# The Bourbaki degree of plane projective curves

Marcos Jardim Universidade Estadual de Campinas (UNICAMP)
Instituto de Matemática, Estatística e Computação Científica (IMECC)
Departamento de Matemática
Rua Sérgio Buarque de Holanda, 651
13083-970 Campinas-SP, Brazil<EMAIL_ADDRESS>,  Abbas Nasrollah Nejad
Department of Mathematics, Institute for Advanced Studies in Basic Sciences
(IASBS), Zanjan 45137-66731, Iran<EMAIL_ADDRESS>and  Aron Simis
Departamento de Matemática, CCEN, Universidade Federal de Pernambuco, Av.
Jornalista Aníbal Fernandes, s/n, Cidade Universitária CEP 50740-560, Recife,
Pernambuco Brazil<EMAIL_ADDRESS>
###### Abstract.

Bourbaki sequences and Bourbaki ideals have been studied by several authors
since its inception sixty years ago circa. Generic Bourbaki sequences have
been thoroughly examined by the senior author with B. Ulrich and W.
Vasconcelos, but due to their nature, no numerical invariant was immediately
available. Recently, J. Herzog, S. Kumashiro, and D. Stamate introduced the
Bourbaki number in the category of graded modules as the shifted degree of a
Bourbaki ideal corresponding to submodules generated in degree at least the
maximal degree of a minimal generator of the given module. The present work
introduces theBourbaki degree as the algebraic multiplicity of a Bourbaki
ideal corresponding to choices of minimal generators of minimal degree. The
main intent is a study of plane curve singularities via this new numerical
invariant. Accordingly, quite naturally, the focus is on the case where the
standing graded module is the first syzygy module of the gradient ideal of a
reduced form $f\in k[x,y,z]$ – i.e., the main component of the module of
logarithmic derivations of the corresponding curve. The overall goal of this
project is to allow for a facet of classification of projective plane curves
based on the behavior of this new numerical invariant, with emphasis on
results about its lower and upper bounds.

###### Key words and phrases:

Plane curve singularities, Bourbaki ideal, syzygies, gradient ideal, graded
free resolution, Cohen–Macaulay ring, free divisor

###### 2020 Mathematics Subject Classification:

Primary: 13A02, 13D02, 13H15. Secondary: 14B05, 14H20, 14H50

## Introduction

Much of recent work on the homological facets of a plane curve $X=V(f)$ has
largely focused on the nature of the gradient ideal $J_{f}$ of $f$. And yet,
this approach is not so new (see, e.g., [29], [30], and possibly earlier),
coming from the need to characterize certain classes of singularities in
purely algebraic terms, that is to say, by drawing upon the core of
commutative algebra. For example, a typical algebraic puzzle asks if there is
a clear-cut watershed between the nature of an ideal $I\subset R$ generated by
three forms of the same degree and the gradient ideal of a reduced form. Of
course, thanks to Dolgachev’s theorem, the Cremona issue makes them as apart
as possible. But, what about the homological behavior? Does it disentangle the
conundrum as to when the strong geometric background of the gradient comes
from a loose choice of three forms?

The commutative algebra comes to help facing this conundrum by bringing out
the related homology. This way, one trades out the hard geometric
classification of singularity types for a “classification” of homological
types referring to Betti numbers and degree shifts in minimal free
resolutions. In terms of a classical algebraic language, one is after syzygies
of graded modules and their degrees. In order to look closer at these syzygies
one may resort to several methods, one of which is to reduce the problem to
the case of homogeneous ideals. The choice of trailing after the idea of the
Bourbaki ideal goes along this pattern.

The notion of a Bourbaki sequence, that basically relates a module to an ideal
via a pivotal submodule, was introduced in [2]. It has soon been considered by
several authors, for various purposes ([1], [17], [24], [21], [31], [37],
[12], [22]). The present work is closer to [22] in that one searches to
extract a numerical invariant out of a Bourbaki sequence in the graded module
category. However, in contrast to loc.cit., where the Bourbaki number of a
graded module $M$ is the shift of the Bourbaki ideal in a graded Bourbaki
sequence corresponding to the choice of an equigenerated submodule generated
in degree at least the degree of a minimal generator of $M$ of maximal degree,
here we consider instead a Bourbaki ideal coming from a Bourbaki sequence
corresponding to choices of minimal generators of minimal possible degree.
Moreover, the numerical invariant considered here is totally different,
namely, it is the degree (multiplicity) of the Bourbaki ideal in the
discussion.

The above is as much nearness as there exists between the two approaches. We
focus on a particular graded module over the polynomial ring $R:=k[x,y,z]$,
having in mind the understanding of projective plane curve theory from this
angle. Namely, we consider the main direct summand of the module of
logarithmic derivations associated with a reduced form $f\in R$. This module
is canonically identified with the first syzygy module $\mbox{\rm
Syz}(R/J_{f})$ of the gradient (Jacobian) ideal $J_{f}\subset R$ of $f$. Since
$R/J_{f}$ determines the singularities of the associated projective plane
curve $X:=V(f)\subset\mathbb{P}_{k}^{2}$, the quest has both algebraic and
geometric facets.

The novelty introduced here is a Bourbaki degree $\operatorname{Bour}(X)$ of
$X$, defined as the degree of $R/I_{\epsilon}$, where $I_{\epsilon}$ is the
Bourbaki ideal associated to a choice of a minimal generator $\epsilon$ of
$\mbox{\rm Syz}(R/J_{f})$ of minimal possible degree. Bourbaki ideals defined
this way have been considered before by Dimca–Sticlaru ([12, Section 5], [13,
Section 3]) and, in a non-explicit way, by du Plessis–Wall ([16]). Both are
closely related to this work but do not single out any new numerical invariant
attached to a Bourbaki sequence, by rather focusing on other invariants, such
as the total Tjurina number and the stability threshold related to the former.
Drawing upon the facilitation of a Bourbaki degree, our work entails further
precision to the relationship among some of these invariants and the curve
geometry. We now briefly describe the sections of the paper.

The first section contains notation and preliminaries for the rest of the
paper. Here we introduce the module $\mbox{\rm Syz}(R/J_{f})$ of first
syzygies of the gradient ideal $J_{f}$ in terms of the module of logarithmic
derivations of $f\subset R=k[x,y,z]$, thereafter elaborating on its basic
properties. It will be our standing graded module throughout the paper.

The second section evolves around the details of a Bourbaki ideal
$I_{\epsilon}\subset R$ associated with the choice of a minimal generator
$\epsilon$ of $\mbox{\rm Syz}(R/J_{f})$. We give a formula for the degree
(multiplicity) of $R/I_{\epsilon}$ in terms of the degree of $R/J_{f}$, the
standard degree $e$ of $\epsilon$ and the homogeneous degree of $f$. This
approach is pretty much the same as the one by Dimca and co-authors, but it
has the advantage for the reader that it collects the basic facts and the
respective proofs in one single theorem in purely algebraic terms, while the
former is somewhat spread out in a few different papers and uses both
algebraic and geometric arguments.

Likewise, we recover the definitions introduced in du Plessis–Wall, where they
make use of a suitable colon ideal.

The tight relationship between the various terms in the formula above of the
degree of $R/I_{\epsilon}$ has encouraged us to call it the Bourbaki degree
$\operatorname{Bour}(X)$ of $X=V(f)\subset{\mathbb{P}}^{2}_{k}$ associated to
$\epsilon$, in the case where the homogeneous degree of the latter is the
initial degree of $\mbox{\rm Syz}(R/J_{f})$. Focusing on the initial degree
complies with Dimca’s approach, although a Bourbaki degree as such has not
been a theme thereof.

As a follow-up to the basic structural theorem and the introduction of the
Bourbaki degree $\operatorname{Bour}(X)$, we give some examples to illustrate
the behavior of the latter. One of these has to do with the maximality
behavior of nodal curves in this respect: besides attaining almost maximal
bound for $\operatorname{Bour}(X)$, it also frequently attains the maximal
possible degree of a generating syzygy. In this thread of line, we conjecture
that the curve $f=(x^{2}-y^{2})z^{d-1}-(x^{d-1}-y^{d-1})x^{2}-y^{d+1}$ (for
any $d\geq 2$) does attain such maximum.

Next, is one main theorem proving the upper bound $\operatorname{Bour}(X)\leq
e^{2}$ for a singular curve $X$, where $e$ is a minimal syzygy homogeneous
degree. Taking into account that the total Tjurina number coincides with the
multiplicity $\deg(R/J_{f})$, this bound implies the main result in [16,
Theorem 3.2]. The theorem also implies that if $e=1$ then $f$ is either free
or nearly free, a result originally due to Dimca and co-authors.

A note of interest in the proof of this theorem is the inclusion of radical
ideals $\sqrt{H}\subset\sqrt{I_{\epsilon}}$, where $H$ denotes the ideal
generated by the coordinates of a syzygy $\epsilon$ of minimal homogeneous
degree. This inclusion might be strict or an equality as the choice of
$\epsilon$ varies.

In the sequel, we establish a few scattered facts in the case where $f$ is
irreducible. Using that the total Milnor number is bounded by $d(d-1)$ in this
case, we improve the structural lower bound to $\operatorname{Bour}(X)$ all
the way to $\operatorname{Bour}(X)\geq d+e(e-d)$. As a corollary, there are no
irreducible free divisors of degree less than $5$, a result that has been
spread around as a clear statement, but no rigorous proof has been given so
far. Finally, along this line, we show that if $f$ is an irreducible free
divisor of degree $5$ then its minimal syzygy degree is $2$, a situation that
repeats itself throughout many examples in the literature (see, e.g., [36],
[26]).

The section ends with a characterization of curves such that
$\operatorname{Bour}(X)\leq 2$, which takes care of the nearly free curves and
some one-plus curves introduced by Dimca and co-authors.

