 Calculation of e1(I) of the base ideal of a rational map
One objective here is to apply some general formulas for the Chern number e1(I) of an ideal
I to the case of the base ideal of a rational map with source P1 = Proj(k[x1, x2]).
Here is a method put together from scattered facts in the literature of Rees algebras
(see [23, Chapter 2]).
Proposition 3.4 Let (R,m) be a Cohen-Macaulay local ring of dimension d, let I be an
m-primary ideal with a minimal reduction J = (a1, . . . , ad). Set R
′ = R/(a1, . . . , ad−1),
I ′ = IR′. Then
(i) e0(I) = e0(I
′) = λ(R/J), e1(I) = e1(I
(ii) r(I ′) < degR′ ≤ e0(I); in particular, for n ≥ r = r(I ′), one has I ′n+1 = adI ′n
(iii) λ(R′/I ′ r+1) = λ(R′/I ′ r) + λ(I ′ r/adI
′ r) = e0(I)(r + 1)− e1(I)
(iv) e1(I) = −λ(R′/I ′ r) + e0(I)r
It would be desirable to develop a direct method suitable for the ideal I = (a, b, c)
generated by forms of R = k[s, t], of degree n. We may assume that a, b for a regular
sequence (i.e. gcd(a, b) = 1). We already know that e0(I) = n
2. For regular rings, one
knows ([15]) that e1(I) ≤ d−12 e0(I), d = dimR. Nevertheless the steps above already lead
to an efficient calculation for two reasons: the multiplicity e0(I) is known at the outset and
it does not really involve the powers of I. Forms of degree up to 10 are handled well by
Maucalay2 ([11]).
4 Sylvester forms in dimension two
We establish the basic notation to be used throughout. R = k[s, t] is a polynomial ring over
the infinite field k, and I ⊂ R = k[s, t] is a codimension 2 ideal generated by 3 forms of the
same degree n+ 1, with free graded resolution
0 −→ R(−n−1−µ)⊕R(2(−n−1)+µ) ϕ−→ R3(−n−1) −→ I −→ 0, ϕ =
α1 β1 γ1
α2 β2 γ2
Then the symmetric algebra of I is S ≃ R[T1, T2, T3]/(f, g) with
f = α1T1 + β1T2 + γ1T3
g = α2T1 + β2T2 + γ2T3.
Starting out from these 2 forms, the defining equations of S, following [9], we obtain by elim-
ination higher degrees forms in the defining ideal of R(I). It will make use of a computer-
assisted methodology to show that these algorithmically specified sets generate the ideal
of definition M of R(I) in several cases of interest–in particular answering some questions
raised [9]. More precisely, the so-called ideal of moving forms M is given when I is gen-
erated by forms of degree at most 5. In arbitrary degree, the algorithm will provide the
elimination equation in significant cases.
4.1 Basic Sylvester forms in dimension 2
Let R = k[s, t] and let F,G ∈ B = R[s, t, T1, T2, T3]. If F,G ∈ (u, v)B, for some ideal
(u, v) ⊂ R, the form derived from
h = ad− bc = det(F,G)(u,v),
will be called a basic Sylvester form.
To explain their naturalness, even for ideals I not necessarily generated by forms, we
give an approach to irreducible decomposition of certain ideals.
Theorem 4.1 Let (R,m) be a Gorenstein local ring and let I be an m–primary ideal. Let
J ⊂ I be an ideal generated by a system of parameters and let E = (J : I)/J be the canonical
module of R/I. If E = (e1, . . . , er), ei 6= 0, and Ii = ann (ei), then Ii is an irreducible ideal
The statement and its proof will apply to ideals of rings of polynomials over a field.
Proof. The module E is the injective envelope of R/I, and therefore it is a faithful R/I–
module (see [2, Section 3.2] for relevant notions). For each ei, Re1 is a nonzero submodule of
E whose socle is contained in the socle of E (which is isomorphic to R/m) and therefore its
annihilator Ii (as an R-ideal) is irreducible. Since the intersection of the Ii is the annihilator
of E, the asserted equality follows. ✷
Corollary 4.2 Let (R,m) be a regular local ring of dimension two and let I be an m–
primary ideal with a free resolution
0 → Rn−1 ϕ−→ Rn −→ I → 0,

an−1,1 · · · an−1,n−1
an,1 · · · an,n−1

and suppose that the last two maximal minors ∆n−1,∆n of ϕ form a regular sequence. If
e1, . . . , en−1 are as above, then
(∆n−1,∆n) : I = In−2(ξ
′) = (e1, . . . , en−1)
and each ideal (∆n−1,∆n) : ei is a complete intersection of codimension 2.
