.

# Jacobian schemes of conic-line arrangements
and eigenschemes

Valentina Beorchia  Dipartimento di Matematica e Geoscienze, Università di
Trieste, Via Valerio 12/1, 34127 Trieste, Italy<EMAIL_ADDRESS>ORCID
0000-0003-3681-9045 and  Rosa M. Miró-Roig  Facultat de Matemàtiques i
Informàtica, Universitat de Barcelona, Gran Via des les Corts Catalanes 585,
08007 Barcelona, Spain<EMAIL_ADDRESS>ORCID 0000-0003-1375-6547

###### Abstract.

The Jacobian scheme of a reduced, singular projective plane curve is the zero-
dimensional scheme, whose homogeneous ideal is generated by the partials of
its defining polynomial. The degree of such a scheme is called the global
Tjurina number and, if the curve is not a set of concurrent lines, some upper
and lower bounds depending on the degree of the curve and the minimal degree
of a Jacobian syzygy, have been given by A.A. du Plessis and C.T.C. Wall.

In this paper we give a complete geometric characterization of conic-line
arrangenents, with global Tjurina number attaining the upper bound.
Furthermore, we characterize conic-line arrangenents attaining the lower bound
for the global Tjurina number, among all curves with a linear Jacobian syzygy.

As an application, we characterize conic-line arrangenents with Jacobian
scheme equal to an eigenscheme of some ternary tensor, and we study the
geometry of their polar maps.

The first author is a member of GNSAGA of INdAM and is supported by the fund
Università degli Studi di Trieste - FRA 2023.

The second author has been partially supported by the grant
PID2019-104844GB-I00

##  1\. Introduction

The Jacobian scheme $\Sigma_{f}$ of a reduced, singular projective plane curve
$C=V(f)\subseteq\mathbb{P}^{2}$ is the zero-dimensional scheme, whose
homogeneous ideal is generated by the partials of $f$. The degree of such a
scheme is called the global Tjurina number $\tau(C)$ and it is equal to
$(d-1)^{2}$, if $C$ consists of concurrent lines, while in all other cases, a
theorem by A.A. du Plessis and C.T.C. Wall in [21] determines the following
bounds on $\tau(C)$ in terms of the minimal degree $r$ of a syzygy between the
three partials:

$(d-1)(d-r-1)\leq\tau(C)\leq(d-1)(d-r-1)+r^{2}.$

In particular, we have $\tau(C)\leq(d-1)(d-2)+1$.

The scheme structure of Jacobian schemes is, in general, not completely
understood, even in the case of irreducible curves. For instance, a class of
curves attaining the bound above for any $r\geq 2$ is given by some rational
cuspidal curves, as shown in [14, Theorem 1.1]. The set of curves attaining
the maximal bound or one less seems to be very rich, and examples having high
genus and many branches have been given in [3, Theorem 3.9 and Theorem 3.11].
As far as we know, the only characterization result is given in the case of a
linear Jacobian syzygy by A.A. du Plessis and C.T.C. Wall in [22, Proposition
1.1], which states that $r=1$ and only if the curve admits a $1$-dimensional
symmetry, i.e. the curve admits a $1$-dimensional algebraic subgroup of
$PGL_{2}(\mathbb{C})$ as automorphism group.

In this paper we focus on the case of conic-line arrangements with a linear
Jacobian syzygy. Some results concerning conic-line arrangements with only
nodes, tacnodes, and ordinary triple points are given in [11], but in degree
$d\geq 5$ one has $r\geq 2$. The Jacobian syzygies and the global Tjurina
number of conics arrangements of conics belonging to particular pencils can be
found in [23], [25] and [8].

The assumption $r=1$ implies $(d-1)(d-2)\leq\tau(C)\leq(d-1)(d-2)+1$, and any
curve attaining the upper bound is reducible by [22, Proposition 1.3].
Concerning the lower bound, there are irreducible curves, with a linear
syzygy, satisfying $\tau(C)=(d-1)(d-2)$, like for instance some Sebastiani -
Thom rational cuspidal curves (see [13, Proposition 2.11]).