The last section focuses on lower bounds to $\operatorname{Bour}(X)$. We
restrict ourselves to quasi-homogeneous singularities, with particular
endeavor for the case of nodal singularities. In the spirit mentioned before
in this Introduction, yet another paradigm of the maximality phenomenon of
nodal singularities now takes place, this time around in regard to the minimal
syzygy degree, which is now at least $d-1$. By drawing upon the known upper
bounds for the number of singular points, we state lower bounds for
$\operatorname{Bour}(X)$ in the reduced and irreducible cases, reflecting the
cases where $e=d-1$ or $e=d$. Next is a final blow about the maximality of
nodal curves, to the effect that a curve having a unique singular point, which
is nodal, is characterized by the value $\operatorname{Bour}(X)=d^{2}-1$.

We end the section with some comments on the relation to Dolgachev’s theorem.
In particular, we give another proof of the case of Dolgachev’s theorem where
one assumes that the curve is quasi-homogeneous. This observation has been
stated before in [27] and [34].

### Acknowledgments

MJ is supported by the CNPQ grant number 305601/2022-9 and the FAPESP Thematic
Project number 2018/21391-1. AAN is partially supported by the FAPESP grant
number 2022/09853-5. AS is partially supported by a CNPq grant number
301131/2019-8. AAN and AS thank for the warm hospitality during their visits
to IMECC-UNICAMP.

##  1\. Preliminaries

###  1.1. Logarithmic derivations

In this short section, we recall the notion of the module (respectively,
sheaf) of logarithmic derivations, frequently referred to as the Derlog module
(respectively, sheaf). For an encompassing treatment, see [28], [33].

Fix a perfect infinite field $k$ of high enough characteristic (as a function
of certain data). For the geometric purpose, we may assume that it is
algebraically closed of characteristic zero. Let $R=k[x_{1},\ldots,x_{m}]$
denote a standard graded polynomial ring. The set of $k$-linear derivations of
$R$ into itself is an $R$-module, denoted $\operatorname{Der}_{k}(R,R)$, or
more simply, $\operatorname{Der}_{k}(R)$. The usual partial derivatives
$\partial_{x_{i}}:=\partial/\partial x_{i}$, as maps from $R$ to $R$, are
$k$-linear derivations and form a free basis of the $R$-module
$\operatorname{Der}_{k}(R)$. Since $R$ is standard graded, then
$\operatorname{Der}_{k}(R)$ admits a natural ${\mathbb{N}}$-grading where

$\operatorname{Der}_{k}(R)_{n}=\sum_{i=1}^{m}R_{n}\partial_{x_{i}},\;n\in{\mathbb{N}},$

where $R_{n}$ denotes the $k$-vector space spanned by the homogeneous
polynomials of degree $n$.

The following notion can be stated more generally for ideals $I\subset R$, but
in this work, we are only interested in the case of a homogeneous principal
ideal $(f)$, where $f\in R_{d+1}$, with $d\geq 1$. Thus, we are looking at a
hypersurface in projective space.

######  Definition 1.1.

The module of logarithmic derivations of $f$ is the submodule

$\operatorname{Der}_{f}(R):=\\{\delta\in\operatorname{Der}_{k}(R)\ |\
\delta(f)\in(f)\\}\subset\operatorname{Der}_{k}(R).$

Since $f\operatorname{Der}_{k}(R)\subset\operatorname{Der}_{f}(R)$, latter has
same rank as $\operatorname{Der}_{k}(R)$

It has yet another notable submodule, namely,

$\operatorname{Der}_{f}(R)^{0}:=\\{\delta\in\operatorname{Der}_{k}(R)\ |\
\delta(f)=0\\}\subset\operatorname{Der}_{k}(R).$

Writing a $k$-linear derivation as
$\delta=\sum_{i=1}^{m}g_{i}\partial_{x_{i}}$ then, upon identification
$\operatorname{Der}_{k}(R)\simeq R^{m}$ via orderly mapping the partial
derivatives to the canonical basis of $R^{m}$, $\operatorname{Der}_{f}(R)^{0}$
becomes the $R$-module $\mathcal{Z}(\partial)$ of (first) syzygies of the set
$\\{\partial f/\partial_{x_{i}}\,|\,1\leq i\leq m\\}$. Since the latter are
canonical generators of the gradient ideal $J_{f}$ of $f$, we rename ${\rm
Syz}(J_{f}):=\mathcal{Z}(\partial)$.

With this notation, there is a remarkable short exact sequence of $R$-modules
(see, e.g., [33, Proposition 3.2])

$0\rightarrow{\rm
Syz}(J_{f})\longrightarrow\operatorname{Der}_{f}(R)\longrightarrow
J_{f}\colon\kern-2.0pt{{}_{R}}(f)\rightarrow 0.$

Assuming, moreover, that char$(k)$ does not divide $d+1$, the above sequence
splits, and the complementary direct summand is identified with the Euler
derivation $\delta_{E}:=\sum_{i=1}^{m}x_{i}\partial_{x_{i}}$:

(1) $\operatorname{Der}_{f}(R)={\rm Syz}(J_{f})\bigoplus R\delta_{E},$

###  1.2. The role of syzygies

Note the graded presentation of the gradient ideal

(2) $0\rightarrow\sum_{e}R\epsilon_{d+e}(-(d+e))\longrightarrow
R^{3}(-d)\stackrel{{\scriptstyle\Theta_{f}}}{{\longrightarrow}}J_{f}\rightarrow
0,$

where $\Theta_{f}$ maps the canonical $i$th basis element to
$\partial_{x_{i}}$, while $\epsilon_{d+e}$ is a syzygy of degree $d+e$ in the
grading of $R^{3}(-d)$, i.e., of standard degree $e$ in the polynomial ring
$R$. Shifting by $d$, we can write

(3) $0\rightarrow{\rm Syz}(J_{f})\longrightarrow
R^{3}\stackrel{{\scriptstyle\Theta_{f}}}{{\longrightarrow}}J_{f}(d)\rightarrow
0.$

Thus, throughout ${\rm Syz}(J_{f})$ is the module of syzygies of $J_{f}$ in a
graded resolution of the latter, with degrees of generators setback to
standard degree.

From now on, assume that $m=3$ and that $f$ is reduced (i.e., $(f)$ is a
radical ideal). For common reasons, we replace the variables by $x,y,z$.
Throughout, $k$ is algebraically closed and char$(k)$ does not divide $d+1$.
In particular, $f\in J_{f}$.

It is natural to assume that the partial derivatives $f_{x},f_{y},f_{z}$ of
$f$ are algebraically independent over $k$, as otherwise in this dimension the
Hesse–Gordan–Noether classical result forces $f_{x},f_{y},f_{z}$ to actually
be $k$-linearly independent, i.e., dependent upon only two variables up to a
linear change of variables (see, e.g., [5, Proposition 2.7]). In other words,
the algebraic dependence of $f_{x},f_{y},f_{z}$ implies that $V(f)$ is a cone.
The hypothesis that $f_{x},f_{y},f_{z}$ of $f$ are algebraically independent
over $k$ can be rephrased, e.g., to the effect that $k$-subalgebra
$k[f_{x},f_{y},f_{z}]\subset R=k[x,y,z]$ has dimension three, or that the
gradient ideal $J_{f}$ has maximal analytic spread, or still that the fiber
cone algebra of $J_{f}$ is (isomorphic to) $k[x,y,z]$.

Since $f$ is reduced, $J_{f}$ has codimension at least two, hence it is
locally Cohen–Macaulay in the punctured spectrum of $R=k[x,y,z]$ (being
primary to the maximal ideal or the ring itself in every localization). Thus,
locally in the punctured spectrum, $J_{f}$ is either a codimension two perfect
ideal or else the localized ring. Then (3) implies that locally in the
punctured spectrum, ${\rm Syz}(J_{f})$ is free in two generators. In
particular, ${\rm Syz}(J_{f})$ has rank two. In addition, being a submodule of
a free module, it is reflexive.

Let $X:=V(f)\subseteq{\mathbb{P}}^{2}$ denote the corresponding reduced
projective plane curve of degree $d+1$, with singular scheme ${\rm
Proj}(R/J_{f})$. Throughout we will be mainly focusing on a singular curve
$X$, so ${\rm codim}(J_{f})=2$ will be the case.

The coherent sheaf associated to the graded $R$-module ${\rm Syz}(J_{f})$ is
denoted $\mathcal{T}_{f}$. With ${\mathcal{J}}_{f}$ denoting the ideal sheaf
associated to the gradient ideal $J_{f}$, (3) reads out as an exact sequence

(4)
$0\rightarrow\mathcal{T}_{f}\longrightarrow{\mathcal{O}}_{{\mathbb{P}}^{2}}\stackrel{{\scriptstyle\Theta_{f}}}{{\longrightarrow}}{\mathcal{J}}_{f}(d)\rightarrow
0.$

This gives an identification

${\rm
Syz}(J_{f})_{n}\simeq\mathrm{H}^{0}({\mathbb{P}}^{2},\mathcal{T}_{f}(n)),\;n\in{\mathbb{N}},$

thus substantiating the terminology vector fields sections annihilating $f$
used in [16].

##  2\. The graded Bourbaki ideal associated with a projective plane curve

We introduce the Bourbaki ideal of a projective plane curve in an algebraic
way, pretty much as in [12, Section 5], with slight modifications to serve our
present purpose. Generalizations to arbitrary dimensions can be found in [21]
or [31] and, more recently, in [22].

###  2.1. Basic theorem

Let $\epsilon\in{\rm Syz}(J_{f})$ be a syzygy of degree $e$ (in the geometric
language as above, a global section of the shifted sheaf
$\mathcal{T}_{f}(e))$. The inclusion $\epsilon\in{\rm Syz}(J_{f})$ induces a
homogeneous injective map
$R(-e)\stackrel{{\scriptstyle\epsilon}}{{\longrightarrow}}{\rm Syz}(J_{f})$
sitting on an exact sequence of graded $R$-modules

(5) $0\rightarrow
R(-e)\stackrel{{\scriptstyle\epsilon}}{{\longrightarrow}}{\rm
Syz}(J_{f})\longrightarrow{\rm coker}(\epsilon)\rightarrow 0.$

Setting $M_{\epsilon}:={\rm coker}(\epsilon)$, one has the following
commutative diagram of graded $R$-modules

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The following is the basic theorem that will guide our results. It can be
considered as an encore to [12, Theorem 5.1], with a different emphasis as
convenient to the sequel.