Proof. The assertion that the irreducible Ii is a complete intersection is a result of Serre,
valid for all two-dimensional regular rings whose projective modules are free. ✷
Remark 4.3 In our applications, I = C(f, g), the content ideal of f, g. In some of these
cases, C(f, g) = (s, t)n, for some n, an ideal which admits the irreducible decomposition
(s, t)n =
(si, tn+1−i).
One can then process f, g through all the pairs {si, tn−i+1}, and collect the determinants for
the next round of elimination. As in the classical Sylvester forms, the inclusion C(f, g) ⊂
(s, t)n may be used anyway to start the process, although without the measure of control
of degrees afforded by the equality of ideals.
4.2 Cohen-Macaulay algebras
We pointed out in Theorem 2.1 that the basic control of Cohen-Macaulayness of a Rees
algebra of an ideal I ⊂ k[s, t] is that its reduction number be at most 1. We next give a
mean of checking this property directly off a free presentation of I.
Theorem 4.4 Let I ⊂ R be an ideal of codimension 2, minimally generated by 3 forms of
the same degree. Let
α1 α2
β1 β2
γ1 γ2
be the Hilbert-Burch presentation matrix of I. Then R is Cohen-Macaulay if and only if
the equalities of ideals of R hold
(α1, β1, γ1) = (α2, β2, γ2) = (u, v),
where u, v are forms.
Proof. Consider the presentation
0 → L −→ S = R[T1, T2, T3]/(f, g) −→ R → 0,
where f, g are the 1-forms [
T1 T2 T3
If R is Cohen-Macaulay, the reduction number of I is 1 by Theorem 2.1, so there must
be a nonzero quadratic form h with coefficients in k in the presentation ideal M of R. In
addition to h, this ideal contains f, g, hence in order to produce such terms its Hilbert-Burch
matrix must be of the form 
p1 p2
q1 q2
where u, v are forms of k[s, t], and the other entries are 1-forms of k[T1, T2, T3]. Since p1, p2
are q1, q2 are pairs of linearly independent 1-forms, the assertion about the ideals defined
by the columns of ϕ follow.
4.3 Base ideals generated in degree 4
This is the case treated by D. Cox in his Luminy lecture ([9]). We accordingly change the
notation to R = k[s, t], I = (f1, f2, f3), forms of degree 4. The field k is infinite, and we
further assume that f1, f2 form a regular sequence so that J = (f1, f2) is a reduction of I
and of (s, t)4. Let
0 → R(−4− µ)⊕R(−8 + µ) ϕ−→ R3(−4) −→ R −→ R/I → 0, ϕ =
α1 α2
β1 β2
γ1 γ2
 (4)
be the Hilbert-Burch presentation of I. We obtain the equations of f1, f2, f3 from this
matrix.
Note that µ is the degree of the first column of ϕ, 4 − µ the other degree. Let us first
consider (as in [9]) the case µ = 2.
Balanced case
We shall now give a computer-assisted treatment of the balanced case, that is when the
resolution (4) of the ideal I has µ = 2 and the content ideal of the syzygies is (s, t)2. Since
k is infinite, it is easy to show that there is a change of variables, T1, T2, T3 → x, y, z, so
that (s2, st, t2) is a syzygy of I. The forms f, g that define the symmetric algebra of I can
then be written
[f g] = [s2 st t2]
where u, v, w are linear forms in x, y, z. Finally, we will assume that the ideal I2
x y z
u v w
has codimension two. Note that this is a generic condition.
We introduce now the equations of I.
• Linear equations f and g:
[f g] = [x y z] ϕ = [x y z]
α1 α2
β1 β2
γ1 γ2
= [s2 st t2]
where u, v, w are linear forms in x, y, z.
• Biforms h1 and h2:
Write Γ1 and Γ2 such that
[f g] = [x y z] ϕ = [ s t2 ] Γ1 = [ s
2 t ] Γ2.
Then h1 = detΓ1 and h2 = detΓ2.
• Implicit equation F = detΘ, where [h1 h2] = [s t] Θ.
Using generic entries for ϕ, in place of the true k-linear forms in old variables x, y, z, we
consider the ideal of k[s, t, x, y, z, u, v, w] defined by
f = s2x+ sty + t2z
g = s2u+ stv + t2w
h1 = −syu− tzu+ sxv + txw
h2 = −szu− tzv + sxw + tyw
F = −z2u2 + yzuv − xzv2 − y2uw + 2xzuw + xyvw − x2w2
Proposition 4.5 If I2
x y z
u v w
specializes to a codimension two ideal of k[x, y, z],
then L = (f, g, h1, h2, F ) ⊂ A = R[x, y, z, u, v, w] specializes to the defining ideal of R.