We give a complete geometric characterization of the conic-line arrangements
attaining the two bounds for $r=1$. Concerning the upper bound, examples are
given by line arrangements consisting of the union of $d-1$ concurrent lines
and one general line, as described in [9, Proposition 4.7(5)]. Our first main
result is the following (see Theorem 3.5):

######  Theorem A.

Let $C=V(f)$ be a conic-line arrangement in $\mathbb{P}^{2}$ of degree $d\geq
5$. Then, $\tau(C)=(d-1)(d-2)+1$ if and only if $C$ is either:

  * •

$\mathcal{L}$: a line arrangement with $d-1$ concurrent lines and a general
line;

  * •

$\mathcal{C}_{1}$: a union of conics belonging to a hyperosculating pencil,
that is with base locus supported in a point;

  * •

$\mathcal{C}\mathcal{L}_{1}$: a union of conics belonging to a hyperosculating
pencil and the tangent line in the hyperosculating point;

  * •

$\mathcal{C}\mathcal{L}_{2}$: the union of conics belonging to a bitangent
pencil and a tangent line in one of the bitangency points;

  * •

$\mathcal{C}\mathcal{L}_{3}$: the union of conics belonging to a bitangent
pencil and the two tangent lines in the bitangency points;

  * •

$\mathcal{C}\mathcal{L}_{4}$: the union of conics belonging to a bitangent
pencil, one tangent line in a tangency point, and the line connecting the two
tangency points;

  * •

$\mathcal{C}\mathcal{L}_{5}$: the union of conics belonging to a bitangent
pencil, the two tangent lines in the bitangency points, and the line
connecting the two tangency points.

In the case $\tau(C)=(d-1)(d-2)$, the characterization of conic-line
arrangements is given by the following result (see Theorem 3.6):

######  Theorem B.

Let $C=V(f)$ be a reduced conic-line arrangement in $\mathbb{P}^{2}$ of degree
$d\geq 6$. Then, $\tau(C)=d^{2}-3d+2$ if and only if $C$ is either:

  * •

${\mathcal{C}}_{2}$: a conic arrangement given by the union of conics
belonging to a bitangent pencil;

  * •

${\mathcal{C}}{\mathcal{L}}_{6}$: a conic-line arrangement given by the union
of conics belonging to a bitangent pencil and the line passing through the two
bitangency points.

As a consequence of our first result, we can determine the degree of the polar
map of conic-line arrangements with quasihomogenous singularities, that is
with total Milnor number $\mu(C)$ (see Definition 2.3) equal to the total
Tjurina number. Indeed, recall that, when $C=V(f)$ is not a set of concurrent
lines, the polar map $\nabla f$ associated with $f$ defines a generically
finite rational map $\nabla f:\mathbb{P}^{2}\dasharrow\mathbb{P}^{2}$ of
degree $(d-1)^{2}-\mu(C)$. Therefore, when $\mu(C)=\tau(C)$ and $\tau(C)$ is
maximal, such a degree is minimal and equal to $d-2$. This occurs in cases
$\mathcal{L}$ and $\mathcal{C}\mathcal{L}_{2}$.

Another related problem is a Torelli-type question, posed by Dolgachev and
Kapranov (see [15]), which asks whether the rank $2$ vector bundle of
logarithmic vector fields $\mathcal{T}\langle C\rangle$, given as the kernel
of the map

$(\partial_{x}f,\partial_{y}f,\partial_{z}f):\mathcal{O}^{\oplus
3}_{\mathbb{P}^{2}}(1)\to\mathcal{J}_{f}(d),$

with $\mathcal{J}_{f}$ the sheafyfied Jacobian ideal, determines uniquely the
curve, see also [12] and [7]. In the cases under consideration this result
does not hold.