######  Theorem 2.1.

Let $\epsilon\in{\rm Syz}(J_{f})$ be a minimal homogeneous generator of degree
$e\geq 1$. Then the following hold.

  1. (a)

$M_{\epsilon}$ is free if and only if $f$ is a free divisor. In this case,
$\deg(R/J_{f})=d^{2}+e(e-d)$.

  2. (b)

If $f$ is not a free divisor, $M_{\epsilon}$ is isomorphic to a proper
homogeneous ideal $I_{\epsilon}\subset R$ of codimension two, such that the
induced isomorphism $M_{\epsilon}\simeq I_{\epsilon}(e-d)$ is homogeneous of
degree zero. In addition, for such an ideal $I_{\epsilon}$ one has

(7)
$\deg(R/I_{\epsilon})=\left\\{\begin{array}[]{ll}d^{2}+e(e-d)-\deg(R/J_{f})&\mbox{if
$f$ is singular}\\\ d^{2}+e(e-d)&\mbox{if $f$ is smooth}.\end{array}\right.$

  3. (c)

If $f$ is not a free divisor, pick a complete set of minimal homogeneous
generators of the $R$-module ${\rm Syz}(J_{f})$ that include $\epsilon$, and
let

(8) $0\rightarrow
F_{1}\stackrel{{\scriptstyle\left[\lambda\atop\psi\right]}}{{\longrightarrow}}R(-e)\oplus
F_{0}\stackrel{{\scriptstyle(\epsilon,\varphi)}}{{\longrightarrow}}{\rm
Syz}(J_{f})\rightarrow 0$

denote a minimal graded free resolution based on these generators, where the
remaining shifts have not been specified. Then a graded minimal free
resolution of $I_{\epsilon}$ is

(9) $0\rightarrow
F_{1}(d-e))\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}F_{0}(d-e)\stackrel{{\scriptstyle
u\circ\varphi}}{{\longrightarrow}}I_{\epsilon}\rightarrow 0.$

In particular, $I_{\epsilon}$ is a codimension $2$ perfect ideal.

###### Proof.

First, $M_{\epsilon}$ is a torsion-free module of rank one. Indeed, quite
generally, if $F\subset E$ is a free submodule of a torsionfree module $E$
then $E/F$ is torsionfree. This is because $F:_{E}(a)=F$ for any $0\neq a\in
R$. Clearly, $\operatorname{rk}E/F=\operatorname{rk}E-\operatorname{rk}F$.

(a) Suppose that $M_{\epsilon}$ is a free $R$-module. Since $M_{\epsilon}$ has
rank one, $M_{\epsilon}\simeq R$. Then the exact sequence (5) splits as
$\mbox{\rm Syz}(J_{f})\simeq R(-e)\oplus R$, that is, $\mbox{\rm Syz}(J_{f})$
is a free $R$-module generated by $\epsilon$ and another minimal generator,
necessarily, of (standard) degree $d-e$. In other words, one gets a graded
isomorphism

$\mbox{\rm Syz}(J_{f})\simeq R(-e)\oplus R(d-e)=R(-e)\oplus R(-(e-d)).$

This means that $J_{f}$ has a Hilbert–Burch free resolution, i.e., it is a
codimension two perfect ideal. In other words, $f$ is a free divisor ([33,
Proposition 3.7]).

The converse is clear.

The second statement is a calculation with the Hilbert polynomial along the
free resolution of $R/J_{f}$:

$0\rightarrow R(-(d+e))\oplus R(-(2d-e))\longrightarrow
R(-d)^{3}\longrightarrow R\longrightarrow R/J_{f}\rightarrow 0.$

(b) Again quite more generally, any finitely generated torsionfree module over
a Noetherian ring $R$, having rank $r$, is isomorphic to a submodule of
$R^{r}$ ([35, Proposition 3.5.3]).

Thus, in the present case, $M_{\epsilon}$ can be embedded as an ideal
$I_{\epsilon}\subset R$. Since $f$ is not a free divisor by assumption, item
(a) implies that $I_{\epsilon}$ is not a free $R$-module, hence it is a proper
ideal that is not principal. Suppose that $I_{\epsilon}$ has height one. Since
$R$ is an UFD, $I_{\epsilon}=a\tilde{I}$, for some $0\neq a\in R$ and an ideal
$\tilde{I}$ of height at least two. Then, $I_{\epsilon}$ and $\tilde{I}$ are
isomorphic as $R$-modules by multiplication by $a^{-1}$, hence
$M_{\epsilon}\simeq\tilde{I}$. Since $M_{\epsilon}$ is graded, $I_{\epsilon}$
is homogeneous and so $a$ and $\tilde{I}$ can be taken to be homogeneous.
Finally, we rename $\tilde{I}$ back to $I_{\epsilon}$.

Next, we claim that, under the assumption of the item, $I_{\epsilon}$ has
codimension exactly two. Indeed, if $I_{\epsilon}))$ has a regular sequence of
length three, then by “décalage” ([35, Proposition 6.2.77]) ,
$\operatorname{Ext}^{1}(I_{\epsilon},R(-e))=\operatorname{Ext}^{2}(R/I_{\epsilon},R(-e))=0$.
Dualizing the exact sequence $0\rightarrow R(-e)\longrightarrow{\rm
Syz}(J_{f})\longrightarrow I_{\epsilon}\rightarrow 0$ into $R(-e)$ yields the
split exact sequence

$0\rightarrow\mbox{\rm Hom}(I_{\epsilon},R(-e))\longrightarrow\mbox{\rm
Hom}({\rm Syz}(J_{f}),R(-e))\longrightarrow\mbox{\rm Hom}(R(-e),R(-e))\simeq
R\rightarrow 0.$

Therefore, ${\rm Syz}(J_{f})^{*}\simeq\mbox{\rm Hom}({\rm
Syz}(J_{f}),R(-e))\simeq R^{2},$ hence ${\rm Syz}(J_{f})$ is free since it is
reflexive. Then $f$ is a free divisor, contradicting the assumption of this
item.

It remains to determine a shift $s$ such that the induced isomorphism
$M_{\epsilon}\simeq I_{\epsilon}(s)$ is graded homogeneous (i.e., of degree
zero). For this, we resort to a Hilbert function calculation.

We use the additive property of the Hilbert polynomial on the lowest
horizontal graded exact sequence in (6). Then we draw on the respective
tautological presentations of $R(d)/J_{f}(d)$ and of $R(s)/I_{\epsilon}(s)$
and use the fact that, since both $R(d)/J_{f}(d)$ and $R(s)/I_{\epsilon}(s)$
are one dimensional, their degrees (multiplicities) coincide with the
corresponding Hilbert polynomials.

Thus, we get:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\operatorname{HP}_{R^{3}}(t)-\operatorname{HP}_{R(-e)}(t)-\operatorname{HP}_{J_{f}(d)}(t)-\operatorname{HP}_{I_{\epsilon}(s)}(t)$
$\displaystyle=$
$\displaystyle\operatorname{HP}_{R^{3}}(t)-\operatorname{HP}_{R(-e)}(t)-\operatorname{HP}_{R(d)}(t)-\operatorname{HP}_{R(s)}(t)+\operatorname{HP}_{R(d)/J_{f}(d)}(t)+\operatorname{HP}_{R(s)/I_{\epsilon}(s)}(t)$
$\displaystyle=$ $\displaystyle
3\binom{t+2}{2}-\binom{t-e+2}{2}-\binom{t+d+2}{2}-\binom{t+s+2}{2}+\operatorname{HP}_{R(d)/J_{f}(d)}(t)+\mathfrak{e}_{0}(R/I_{\epsilon})$
$\displaystyle=$
$\displaystyle(e-d-s)t+\frac{1}{2}(-e^{2}-d^{2}-s^{2}+3e-3d-3s)+\operatorname{HP}_{R(d)/J_{f}(d)}(t)+\mathfrak{e}_{0}(R/I_{\epsilon}),$

because $\dim R/I_{\epsilon}=2$, where $\mathfrak{e}_{0}$ denotes the
algebraic multiplicity of a homogeneous quotient $R/I$. Since we are thinking
of $t$ as a variable over $k$, and
$[\operatorname{HP}_{R(d)/J_{f}(d)}(t)]_{t}=0$ for $t\geq 1$ as $\dim
R/J_{f}\leq 1$, it follows that $s=e-d$ and, additionally,

$\displaystyle\mathfrak{e}_{0}(R/I_{\epsilon})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(d^{2}+e^{2}+s^{2}-3(e-d-s)\right)-\operatorname{HP}_{R(d)/J_{f}(d)}(t)$
$\displaystyle=$ $\displaystyle
d^{2}+e(e-d)-\operatorname{HP}_{R(d)/J_{f}(d)}(t).$

Now, replacing the multiplicity by the degree in case it is defined, we
conclude:

$\deg(R/I_{\epsilon})=\left\\{\begin{array}[]{ll}d^{2}+e(e-d)-\deg(R/J_{f})&\mbox{if
$f$ is singular}\\\ d^{2}+e(e-d)&\mbox{if $f$ is smooth}.\end{array}\right.$

(c) With the assumed notation of the item, one has te following commutative
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where $G$ is the kernel of $\overline{(\epsilon,\varphi)}$. By the snake
lemma, the leftmost vertical map $F_{1}\longrightarrow G$ is an isomorphism.
Since the map $\overline{(\epsilon,\varphi)}$ is identified with
$u\circ\varphi$, then the lowest horizontal exact sequence gives free
resolution of ${\rm Syz}(J_{f})/R(-e)=M_{\epsilon}\simeq I_{\epsilon}(e-d)$.
Shifting by $-(e-d)=d-e$ gives the stated free resolution of $I_{\epsilon}$..

In particular, since $I_{\epsilon}$ has codimension two, it is identified with
the ideal of maximal minors of a Hilbert–Burch matrix representing the rank
two map $F_{1}\longrightarrow F_{0}$. Thus, it is a perfect (Cohen–Macaulay)
ideal.