Proof. Macaulay2 ([11]) gives a resolution
0 → A d2−→ A5 −→ A5 −→ L → 0
where

zv − yw
zu− xw
−yu+ xv

The assumption on I2
x y z
u v w
says that the entries of d2 generate an ideal of
codimension four and thus implies that the specialization LS has projective dimension two
and that it is unmixed. Since LS 6⊂ (s, t)S, there is an element q ∈ (s, t)R that is regular
modulo S/LS. If
LS = Q1 ∩ · · · ∩Qr
is the primary decomposition of LS, the localization LSq has the corresponding decompo-
sition since q is not contained in any of the
Qi. But now Symq = Rq, so LSq = (f, g)u,
as Iq = Rq. ✷
Non-balanced case
We shall now give a similar computer-assisted treatment of the non-balanced case, that is
when the resolution (4) of the ideal I has µ = 3. This implies that the content ideal of the
syzygies is (s, t). Let us first indicate how the proposed algorithm would behave.
• Write the forms f, g as
f = as+ bt
g = cs+ dt,
where
x y z
u v w
• The next form is the Jacobian of f, g with respect to (s, t)
h1 = det(f, g)(s,t) = ad− bc = −bxs2 − byst− bzt2 + aus2 + avst+ awt2.
• The next two generators
h2 = det(f, h1)(s,t) = b
2xs+ b2yt− abzt− abus− abvt+ a2wt
and the elimination equation
h3 = det(f, h2)(s,t) = −b3x+ ab2y − a2bz + ab2u− a2bv + a3w.
Proposition 4.6 L = (f, g, h1, h2, h3) ⊂ A = k[s, t, x, y, z, u, v, w] specializes to the defin-
ing ideal of R.
Proof. Macaulay2 ([11]) gives the following resolution of L
0 → A2 ϕ−→ A6 ψ−→ A5 −→ L → 0,
x + abu −b
y + abz + abv − a
w −bsx− bty + asu + atv −btz + atw −s
x− sty − t
u− stv − t
t −s 0 0 0 0
a b t −s 0 0
0 0 a b t −s
0 0 0 0 a b
The ideal of 2 × 2 minors of ϕ has codimension 4, even after we specialize from A to
S in the natural manner. Since LS has projective dimension two, it will be unmixed. As
LS 6⊂ (s, t), there is an element u ∈ (s, t)R that is regular modulo S/LS. If
LS = Q1 ∩ · · · ∩Qr
is the primary decomposition of LS, the localization LSu has the corresponding decompo-
sition since u is not contained in any of the
Qi. But now Symu = Ru, so LSu = (f, g)u,
as Iu = Ru. ✷
4.4 Degree 5 and above
It may be worthwhile to extend this to arbitrary degree, that is assume that I is defined
by 3 forms of degree n+1 (for convenience in the notation to follow). We first consider the
case µ = 1. Using the procedure above, we would obtain the sequence of polynomials in
A = R[a, b, x1, . . . , xn, y1, . . . , yn]
• Write the forms f, g as
f = as+ bt
g = cs+ dt,
where
x1 · · · xn
y1 · · · yn

sn−2t
stn−2

• The next form is the Jacobian of f, g with respect to (s, t)
h1 = det(f, g)(s,t) = ad− bc
• Successively we would set
hi+1 = det(f, hi)(s,t), 1 < n.
• The polynomial
hn = det(f, hn−1)(s,t)
is the elimination equation.
Proposition 4.7 L = (f, g, h1, . . . , h5) ⊂ A specializes to the defining ideal of R.
In Macaulay2, we checked the degrees 5 and 6 cases. In both cases, the ideal L (which
has one more generator in degree 6) has a projective resolution of length 2 and the ideal of
maximal minors of the last map has codimension four.
Conjecture 4.8 For arbitrary n, L = (f, g, h1, . . . , hn) ⊂ A has projective dimension two
and specializes to the defining ideal of R.
In degree 5, the interesting case is when the Hilbert-Burch matrix φ has degrees 2 and
3. Let us describe the proposed generators. For simplicity, by a change of coordinates, we
assume that the coordinates of the degree 2 column of ϕ are s2, st, t2
f = s2x+ sty + t2z
g = (s3w1 + s
2tw2 + st
2w3 + t
3w4)x+ (s
3w5 + s
2tw6 + st
2w7 + t
3w8)y
+ (s3w9 + s
2tw10 + st
2w11 + t
3w12)z
Let [
x y z
sA sB + tC tD
 = φ
x ys+ zt
sA+ tB stC + t2D
xs+ yt z
s2A+ stB sC + tD
where A,B,C,D are k-linear forms in x, y, z.