Finally, we observe that in the case of maximal Tjurina number and linear
syzygy given by three linearly independent forms, the Jacobian schemes
$\Sigma_{f}$ turn out to be also eigenschemes (for the definition see 2.6) of
suitable partially symmetric tensors of order $d-1$. This happens in the cases
$\mathcal{L}$ and $\mathcal{C}\mathcal{L}_{2}$, and we can apply the results
of [20] and [4]. In particular, such schemes arise also as zeroes of a section
of the twisted tangent bundle $\mathcal{T}_{\mathbb{P}^{2}}(d-3)$, and we have
that the blow-up ${\rm
Bl}_{\Sigma_{f}}\mathbb{P}^{2}\subset\mathbb{P}^{2}\times\mathbb{P}^{2}$ is a
complete intersection of the projective bundle
$\mathbb{P}(\mathcal{T}_{\mathbb{P}^{2}})$ and a divisor of bidegree
$(1,d-2)$, the possible contracted curves by the polar map are only lines, and
as a consequence no subscheme of degree $k(d-1)$ is contained in a curve of
degree $k$, for any $2\leq k\leq d-2$. We further specify that in the line
arrangement case $\mathcal{L}$, the contracted lines are precisely the
components of $V(f)$, while in the conic-line arrangement, the only contracted
line is the tangent line appearing in the configuration $V(f)$.

The study of singular curves with a minimal Jacobian syzygy of higher degree
seem to be very involved and wild. We believe that the approach concerning the
study of the fibers of the polar map deserves further investigations.

The techniques involved in our study rely on a result relating the syzygy
module of a product of polynomials, with no common factor, given in [10,
Theorem 5.1 and Corollary 5.3], and on the characterization of the Hilbert-
Burch matrix in the case of particular linear Jacobian syzygies given in [5,
Theorem 3.5]. The proofs of our main results are based on a careful analysis
of the possible Jacobian syzygies of any subcurve of the considered curves.

The organization of the paper is the following: in the next section we recall
definitions and preliminary results regarding Jacobian ideals, Jacobian
sygyzies and eigenschemes of ternary tensors.

In Section 3 we determine the geometric classification of Jacobian schemes
corresponding to Jacobian ideals with a linear syzygy.

Finally, in Section 4, we characterize conic-line arrangements with a Jacobian
scheme, which is also an eigenscheme of some ternary tensor. For such curves
we describe the degree $d-2$ generically finite polar map, characterizing its
contracted curves.

Acknowledgement. Most of this work was done while the first author was a guest
of the Universitat de Barcelona, and she would like to thank the people of the
Departament de Matemàtiques i Informàtica for their warm hospitality.

##  2\. Preliminaries

This section contains the basic definitions and results on Jacobian ideals
associated to reduced singular plane curves as well as on eigenschemes, and it
lays the groundwork for the results in the later sections.

From now on, we fix the polynomial ring $R=\mathbb{C}[x,y,z]$ and we denote by
$C=V(f)$ a reduced curve of degree $d$ in the complex projective plane
$\mathbb{P}^{2}=\operatorname{Proj}(R)$ defined by a homogeneous polynomial
$f\in R_{d}$.

###  2.1. Jacobian ideal of a reduced curve

The Jacobian ideal $J_{f}$ of a reduced singular plane curve $C=V(f)$ of
degree $d$ is definied as the homogeneous ideal in $R$ generated by the 3
partial derivatives $\partial_{x}f$, $\partial_{y}f$ and $\partial_{z}f$. We
denote by ${\rm Syz}(J_{f})$ the graded $R$-module of all Jacobain relations
for $f$, i.e.,

${\rm Syz}(J_{f}):=\\{(a,b,c)\in R^{3}\mid
a\partial_{x}f+b\partial_{y}f+c\partial_{z}f=0\\}.$