######  Remark 2.2.

In wrapping up the proof of item (b) above, we set
$\operatorname{HP}_{R(d)/J_{f}(d)}(t)=0$ when $J_{f}$ is an
${\mathfrak{m}}$-primary ideal. However, this is not saying that the algebraic
multiplicity of the ring $R/J_{f}$ is zero, since, for an
${\mathfrak{m}}$-primary ideal $\mathfrak{a}\subset R$, the multiplicity of
$R/\mathfrak{a}$ is defined as the length of $R/J_{f}$. In particular, this
should not be confused with the geometric degree of ${\rm Proj}(R/J_{f})$,
which is zero because the latter is the empty set.

######  Corollary 2.3.

If $f$ is not a free divisor, let $I_{\epsilon}$ stand by means of its graded
resolution as in Theorem 2.1 (c). Then $I_{\epsilon}=I_{v}$, where $I_{v}$ is
the ideal of $R=k[x,y,z]$ as defined in [16, After Lemma 1.1].

###### Proof.

This is an immediate consequence of the shape of $I_{\epsilon}$ via item (c)
of Theorem 2.1 and [16, Proposition 2.1].

The exact sequence $0\rightarrow R(-e)\longrightarrow{\rm
Syz}(J_{f})\longrightarrow I_{\epsilon}(e-d)\rightarrow 0$ is called a
Bourbaki sequence of the curve $X=V(f)$, associated to the generating syzygy
degree $e$. The ideal $I_{\epsilon}(e-d)$ is a Bourbaki ideal of of the curve
$X=V(f)$. As such, it depends on the choice of a minimal generator of a given
degree. If $f$ is free, the obtained Bourbaki ideal is trivially $R(e-d)$,
considered as an ideal.

Recall that, for a graded module $M=\bigoplus_{i\geq 0}M_{i}$ over an
${\mathbb{N}}$-graded Noetherian ring, its initial degree is defined to be
$\mathrm{indeg}(M):=\min\\{i\ |\ M_{i}\neq 0\\}.$ Here the focus is on the
case where $R$ is the standard graded polynomial ring $k[x,y,z]$

######  Definition 2.4.

The Bourbaki degree of $f$ is the co-degree of the Bourbaki ideal of ${\rm
Syz}(J_{f})$ with respect to a minimal generator $\epsilon$ of standard
initial degree. In other words, it is the degree (multiplicity) of the
$R$-module $R/I_{\epsilon}$.

Note that the Bourbaki degree does not depend on the choice of minimal
generator chosen, as long as it is of initial degree. Because of this
independence, we denote it by $\mathrm{Bour}(f)$. By setting $X=V(f)$, the
projective scheme $\mathrm{Proj}(R/I_{\epsilon})$ is called a Bourbaki scheme
of $X$ and we will often write $\mathrm{Bour}(f)=\mathrm{Bour}(X)$.

If $X$ is a free divisor, as has been seen, $I_{\epsilon}$ is identified with
$R$. Thus, we set quite naturally $\mathrm{Bour}(X)=0$.

######  Remark 2.5.

(1) Theorem 2.1 implies the following formula regardless for any reduced
singular curve $X=V(f)\subset{\mathbb{P}}^{2}$:

(11) $\operatorname{Bour}(X)=d^{2}+e(e-d)-\deg(R/J_{f})$

Geometrically, this formula is still kosher when $X$ is smooth, provided one
thinks of $\deg(R/J_{f})$ as the geometric degree of ${\rm Proj}(R/J_{f})$,
which is zero.

(2) Supposing that $V(f)$ is not smooth, let $J_{f}=J_{f}^{\rm
un}\cap\mathfrak{M}$ denote a primary decomposition of the ideal $J_{f}$,
where $J_{f}^{\rm un}$ is its minimal component (“un” for unmixed part), while
$\mathfrak{M}$ denotes an irrelevant component. Clearly, $J_{f}^{\rm
un}=J_{f}^{\rm sat}:=J_{f}\colon(x,y,z)^{\infty}$. Thus, $J_{f}^{\rm un}$
defines the singular locus of $V(f)$ scheme theoretically. Algebraically,
since $\operatorname{depth}R/J_{f}^{\rm un}=1$, the ideal $J_{f}^{\rm un}$ is
Cohen–Macaulay, thus yielding yet another codimension two Cohen–Macaulay
ideal. In particular, by (11), we are being told that the ubiquitous numerical
expression $d^{2}+e(e-d)$ is the sum of the algebraic co-degrees of two such
ideals.

Setting ${\mathfrak{m}}=(x,y,z)$, the cohomology module
$H_{{\mathfrak{m}}}(R/J_{f})=J_{f}^{\rm un}/J_{f}$ is of finite length, so
eventually $[J_{f}^{\rm un}]_{l}=[J_{f}]_{l}$ for $l>>0$. The least such $l$
is the saturation number of $J_{f}$, denoted ${\rm sat}(J_{f})$. A nontrivial
numerical relation has been established in [20, Theorem 1.2], to the effect
that ${\rm indeg}(H_{{\mathfrak{m}}}(R/J_{f}))=3(d-3)-{\rm sat}(J_{f})$. The
algebraic–combinatorial side of ${\rm indeg}(H_{{\mathfrak{m}}}(R/J_{f}))$ has
been explored in [3], while the more geometrically inclined facet has been
considered in [20] and [36], concerning a watershed bound related to plane
Cremona maps, to wit, ${\rm indeg}(H_{{\mathfrak{m}}}(R/J_{f}))\geq d+1$. It
would be interesting to understand the role of this lower bound in the present
case of a gradient ideal of a plane curve.

###  2.2. Propaedeutic examples

As a rule, when bringing up examples, we assume that the ground field
characteristic is well-behaved with respect to the essential exponents.

######  Example 2.6.

Let $f=(x^{2}-y^{2})z^{d-1}-(x^{d-1}-y^{d-1})x^{2}-y^{d+1}$ ($d\geq 2$).

It is clear that the point $p=(0:0:1)$ is a singular point. The local
expression of $f$ at $(x,y)$ tells it is a simple node. Now,
$f_{z}=(d-1)(x^{2}-y^{2})z^{d-2}$. Therefore, any minimal prime $\wp$ of the
gradient ideal $J_{f}$ must contain either $x+y,x-y$, or $z$. If $z\in\wp$,
then $\wp$ must contain the respective pure $\\{x,y\\}$-parts of both $f_{x}$
and $f_{y}$. A closer inspection shows that these have no common factors in
$k[x,y]$. Therefore, ${\rm ht}\,\wp=3$, which is absurd. If, instead, say,
$x-y\in\wp$, then again upon inspection $\wp$ contains the ideal
$(x-y,y^{d},zy^{d-1})$. Since the radical of the latter is $(x,y)$, then
$\wp=(x,y)$. Wrapping up, we have argued that $p=(0:0:1)$ is the unique
singular point of $X=V(f)$. Moreover, locally at $(x,y)$, the gradient ideal
is easily seen to be generated by $x,y$. Therefore, the associativity formula
for the degree (alternatively, the Tjurina number (see Proposition 18)) gives
that $\deg(R/J_{f})=1$.

This example is treated in more generality in Proposition 3.8 regarding the
Bourbaki degree, and hence, the emphasis is on the standard initial degree of
$\mbox{\rm Syz}(R/J_{f})$. The latter has been established in [10, Example 2.2
(i)]. The present example emphasizes instead the maximal generating degree of
$\mbox{\rm Syz}(R/J_{f})$. In virtue of the basic assumption in the Bourbaki
treatment in [22], this may have interest. Concerning this, we pose the
following questions:

######  Conjecture 2.7.

Let $f$ be as above.

  1. (1)

The maximal degree of a minimal generator of $\mbox{\rm Syz}(J_{f})$ is $2d-2$
(maximal possible according to [4, Corollary 11]).

  2. (2)

The minimal graded free resolution of $J_{f}$ is of the form

(12) $0\rightarrow R(-(3d-1))^{2}\longrightarrow R(-2d)^{3}\oplus
R(-(3d-2))\longrightarrow R(-d)^{3}\longrightarrow R,$

where the generating syzygies in degree $2d$ are the Koszul syzygies. In
particular, the regularity of $R/J_{f}$ is $3d$.

The degree of the Bourbaki ideal with respect to the syzygy of (standard)
degree $2d-2$ is $3(d-1)^{2}$, much larger than the Bourbaki degree as
introduced here, which is $d^{2}-1$ (they coincide if and only if $f$ is
cubic). Concerning the issue of an upper bound for degrees of minimal
generating syzygies, [23, Proposition 6], has a general bound for the degree
of generating syzygies of homogeneous matrices – in the case of equigenerated
ideals in $k[x,y,z]$ it amounts to $3/2(d^{2}+d-2)$. In higher dimension an
upper bound was established for the syzygies of the gradient ideal in [4,
Corollary 11] in the case of isolated singularities.

The conjecture has been verified affirmatively, with computer assistance, for
all sufficiently large values of $d$.