h1 = det(B1)
= s2(−yA) + st(xC − yB − zA) + t2(xD − zB)
= s2(−yA) + t(xCs− yBs− zAs+ xDt− zBt)
= s(−yAs+ xCt− yBt− zAt) + t2(xD − zB),
h2 = det(B2)
= s2(xC − zA) + st(xD + yC − zB) + t2(yD)
= s2(xC − zA) + t(xDs+ yCs− zBs+ yDt)
= s(xCs− zAs + xDt+ yCt− zBt) + t2(yD).
x ys+ zt
−yA xCs− yBs− zAs + xDt− zBt
xs+ yt z
−yAs+ xCt− yBt− zAt xD − zB
x ys+ zt
xC − zA xDs+ yCs− zBs+ yDt
xs+ yt z
xCs− zAs+ xDt+ yCt− zBt yD
c1 = det(C1) = x
2(Cs+Dt) + xy(−Bs) + xz(−As−Bt) + yz(At) + y2(As)
c2 = det(C2) = x
2(Ds) + xy(Dt) + xz(−Bs−Ct) + yz(As) + z2(At)
c3 = det(C3) = x
2(Ds) + xy(Dt) + xz(−Bs−Ct) + yz(As) + z2(At)
c4 = det(C4) = xy(Ds) + xz(−Cs−Dt) + yz(−Ct) + z2(As +Bt) + y2(D)
x y z
−yA xC − yB − zA xD − zB
xC − zA xD + yC − zB yD
Then F = −x3D2+x2yCD+xy2(−BD)+x2z(2BD−C2)+xz2(2AC−B2)+xyz(BC−
3AD) + y2z(−AC)+ yz2(AB)+ y3(AD) + z3(−A2), an equation of degree 5. In particular,
the parametrization is birational.
Proposition 4.9 L = (f, g, h1, h2, c1, c2, c4, F ) specializes to the defining ideal of R.
Proof. Using Macaulay2, the ideal L has a resolution:
0 −→ S1 d3−→ S6 d2−→ S12 d1−→ S8 −→ L −→ 0.
d3 = [−z y x − t s 0]t
y z 0 0 0 0
x 0 z 0 0 0
−v 0 0 z 0 x2w4 − xzw7 + xyw8 + xzw12
u 0 0 0 z −xzw3 + xyw4 + z
2w6 − yzw7 + y
2w8 − xzw8 − z
2w11 + yzw12
0 x −y 0 0 0
0 −v 0 −y 0 xzw1 − x
2w3 + yzw5 + z
2w9 − xzw11
0 u 0 0 −y xzw2 − x
2w4 + z
2w10 − xzw12
0 0 u 0 −x xzw1 + yzw5 − xzw6 + x
2w8 + z
0 0 0 u v 0
0 0 v x 0 −xyw1 + x
2w2 − y
2w5 + xyw6 − x
2w7 − yzw9 + xzw10
0 0 0 0 0 −t
0 0 0 0 0 s
The ideals of maximal minors give codim I1(d3) = 5 and codim I5(d2) = 4 after special-
ization. As we have been arguing, this suffices to show that the specialization is a prime
ideal of codimension two. ✷
Elimination forms in higher degree
In degrees greater than 5, the methods above are not very suitable. However, in several
cases they are still supple enough to produce the elimination equation. We have already
seen this when one of the syzygies is of degree 1. Let us describe two other cases.
• Degree n = 2p, f and g both of degree p. We use the decomposition
(s, t)p =
(si, tp+1−i).
For each 1 ≤ i ≤ p, let
hi = det(f, g)(si,tp+1−i).
These are quadratic polynomials with coefficients in (s, t)p−1. We set
[h1, · · · , hp] = [sp−1, · · · , tp−1] ·A,
where A is a p × p matrix whose entries are 2-forms in k[x, y, z]. The Sylvester form of
degree n, F = det(A), is the required elimination equation.
• Degree n = 2p+ 1, f of degree p. We use the decomposition
(s, t)p =
(si, tp+1−i).
For each 1 ≤ i ≤ p, let
hi = det(f, g)(si,tp+1−i).
These are quadratic polynomials with coefficients in (s, t)p. We set
[f, h1, · · · , hp] = [sp, · · · , tp] ·B,
where A is a (p + 1) × (p + 1) matrix with one column whose entries are linear forms and
the remaining columns with entries 2-forms in k[x, y, z]. The Sylvester form F = det(B) is
the required elimination equation.
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	Introduction
	Preliminaries on symmetric and Rees algebras
	Algebraic invariants in rational parametrizations
	Elimination degrees and birationality
	Calculation of e1(I) of the base ideal of a rational map
	Sylvester forms in dimension two
	Basic Sylvester forms in dimension 2
	Cohen-Macaulay algebras
	Base ideals generated in degree 4
	Degree 5 and above