We will denote by ${\rm Syz}(J_{f})_{t}$ the homogeneous part of degree $t$ of
the graded $R$-module ${\rm Syz}(J_{f})$; for any $t\geq 0$, we have that
${\rm Syz}(J_{f})_{t}$ is a $\mathbb{C}$-vector space of finite dimension. The
minimal degree of a Jacobian syzygy for $f$ is the integer ${\rm mrd}(f)$
defined to be the smallest integer $r$ such that there is a nontrivial
relation $a\partial_{x}f+b\partial_{y}+c\partial_{z}=0$ among the partial
derivatives $\partial_{x}f$, $\partial_{y}f$ and $\partial_{z}f$ of $f$ with
coefficients $a,b,c\in R_{r}$. More precisely, we have:

${\rm mrd}(f)=min\\{n\in\mathbb{N}\mid{\rm Syz}(J_{f})_{n}\neq 0\\}.$

It is well known that ${\rm mrd}(f)=0$, i.e. the three partials
$\partial_{x}f$, $\partial_{y}f$ and $\partial_{z}f$ are linearly dependent,
if and only if $C$ is a union of lines passing through one point
$p\in\mathbb{P}^{2}$. So, we will always assume that ${\rm mrd}(f)>0$ and one
of our goals will be to give a geometric classification of conic-line
arrangements $C=V(f)$ of degree $d$ with ${\rm mrd}(f)=1$, see Theorems 3.5
and 3.6.

######  Definition 2.1.

Let $C=V(f)$ be a reduced singular plane curve of degree $d$. We say that $C$
is free if the graded $R$-module ${\rm Syz}(J_{f})$ of all Jacobian relations
for $f$ is a free $R$-module, i.e.

(2.1) ${\rm Syz}(J_{f})=R(-d_{1})\oplus R(-d_{2})$

with $d_{1}+d_{2}=d-1$. In this case $(d_{1},d_{2})$ are called the exponents
of $C$.

We say that $C$ is nearly free if the minimal free resolution of ${\rm
Syz}(J_{f})$ looks like:

(2.2) $0\longrightarrow R(-d-d_{2})\longrightarrow R(1-d-d_{1})\oplus
R(1-d-d_{2})^{2}\longrightarrow{\rm Syz}(J_{f})\longrightarrow 0$

with $d_{1}\leq d_{2}$ and $d_{1}+d_{2}=d$.

######  Example 2.2.

(1) The rational cuspidal quintic $C\subset\mathbb{P}^{2}$ of equation
$C=V(f)=V(y^{4}z+x^{5}+x^{2}y^{3})$ is free. Indeed,
$J_{f}=(5x^{4}+2xy^{3},3x^{2}y^{2}+4y^{3}z,y^{4})\subset R$ and it has a
minimal free $R$-resolution of the following type:

$0\longrightarrow R(-6)^{2}\longrightarrow R(-4)^{3}\longrightarrow
J_{f}\longrightarrow 0.$

We have ${\rm mrd}(f)=2$, $\deg(J_{f})=12$ and $C$ is free. (2) The rational
cuspidal quintic $C\subset\mathbb{P}^{2}$ of equation $C=V(f)=V(y^{4}z+x^{5})$
is nearly free. Indeed, $J_{f}=(5x^{4},4y^{3}z,y^{4})\subset R$ and it has a
minimal free $R$-resolution of the following type:

$0\longrightarrow R(-9)\longrightarrow R(-5)\oplus R(-8)^{2}\longrightarrow
R(-4)^{3}\longrightarrow J_{f}\longrightarrow 0.$

We have ${\rm mrd}(f)=1$, $\deg(J_{f})=12$ and $C$ is not free but it is
nearly free.