######  Example 2.8.

Let $X\subseteq{\mathbb{P}}_{k}^{2}$ be the quintic curve defined by the
polynomial $f=x^{5}+x^{4}y+x^{3}z^{2}+y^{2}z^{3}$. A computation with [6]
shows that $\mbox{\rm Syz}(J_{f})$ is minimally generated by two syzygies
${\epsilon}_{1}$, and ${\epsilon}_{2}$ of standard degree $3$ and two
additional ones of standard degree $4$. Moreover, $\mbox{\rm Syz}(J_{f})$ has
the following minimal graded resolution

$0\rightarrow
R^{2}(-5)\stackrel{{\scriptstyle\begin{bmatrix}4y^{2}&3x^{2}-10yz\\\
-6z&-9y\\\ -x&-2z\\\
-3z^{2}&-10xz-8yz\end{bmatrix}}}{{\longrightarrow}}R(-3)\oplus(R(-3)\oplus
R^{2}(-4))\rightarrow\mbox{\rm Syz}(J_{f})\rightarrow 0,$

where, say, $\epsilon_{1}$ is chosen as the first term in an ordered set of
minimal generators of $\mbox{\rm Syz}(J_{f})$. By Theorem 2.1, the minimal
graded resolution of the Bourbaki ideal $I_{{\epsilon}_{1}}$ is

$0\rightarrow R^{2}(-4)\stackrel{{\scriptstyle\begin{bmatrix}-6z&-9y\\\
-x&-2z\\\ 3z^{2}&-10xz-8yz\end{bmatrix}}}{{\longrightarrow}}R^{2}(-3)\oplus
R(-2)\rightarrow I_{{\epsilon}_{1}}\rightarrow 0.$

By the above exact sequence, the Hilbert Series of $R/I_{{\epsilon}_{1}}$ is

$\mathrm{HS}_{R/I_{{\epsilon}_{1}}}(t)=\dfrac{1-t^{2}-2t^{3}+2t^{4}}{(1-t)^{3}}=\dfrac{1+2t+2t^{2}}{(1-t)}.$

Therefore, $\mathrm{Bour}(X)=1+2+2=5$ and $\deg(R/J_{f})=8$.

Note that, had we chosen $\epsilon_{2}$ instead, the resulting Bourbaki ideal
$I_{\epsilon_{2}}$ would have the same Betti numbers and shifts, hence the
same value for the Bourbaki degree.

######  Example 2.9.

Let $X=V(f)$ be a sextic curve defined by $f=xy^{4}z+x^{6}+y^{6}$. A
computation with [6] shows that $\mbox{\rm Syz}(J_{f})$ is generated by the
column vectors of the matrix

$\begin{bmatrix}0&-6y^{3}z&-6y^{4}\\\ xy&9x^{4}+y^{2}z^{2}&y^{3}z\\\
-6y^{2}-4xz&-54x^{3}y-4yz^{3}&36x^{4}-4y^{2}z^{2}\end{bmatrix}$

with graded minimal resolution

$0\rightarrow R(-5)\stackrel{{\scriptstyle\begin{bmatrix}9x^{3}\\\ -y\\\
z\end{bmatrix}}}{{\longrightarrow}}R(-2)\oplus R^{2}(-4)\rightarrow\mbox{\rm
Syz}(J_{f})\rightarrow 0.$

Choosing $\epsilon$ as the unique column vector of degree $2$, by Theorem 2.1
the minimal graded resolution of $I_{\epsilon}$ is

$0\rightarrow R(-2)\stackrel{{\scriptstyle\begin{bmatrix}z\\\ -y\\\
\end{bmatrix}}}{{\longrightarrow}}R^{2}(-1)\rightarrow I_{\epsilon}\rightarrow
0.$

Therefore, the Bourbaki scheme ${\rm Proj}\,(R/I_{\epsilon})$ is a reduced and
irreducible complete intersection of degree one, hence $\mathrm{Bour}(X)=1$
and $\deg(R/J_{f})=18$.

This is an example falling within a large class of curves introduced by Dimca
and co-authors under the name nearly free curves ([8]) – see Proposition 2.18
for further details. It is worth noting that the Bourbaki scheme associated
with the other syzygies is not even reduced and has degree $3$, pointing to
the fact that the Bourbaki invariants considered in [22] may be different and
possibly less tight.

###  2.3. A distinguished upper bound

We give an upper bound for the Bourbaki degree solely in terms of the initial
degree of the syzygies of the gradient ideal of a curve.

######  Theorem 2.10.

Let $X=V(f)$ be a singular curve of degree $d+1$ and let
$e=\mathrm{indeg}(\mbox{\rm Syz}(J_{f}))$. Then

(13) $\operatorname{Bour}(X)\leq e^{2}.$

###### Proof.

Dualize into $R$ the lowest horizontal exact sequence in (6), with
$M_{\epsilon}=I_{\epsilon}(e-d)$. Observing that
$\operatorname{Ext}^{2}(R^{3}/\tilde{\epsilon}(R(-e)),\,R)=0$ because
$R^{3}/\tilde{\epsilon}(R(-e))$ has homological dimension one, entails the
short exact sequence

(14)
$\operatorname{Ext}^{1}(R^{3}/\tilde{\epsilon}(R(-e)),\,R)\longrightarrow\operatorname{Ext}^{1}(I_{\epsilon}(e-d),R)\longrightarrow\operatorname{Ext}^{2}(J_{f},R)\rightarrow
0.$

Since $J_{f}$ is Cohen–Macaulay locally on the punctured spectrum,
$\operatorname{Ext}^{2}(J_{f},R)$ vanishes locally on the punctured spectrum.

Claim 1:
$\operatorname{Ext}^{1}(R^{3}/\tilde{\epsilon}(R(-e)),\,R)\simeq(R/H)(e)$,
where $H\subset R$ is the ideal generated by the coordinates of $\epsilon$
regarded as column vector.

To see this, dualize into $R$ the mid vertical exact sequence in (6) to get
the exact sequence

$0\rightarrow(R^{3}/\tilde{\epsilon}(R(-e)))^{*}\rightarrow
R^{3}\stackrel{{\scriptstyle\tilde{\epsilon}^{t}}}{{\rightarrow}}R(e)\rightarrow\operatorname{Ext}^{1}(R^{3}/\tilde{\epsilon}(R(-e)),\,R)\rightarrow
0,$

where the image of the map $\tilde{\epsilon}^{t}$ is $H(e)$, with $H$ as
explained. Thus, we obtain the following short exact sequence

$0\rightarrow H(e)\rightarrow
R(e)\rightarrow\operatorname{Ext}^{1}(R^{3}/\tilde{\epsilon}(R(-e)),\,R)\rightarrow
0,$

giving
$\operatorname{Ext}^{1}(R^{3}/\tilde{\epsilon}(R(-e)),\,R)\simeq(R/H)(e)$
canonically.

So much for the claim. We then have:

Claim 2: Every minimal prime of $R/I_{\epsilon}$ is a minimal prime of $R/H$,
and moreover, for any such prime one has a surjective map

(15) $(R/H)(e)_{\wp}\twoheadrightarrow R/I_{\epsilon}(e-d)_{\wp}.$

Indeed, substituting the result of Claim 1 in (14) and localizing in the
punctured spectrum, we obtain surjective maps

$(R/H)(e)_{\wp}\twoheadrightarrow\operatorname{Ext}^{1}(I_{\epsilon}(e-d),R)_{\wp}$

throughout all primes $\wp\neq{\mathfrak{m}}$. In particular, this holds if
$\wp$ is a minimal prime of $R/I_{\epsilon}$. In addition, it forces any such
prime to be a minimal prime of $R/H$; in fact, it lies on the support of
$R/I_{\epsilon}$, while similarly, the support of $R/H$ in codimension two is
the set of minimal primes of $R/H$.

Next, recall that the module $\mbox{\rm Syz}(R/J_{f})$ is free of rank two
locally on the punctured spectrum. Therefore, the Bourbaki sequence implies
that the Bourbaki ideal $I_{\epsilon}(e-d)$ is generically a complete
intersection (of two generators). Then, as is well-known,
$\operatorname{Ext}^{1}(I_{\epsilon}(e-d),R)_{\wp}\simeq
R/I_{\epsilon}(e-d)_{\wp}$ for every minimal prime of the latter.

This takes care of Claim 2.

Now, note the length inequality from (15)

$\ell((R/H)(e)_{\wp})\geq\ell(R/I_{\epsilon}(e-d)_{\wp}).$

As a result, we get $\deg(R/I_{\epsilon}(e-d))\leq\deg(R/H)$ via the
associativity formula for degrees. The stated bound now follows since
$\operatorname{Bour}(X)=\deg(R/I_{\epsilon}(e-d))$ and $\deg(R/H)\leq e^{2}$
because $H$ is generated by three forms of degree $e$.

######  Corollary 2.11.

[8, Corollary 1.4 (ii)] Notation as above. If $e=1$ then the curve $X$ is
either a free divisor or a nearly free curve.

###### Proof.

By Theorem 2.10, $\operatorname{Bour}(X)=0$ or $1$. As accounted for before,
these values characterize the two types of curves in the statement.

######  Example 2.12.

Perhaps the simplest example where the above upper bound is attained is the
higher cusp singularity defined by the polynomial $f=y^{d}z+x^{d+1}$. One has
$J_{f}=(x^{d},y^{d-1}z,y^{d})$. An application of the Buchsbaum–Eisenbud
acyclicity criterion yields the minimal resolution

$0\rightarrow
R(-2d-1)\stackrel{{\scriptstyle\varphi_{2}}}{{\longrightarrow}}R(-d-1)\oplus
R^{2}(-2d)\stackrel{{\scriptstyle\varphi_{1}}}{{\longrightarrow}}R^{3}(-d)\rightarrow
R\rightarrow R/J_{f}\rightarrow 0,$

where $\varphi_{1}$ and $\varphi_{2}$ are respectively defined by the matrices

$\begin{bmatrix}0&-y^{d}&-y^{d-1}z\\\ y&0&x^{d}\\\
z&x^{d}&0\end{bmatrix}\>\text{\rm and}\>\begin{bmatrix}x^{d}\\\ z\\\
-y\end{bmatrix}.$

By Theorem 2.1 (c), it follows that the Bourbaki ideal corresponding to the
initial degree ($=1$) is generated by $y,z$, hence $\operatorname{Bour}(X)=1$.
We note that cusp singularities are a special case of nearly free curves. In
fact, as we will show in Proposition 2.18, the value
$\operatorname{Bour}(X)=1$ characterizes such curves. This example is actually
a member of the curve family studied in [30, Example and Theorem 3.5]. In the
latter the above bound is again attained for singular members and, moreover,
the minimal syzygy degree $e$ is quite arbitrary.

######  Example 2.13.

By Claim 2 in the proof of the above theorem, one has an inclusion of radicals
$\sqrt{H}\subset\sqrt{I_{\epsilon}(e-d)}$. The nature of this inclusion is not
independent of the choice of $\epsilon$. The following example illustrates its
behavior:

$f=x^{2}y^{2}+x^{2}z^{2}+y^{2}z^{2}+2xyz((1/2)x+y+z).$

A run of [6] yields that $\mbox{\rm Syz}(R/J_{f})$ admits two independent
minimal generators in the initial degree $2$ such that, for one of them, the
above inclusion of radicals is strict, while for the other one, it is an
equality.

###  2.4. Special results

In this part, we collect a few results for the Bourbaki degree of certain
classes of curves.

The following result characterizes smooth curves in terms of the Bourbaki
degree.

######  Proposition 2.14.

Let $X$ be a reduced curve of degree $d+1\geq 2$. Then
$\mathrm{Bour}(X)=d^{2}$ if and only if $X$ is smooth.

###### Proof.

If $X$ is smooth, then $J_{f}$ is generated by an $R$-sequence, hence ${\rm
Syz}(f)$ is generated by the Koszul relations. Therefore, $e=d$. In this case,
the result follows from (7). Conversely, let $\mathrm{Bour}(X)=d^{2}$ and
assume that $X$ is singular. Then (7) yields $\deg(R/J_{f})=e(e-d)$. Since $e$
is the initial degree of ${\rm Syz}(J_{f})$ then $e\leq d$, hence
$\deg(R/J_{f})\leq 0$, which is impossible with $\dim R/J_{f}=1$.

In the non-smooth case, if $f$ irreducible, one can add:

######  Proposition 2.15.

Let $f\in R=k[x,y,z]$ be a homogeneous polynomial of degree $d+1\geq 2$ and
let $e:={\rm indeg}(\mbox{\rm Syz}(R/J_{f}))$. Supposing that $f$ is
irreducible, one has:

  1. (i)

If $f$ is not a free divisor then $\operatorname{Bour}(X)\geq e(e-d)+d$.

  2. (ii)

If $f$ is a free divisor then $e\geq 2$ and $d\geq 4;$ in particular, there
are no irreducible free divisors of degree $2\leq d+1\leq 4$. Moreover, if
$d+1=5$ then $e=2$.

  3. (iii)

If $d\geq 2$ and $\operatorname{Bour}(X)=1$ then $e<d$.

###### Proof.

(i) We have $\deg(R/J_{f})=\tau(X)\leq\mu(X)\leq d(d-1)$, where the last
inequality is stated in [7, Chapter 4, Section 4, (4.5)] when $f$ is
irreducible. Now apply (7).

(ii) If $f$ is free then by Theorem 2.1 (a), one has
$\deg(R/J_{f})=d^{2}+e(e-d)$ and, on the other hand, as above,
$\deg(R/J_{f})\leq d(d-1)$. Therefore, $e\geq 2$.

For the additional statement, one can verify that $d\leq 3$ leads to a
contradiction with the inequality $e(e-d)+d\leq 0$.

(iii) Again, by the assumption and the above inequality, we have $e(e-d)+d\leq
1$. Another calculation with $e\geq d$ (hence, $e=d$) and $d\geq 2$ leads to a
contradiction.

######  Remark 2.16.

Note that (ii) may fail if $f$ is not irreducible, e.g., $f=xyz$.

Recall the following class of curves introduced in [11].

######  Definition 2.17.

A reduced curve $X=V(f)$ of degree $d+1$ is called nearly free if $R/J_{f}$
has a graded minimal free resolution of the form

(16) $0\rightarrow R(-d-a_{2}-1)\rightarrow R(-d-a_{1})\oplus
R^{2}(-d-a_{2})\rightarrow R^{3}(-d)\rightarrow R\rightarrow
R/J_{f}\rightarrow 0,$

for some integers $1\leq a_{1}\leq a_{2}$.

In analogy to the case of free curves, these shifts are called the exponents
of $X$, and satisfy the equality $a_{1}+a_{2}=d+1$ – in particular, one has
$a_{1}\leq(d+1)/2$.

The following result characterizes nearly free curves in terms of the Bourbaki
degree.

######  Proposition 2.18.

Let $X=V(f)$ be a reduced curve of degree $d+1$. Then
$\operatorname{Bour}(X)=1$ if and only if $X$ is nearly free.

###### Proof.

Assume that $\operatorname{Bour}(X)=1$. Since a Bourbaki ideal is a
codimension two perfect ideal, the Bourbaki ideal $I_{\epsilon}$ of $\mbox{\rm
Syz}(J_{f})$ associated to a syzygy $\epsilon$ of degree
$e=\mathrm{indeg}(\mbox{\rm Syz}(R/J_{f}))$ is a complete intersection
generated by two linear forms and has the graded minimal free resolution

$0\rightarrow R(-2)\rightarrow R(-1)\oplus R(-1)\rightarrow
I_{\epsilon}\rightarrow 0.$

Now we apply part (c) of Theorem 2.1. One has

$0\rightarrow R((e-d)-2)\rightarrow R^{2}((e-d)-1)\oplus
R(-e)\rightarrow\mbox{\rm Syz}(R/J_{f})\rightarrow 0,$

which forces that $d-e+1\geq e$. Therefore, we get

$0\rightarrow R(-d-(d-e+1)-1)\rightarrow R(-d-e)\oplus
R^{2}(-d-(d-e+1))\rightarrow R^{3}(-d)\rightarrow R\rightarrow
R/J_{f}\rightarrow 0,$

which proves the assertion.

Conversely, assume that $X$ is nearly free with exponent $a_{1}\leq a_{2}$.
Then by (16), the $R$-module $\mbox{\rm Syz}(R/J_{f})$ has the graded minimal
resolution of the form

$0\rightarrow R(-a_{2}-1)\stackrel{{\scriptstyle\begin{bmatrix}h\\\
\ell_{1}\\\ \ell_{2}\end{bmatrix}}}{{\rightarrow}}R(-a_{1})\oplus
R^{2}(-a_{2})\rightarrow\mbox{\rm Syz}(R/J_{f})\rightarrow 0,$

where $\ell_{1},\ell_{2}$ are linear forms. By Theorem 2.1 (c), choosing the
unique generator $\epsilon$ of standard degree $a_{1}$, we get the minimal
graded resolution of $I_{\epsilon}$

$0\rightarrow
R(-a_{2}-(a_{1}-d)-1)\stackrel{{\scriptstyle\begin{bmatrix}\ell_{1}\\\
\ell_{2}\end{bmatrix}}}{{\rightarrow}}R^{2}(-a_{2}-(a_{1}-d))\rightarrow
I_{\epsilon}\rightarrow 0,$

which forces that $a_{2}+(a_{1}-d)=1$ as $I_{\epsilon}$ is a codimension $2$
perfect ideal. Thus, $I_{\epsilon}$ is generated by two linear forms, hence
$\operatorname{Bour}(X)=\deg(R/I_{\epsilon})=1$.

Since the definition of nearly free curves is given in terms of the shape of
the corresponding free resolution, one might expect that larger values of
$\operatorname{Bour}(X)$ are responsible for certain shapes of the resolution
as for $\operatorname{Bour}(X)=1$. This sort of approach has been the main
thread of recent work of Dimca–Sticlaru, by stressing numerical invariants
other than the one here. The following proposition is an example of how one
may proceed one step further with a case embodied in the event of $3$-syzygy
curves as in [13, Proposition 3.1].

######  Proposition 2.19.

Let $X=V(f)$ be a singular singular curve of degree $d+1$ and let $e={\rm
indeg}(\mbox{\rm Syz}(J_{f}))$. Then $\operatorname{Bour}(X)=2$ if and only if
$R/J_{f}$ has a graded minimal free resolution of the form

$\displaystyle 0\rightarrow R(-(d+(d-e+3)))\kern-7.0pt$
$\displaystyle\rightarrow$ $\displaystyle\kern-7.0ptR(-(d+(d-e+2)))\oplus
R(-(d+(d-e+1)))\oplus R(-(d+e))$
$\displaystyle\stackrel{{\scriptstyle\varphi}}{{\rightarrow}}$ $\displaystyle
R^{3}(-d)\rightarrow R\rightarrow R/J_{f}\rightarrow 0,$

where $e\leq d+1/2$.

###### Proof.

Suppose that $R/J_{f}$ has the graded minimal resolution as stated. Then,
$\mbox{\rm Syz}(J_{f})={\rm im}(\varphi)$ has the graded minimal free
resolution

$0\rightarrow R(-(d-e+3))\rightarrow R(-(d-e+2))\oplus R(-(d-e+1))\oplus
R(-e)\rightarrow\mbox{\rm Syz}(J_{f})\rightarrow 0.$

By Theorem 2.1 (c) the graded minimal resolution of $I_{\epsilon}$ is of the
form

(17) $0\rightarrow R(-3)\rightarrow R(-2)\oplus R(-1)\rightarrow
I_{\epsilon}\rightarrow 0.$

Hence

$\mathrm{HS}_{R/I_{{\epsilon}}}(t)=\dfrac{1-t-t^{2}+t^{3}}{(1-t)^{3}}=\dfrac{1+t}{(1-t)}.$

Therefore, $\operatorname{Bour}(X)=\deg(R/I_{\epsilon})=2$. Conversely, assume
that the Bourbaki number of $X$ is two. In this degree the codimension two
perfect ideal $I_{\epsilon}$ is necessarily a complete intersection generated
by a linear form and a quadratic form, which implies the graded minimal free
resolution

$0\rightarrow R^{3}(-(d-e+3))\rightarrow R(-(d-e+2))\oplus
R(-(d-e+1))\rightarrow I_{\epsilon}(e-d),$

which forces that $e\leq(d+1)/2$. By the proof of Theorem 2.1 (c), it gives a
minimal free resolution of $\mbox{\rm Syz}(J_{f})$, which extends to the
stated resolution of $R/J_{f}$ by shifting by $-d$.

We note that a reduced curve whose gradient ideal admits a free resolution as
in the above proposition is a special case of both a $3$-syzygy curve and a
one-plus curve – the latter having been characterized in terms of the shape of
the free resolution of the corresponding gradient ideal in [13, Theorem 2.3],
partially drawing on [20].

It is quite obvious that, in terms of the Bourbaki degree, the result of [13,
Proposition 3.1] reads to the effect that a $3$-syzygy curve is characterized
by $\operatorname{Bour}(X)$ being the product of two suitable integers $\geq
1$. It remains the question as to when such integers validate the effective
existence of a corresponding $3$-syzygy curve. For the case where these are
equal integers, the singular curves in the family in [30, Example] respond to
this question.

######  Remark 2.20.

The inverse question, as to whether or when a given homogeneous codimension
two Cohen–Macaulay ideal $I\subset R$ is the Bourbaki ideal of some
torsionfree maximal Cohen–Macaulay module $M$, has been solved in [21], under
the additional requirement that the sought Bourbaki sequence be of the form
$0\rightarrow F\rightarrow M\rightarrow I\rightarrow 0$, where $F$ is free of
rank equal to the type of $I$ (i.e., the number of columns in an $(n+1)\times
n$ matrix defining $I$). In addition, the solution is essentially unique.

To relate to our setup, we have to observe two points: first, in our typical
Bourbaki sequence, the departing module $\mbox{\rm Syz}(R/J_{f}$ is
torsionfree – hence, has maximal dimension – but it has depth $2$, so is not
Cohen–Macaulay. Thus, an inverse question would have to change gear at this
point, possibly renouncing to any uniqueness statement. Second, the free
kernel in our Bourbaki sequence has rank one. Thus, to engage in the above,
even forgetting about the Cohen–Macaulay property, we would impose that $I$
has type one, which means that $I$ is a complete intersection of two forms.

Then, in this situation, pretty much like in the proof of Proposition 2.18
(first direction), we can try to reconstruct a certain free resolution and
expect it to be the resolution of a torsionfree $R$-module isomorphic to
$\mbox{\rm Syz}(R/J_{f})$, for some $f\in R$. Though it sounds quite loose,
this is what is happening in the proof of the theorem when the two forms are
linear.

Of course, in our setup, the natural question would be as to when a given
homogeneous codimension two Cohen–Macaulay ideal $I\subset R$ is the Bourbaki
ideal of some reduced curve $X$, for a suitable generating syzygy degree. At
the moment, we even lack some natural necessary conditions.

##  3\. Lower bounds to the Bourbaki degree

For any two positive integers $d,e$, one has
$d^{2}+e(e-d)>d^{2}-2ed+e^{2}=(d-e)^{2}\geq 0$. Therefore, (7) implies that
$d^{2}+e(e-d)$ is an upper bound for the degree of a Bourbaki ideal associated
with any choice among the minimal generators of $\mbox{\rm Syz}(J_{f})$. In
particular, the Bourbaki degree of a plane curve of degree $d+1$ is bounded
above by $d^{2}+e(e-d)$, where $e={\rm indeg}(\mbox{\rm Syz}(J_{f}))$ – note
that this bound bounces between the two extremes $d^{2}$ and $d(d-1)+1$.

In this section, we deal with finding a corresponding lower bound.

###  3.1. Milnor and Tjurina numbers

We provide a brief retrospection of the Milnor and Tjurina numbers in the case
of a reduced singular curve $X=V(f)\subseteq{\mathbb{P}}^{2}$ of degree
$d+1\geq 2$.

Let $p\in\mathrm{Sing}(X)$. Viewing $p$ as a point of ${\mathbb{P}}^{2}$, its
ideal is a minimal prime $\wp$ of the one-dimensional ring $R/J_{f}$.Thus,
assuming that $k$ is algebraically closed, $\wp$ is generated by two linear
forms. By a projective transformation, assume, say, that $p=[0:0:1]$.
Considering the affine chart $U_{z}={\mathbb{A}}_{k}^{2}$ with coordinate ring
$A=k[T,U]=k[x/z,y/z]$, $p$ corresponds to the point whose ideal is the maximal
ideal ${\mathfrak{p}}:=(T,U)$. Set accordingly, $F(T,U):=f(x/z,y/z,1)$, and
let $J_{F}=(F_{X},F_{Y})$ be the gradient ideal of $F$ in $k[T,U]$ and
$\tilde{J}_{F}:=(F,F_{T},F_{U})$ its “full” Jacobian ideal.

######  Definition 3.1.

The Milnor number (respectively, the Tjurina number) $\mu_{{\mathfrak{p}}}(F)$
(respectively, $\tau_{{\mathfrak{p}}}(F)$) of $F\in k[T,U]$ at
${\mathfrak{p}}$ is $\dim_{k}A_{{\mathfrak{p}}}/(J_{F})_{{\mathfrak{p}}}$
(respectively, $\dim_{k}A_{{\mathfrak{p}}}/(\tilde{J}_{F})_{{\mathfrak{p}}}$.
We often write $\mu_{{\mathfrak{p}}}(X)$ (respectively,
$\tau_{{\mathfrak{p}}}(X)$) if no confusion arises.

We call $M_{{\mathfrak{p}}}(F):=A_{{\mathfrak{p}}}/(J_{F})_{{\mathfrak{p}}}$
and
$T_{{\mathfrak{p}}}(F):=A_{{\mathfrak{p}}}/(\tilde{J}_{F})_{{\mathfrak{p}}}$
the Milnor local algebra and the Tjurina local algebra at ${\mathfrak{p}}$,
respectively. The total Milnor number (respectively, total Tjurina number) of
$V(f)$ is the summation of the local Milnor numbers (respectively, local
Tjurina numbers) at all singular points of $X=V(f)$. The latter will be
denoted by $\mu(X)$ and $\tau(X)$, respectively.

The following is well-known, but we include a proof:

######  Proposition 3.2.

($k$ is algebraically closed and char$(k)$ does not divide $d+1$). Let $f\in
R$ be a reduced form of degree $d+1$ defining a singular curve. Then, with the
above notation, one has

(18) $\deg(R/J_{f})=\tau(X).$

###### Proof.

By a linear change of coordinates, we may assume that $z\notin\wp$, for every
minimal prime $\wp$ of $R/J_{f}$. By the associativity formula, one has:

$\displaystyle\deg(R/J_{f})$ $\displaystyle=$
$\displaystyle\sum_{\wp}\ell(R_{\wp}/\mathcal{P}_{\wp}),\;\text{\rm where
$\mathcal{P}$ is the ${\wp}$-primary component}$ $\displaystyle=$
$\displaystyle\sum_{\wp}\ell(R_{\wp}/(J_{f})_{\wp})=\sum_{\wp}\ell(R_{\wp}/(f_{x},f_{y},f_{z})_{\wp})$
$\displaystyle=$ $\displaystyle\sum_{\wp}\ell(R_{\wp}/(f,f_{x},f_{y})_{\wp})$
$\displaystyle=$ $\displaystyle\tau(X)$

where $z\notin{\wp}$ and the Euler relation $(d+1)f=f=xf_{x}+yf_{y}+zf_{z}$
imply that $f_{z}\in(f,f_{x},f_{y})$.

The following consequence gives another proof of [16, Theorem 3.2].

######  Theorem 3.3.

Let $X=V(f)$ be a reduced singular curve of degree $d+1$ and let
$e=\mathrm{indeg}(\mbox{\rm Syz}(J_{f}))$. Then

$d(d-e)\leq\tau(X)\leq d^{2}+e(e-d).$

###### Proof.

Since $V(f)$ is singular, by Proposition 3.2 one can replace $\tau(X)$ by
$\deg(R/J_{f})$. Then, the upper bound follows form (7), and the lower bound
follows form Theorem 2.10.

A natural question is whether one can compute the Bourbaki degree of a curve
in terms of the Bourbaki degrees of its components.

Next is the picture in the case of two smooth curves.

######  Proposition 3.4.

If $X_{1}=V(f_{1})$ and $X_{2}=V(f_{2})$ are smooth curves intersecting
transversally, then

$\operatorname{Bour}(X_{1}\cup
X_{2})=\operatorname{Bour}(X_{1})+\operatorname{Bour}(X_{2})+(\deg
f_{1}-1)(\deg f_{2}-1).$

###### Proof.

Because of transversality, the intersection multiplicity of the two curves at
every intersection point is one. Then, by Bézout theorem and (18), one has
$\deg(R/J_{f_{1}f_{2}})=\deg f_{1}\deg f_{2}$. On the other hand, the singular
points of $X_{1}\cup X_{2}$ are nodal. Now, for a non-irreducible nodal curve
$V(g)$, the initial degree of $\mbox{\rm Syz}(R/J_{g})$ is $\deg f-2$ ([10,
Example 2.2 (i)]). Then the result follows from (2.1).

###  3.2. Quasi-homogeneous singularities

We refer to [36, Section 2.2] for the details of this subsection.

Quite generally, a polynomial $f\in k[x_{1},\ldots,x_{n}]$ is Eulerian if
$f\in J_{f}=(f_{x_{1}},\ldots,f_{x_{n}})$. The Euler relation says that a
homogeneous polynomial whose degree does not divide char$(k)$ is Eulerian. The
notion of a quasi-homogeneous or weighted homogeneous polynomial is as near as
one can get to the Eulerian property of a homogeneous polynomial.

######  Definition 3.5.

$f$ is said to be quasi-homogeneous if one has

$\lambda f=\sum_{i=1}^{n}w_{i}x_{i}f_{x_{i}},$

for some integer $\lambda>0$ not dividing char$(k)$ and integers $w_{i}>0$
whose $\gcd$ is one.

The integers $w_{i}$ are called (integer) weights, while $w_{i}/\lambda$ are
called rational weights.

We will henceforth only consider quasi-homogeneous singularities in affine
dimension two. Let $X=V(f)\subset{\mathbb{P}}^{2}$ as before, with $f$
reduced. A singular point $p\in\operatorname{Sing}{X}$ is called quasi-
homogeneous singularity if $\mu_{p}(X)=\tau_{p}(X)$. If $X$ has only quasi-
homogeneous singularities, then $\mu(X)=\tau(X)$. The Arnold exponent of a
quasi-homogeneous singularity $p\in X$ with rational weights
$(w_{1}/\lambda,w_{2}/\lambda)$ is defined to be
$\alpha_{F}=(w_{1}+w_{2})/\lambda$.

Notable examples of quasi-homogeneous singularities are the simple
singularities.

We first list a couple of results to be used next.

######  Theorem 3.6.

([10, Theorem 2.1]) Let $X$ be a reduced curve of degree $d+1$ having only
quasi-homogeneous singularities. Then $\mathrm{indeg}(\mbox{\rm
Syz}(R/J_{f}))\geq\alpha(d+1)-2$, where $\alpha$ is the minimum of the Arnold
exponents of the singular points of $X$.

The following estimate holds for a reduced plane curve of degree $d+1$ :

(19) $|\operatorname{Sing}V(f)|\leq d(d+1)/2.$

This follows from the inequalities

$2\,|\operatorname{Sing}V(f)|\leq\sum_{p\in|\operatorname{Sing}V(f)|}m_{p}(f)(m_{p}(f)-1)=\sum_{p\in|\operatorname{Sing}V(f)|}m_{p}(f)m_{p}(f_{x})\leq
d(d+1),$

where $m_{p}(f)$ (respectively, $m_{p}(f_{x})$) denotes the multiplicity of
$p$ on $V(f)$ (respectively, on $V(f_{x})$). For the last inequality see [18,
Section 5.3, Corollary 1].

Moreover, if $f$ is irreducible then ([19, Lemma 18.5] shows:

(20) $|\operatorname{Sing}V(f)|\leq d(d-1)/2.$

In the sequel, by a nodal curve we understand a reduced curve whose singular
points are simple nodes, one of the well-known simple singularities.

######  Proposition 3.7.

Let $X=V(f)\subset{\mathbb{P}}^{2}$ denote a reduced nodal curve of degree
$d+1\geq 2$. One has:

  1. (i)

$e:={\rm indeg}(\mbox{\rm Syz}(R/J_{f}))\geq d-1$.

  2. (ii)

The following bounds hold:

$\mathrm{Bour}(X)\geq\left\\{\begin{array}[]{ll}d(d-1)/2&\mbox{if $e=d$}\\\
d(d-3)/2-1&\mbox{if $e=d-1$}.\end{array}\right.$

If, moreover, $f$ is irreducible then

$\mathrm{Bour}(X)\geq\left\\{\begin{array}[]{ll}d(d+1)/2&\mbox{if $e=d$}\\\
1+d(d-1)/2&\mbox{if $e=d-1$}.\end{array}\right.$

  3. (iii)

If $f$ is irreducible then $X$ is not a free divisor ([32, Theorem 2.1]). If
$X$ is a free divisor then $\deg f\leq 3$ and, moreover, $f=xy$ or else
$f=xyz$ up to a linear change of variables.

###### Proof.

A nodal singularity is quasi-homogeneous with Milnor number one. Therefore,
$\tau(X)=\mu(X)=|\operatorname{Sing}{X}|$, hence
$\deg(R/J_{f})=|\operatorname{Sing}{X}|$ by Proposition 18.

(i) The Arnold exponent of a nodal singularity is one ([10, Example 2.2 (i)]).
Therefore, $e\geq d-1$ by Theorem 3.6.

(ii) Drawing upon (19) one has

$\displaystyle\mathrm{Bour}(X)$ $\displaystyle\geq$ $\displaystyle
d^{2}+e(e-d)-d(d+1)/2=d(d-1)/2+e(e-d)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}d(d-1)/2&\mbox{if $e=d$}\\\
d(d-3)/2-1&\mbox{if $e=d-1$}.\end{array}\right.$

Similarly, if $f$ is irreducible then (20) gives

$\mathrm{Bour}(X)\geq\left\\{\begin{array}[]{ll}d(d+1)/2&\mbox{if $e=d$}\\\
1+d(d-1)/2&\mbox{if $e=d-1$}.\end{array}\right.$

(iii) If $f$ is irreducible, with $\operatorname{Bour}(X)=0$, then the
corresponding bounds in (ii) force $d\leq 1$, hence $d=1$ as we assume
$d+1\geq 2$. This is a contradiction as $f$ cannot be both a free divisor and
a smooth curve of degree $\geq 2$. Thus, if $f$ is a free divisor, we can only
use the first set of lower bounds in (ii). The latter forces $d\leq 2$. Thus,
$f$ is either a quadric or else a cubic. Since we are assuming that $X$ is
nodal, then $f$ must be either the union of two distinct lines or else the
union of three non-concurrent lines.

######  Theorem 3.8.

Let $X$ be an irreducible curve of degree $d+1$. Then
$\operatorname{Bour}(X)=d^{2}-1$ if and only if $X$ has a unique singular
point and this is a nodal point.

###### Proof.

Assume that $\operatorname{Bour}(X)=d^{2}-1$. Suppose that $X$ has more than
one singular point. Then $\tau(X)>1$, hence by (11) and (18),, one has

$d^{2}-1=\operatorname{Bour}(X)=d^{2}+e(e-d)-\tau(X)<d^{2}-e(e-d)-1,$

hence $e(e-d)>0$, a contradiction since $e\leq d$.

It follows that $X$ has a unique singular point because otherwise
$\operatorname{Bour}(X)=d^{2}$ by Proposition 2.14.

Now, suppose the unique singular point $p$ of $X$ is not nodal. Then we claim
that $\tau(X)>1$. Indeed, otherwise, $\tau(X)=1$. That is, locally at $p$, the
multiplicity of $(f,J_{f})$ is one. By [25, Theorem 40.6], $(f,J_{f})$ is the
maximal ideal $\mathfrak{m}$ locally at $p$. Since $f\in\mathfrak{m}^{2}$,
this is a contradiction.

Next, by (11), one has $2\leq\tau(X)=e(e-d)+d^{2}-(d^{2}-1)=e(e-d)+1$. Then
$e(e-d)\geq 1$, which is a contradiction.

Now, suppose that $X$ has a unique singular point and this is a nodal point.
Then, $\tau(X)=\mu(X)=1$. By [10, Example 2.2 (i)], the initial degree $e$ of
$\mbox{\rm Syz}(R/J_{f})$ coincides with $d$. Again, by (7), it follows that

$\operatorname{Bour}(X)=e(e-d)+d^{2}-\tau(X)=d^{2}-\mu(X)=d^{2}-1,$

as required.

###  3.3. Relation to a theorem of Dolgachev

Let $X\subset{\mathbb{P}}^{2}$ be a curve of degree $d+1$ defined by the
reduced homogeneous polynomial $f\in R=k[x,y,z]$. The polar degree of $X$,
denoted by $\deg(\nabla(f))$, is the degree of the rational polar map

$\nabla(f)\colon{\mathbb{P}}^{2}\dashrightarrow{\mathbb{P}}^{2},\quad
p\mapsto[f_{x}(p):f_{y}(p):f_{z}(p)].$

By [9, section 3], the polar degree is given by the formula

(22) $\deg(\nabla(f))=d^{2}-\mu(X).$

The curve $X$ is called homaloidal if $\deg(\nabla(f))=1$, i.e. if the polar
map is a Cremona map of ${\mathbb{P}}^{2}$. Now, drawing upon (11), one has

(23) $e^{2}-ed+\deg(\nabla(f))\leq\operatorname{Bour}(X),$

with equality when $X$ has only quasi-homogeneous singularities.

Recall the theorem of Dolgachev [14] which tells us that a reduced homaloidal
curve of ${\mathbb{P}}^{2}$ has degree at most three. Once this is known, it
becomes an easy exercise to verify the types of such homaloidal quadrics and
cubics. In particular, any such cubic is either conjugate to $f=xyz$ or
$f=x(y^{2}-xz)$. In particular, these are free divisors of exponents $(1,1)$.

We isolate the following case of Dolgachev’s theorem, with an argument via the
Bourbaki number:

######  Proposition 3.9.

Let $X=V(f)\subset{\mathbb{P}}^{2}$ stand for a reduced homaloidal curve of
${\mathbb{P}}^{2}$. If the singularities of $X$ are quasi-homogeneous then $X$
is either a free divisor of $\deg f=3$ or a non-degenerate quadric.

###### Proof.

By the above preliminaries, under the present assumptions, one has
$\operatorname{Bour}(X)=e(e-d)+1$. Since lives in the initial degree of
$\mbox{\rm Syz}(R/J_{f})$, then $1\leq e\leq d$. Since
$\operatorname{Bour}(X)\geq 0$, it follows that $e(e-d)\geq-1$. Thus, either,
say, $e(e-d)=-1$., forcing $e=1,d=2$ Moreover, $\operatorname{Bour}(X)=0$,
which means that $X$ is a free divisor, necessarily of exponents $(1,1)$.

Else, $e(e-d)\geq 0$, hence forcefully, $e(e-d)=0$. This means that $e=d$ and
$\operatorname{Bour}(X)=1$. By Proposition 2.18, the latter means that $X$ is
a nearly-free divisor. But, the initial degree of $\mbox{\rm Syz}(R/J_{f})$ is
the degree of a Koszul relation, and there are three of these which are
independent. Since $\mbox{\rm Syz}(R/J_{f})$ is minimally generated by three
elements for a nearly-free curve $f$, we must conclude that $f$ is smooth.
But, for the latter, $\operatorname{Bour}(X)=d^{2}$ (Proposition 2.14) hence
$d^{2}=1$, which means that $f$ is a smooth quadric.

######  Remark 3.10.

Since quasi-homogeneity and the generically complete intersection property are
interchangeable in our context, the above is a rephrasing of [15, Corollary
3.6]. The generalization of either of these two assumptions to arbitrary
dimensions has some impact on the degree of a homaloidal hypersurface (see
[27], [34]).

######  Question 3.11.

  1. (1)

Can one anticipate a rough classification of plane curves of low polar degree
by drawing upon the above relation between the Bourbaki degree and the polar
degree?

  2. (2)

A conjecture of A. Dimca and G. Sticlaru claims that any rational cuspidal
plane curve is either free or nearly free. If $X$ is rational and cuspidal,
then $\deg(\nabla(f))=d$, in particular,
$e^{2}-ed+d\leq\operatorname{Bour}(X).$ Can these data help advance a facet of
that conjecture?

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