(3) The nodal quintic $C\subset\mathbb{P}^{2}$ of equation
$C=V((x^{2}+y^{2}+z^{2})(x^{3}+y^{3}+z^{3}))=0$ is neither free nor nearly
free. Indeed,
$J_{f}=(5x^{4}+3x^{2}(y^{2}+z^{2})+2x(y^{3}+z^{3}),2x^{3}y+3x^{2}y^{2}+5y^{4}+3y^{2}z^{2}+2yz^{3},2x^{3}z+2y^{3}z+3x^{2}z^{2}+3y^{2}z^{2}+5z^{4})\subset
R$ and it has a minimal free $R$-resolution of the following type:

$0\longrightarrow R(-9)\oplus R(-10)\longrightarrow R(-7)\oplus
R(-8)^{3}\longrightarrow R(-4)^{3}\longrightarrow J_{f}\longrightarrow 0.$

We have ${\rm mrd}(f)=3$ and $\deg(J_{f})=6$.

In general, the condition that a reduced singular curve $C=V(f)$ in
$\mathbb{P}^{2}$ is free is equivalent to the Jacobian ideal $J_{f}$ of $f$
being arithmetically Cohen-Macaulay of codimension two; such ideals are
completely described by the Hilbert-Burch theorem [16]: if $I=\langle
g_{1},\ldots,g_{m}\rangle\subset R$ is a Cohen-Macaulay ideal of codimension
two, then $I$ is defined by the maximal minors of the $(m+1)\times m$ matrix
of the first syzygies of the ideal $I$. Combining this with Euler’s formula
for a homogeneous polynomial, we get that a free curve $C=V(f)$ in
$\mathbb{P}^{2}$ has a very constrained structure: $f=\det(M)$ for a $3\times
3$ matrix $M$, with one row consisting of the 3 variables, and the remaining
$2$ rows are the minimal first syzygies of $J_{f}$.

Free curves are related with the total Tjurina number. We first recall some
notions from singularity theory.

Let $C=V(f)\subset\mathbb{A}^{2}$ be a reduced, not necessarily irreducible,
plane curve and fix a singular point $p\in C$. Let $\mathbb{C}\\{x,y\\}$
denote the ring of convergent power series.

######  Definition 2.3.

The Milnor number of a reduced plane curve $C=V(f)$ at $(0,0)\in C$ is

$\mu_{(0,0)}(C)=\dim\mathbb{C}\\{x,y\\}/\langle\partial_{x}f,\partial_{y}f\rangle.$

The Tjurina number of a reduced plane curve $C=V(f)$ at $(0,0)\in C$ is

$\tau_{(0,0)}(C)=\dim\mathbb{C}\\{x,y\\}/\langle\partial_{x}f,\partial_{y}f,f\rangle.$

To define $\mu_{p}(C)$ and $\tau_{p}(C)$ for an arbitrary point $p$, translate
$p$ in the origin.

We clearly have $\tau_{(0,0)}(C)\leq\mu_{(0,0)}(C)$. For a projective plane
curve $C=V(f)\subset\mathbb{P}^{2}$, it holds:

$\tau(C):=\deg J_{f}=\sum_{P\in Sing(C)}\tau_{p}(C)$

where $J_{f}$ is the Jacobian ideal. We call to $\tau(C)$ the total Tjurina
number of $C$.

A nice result of du Plessis and Wall gives upper and lower bounds for the
total Tjurina number $\tau(C)$ of a reduced plane curve
$C=V(f)\subset\mathbb{P}^{2}$ in terms of its degree $d$ and the minimal
degree ${\rm mrd}(f)$ of a syzygy of its Jacobian ideal $J_{f}$, and relates
the freeness of a curve with $\tau(C)$. More precisely, we have:

######  Proposition 2.4.

Let $C=V(f)$ be a reduced singular plane curve of degree $d$ and let $r:={\rm
mrd}(f)$. It holds:

$(d-1)(d-r-1)\leq\tau(C)\leq(d-1)(d-r-1)+r^{2}.$

Moreover, if $\tau(C)=(d-1)(d-r-1)+r^{2}$, then the curve $C$ is free, and
such as condition is also sufficient if $d>2r$.

###### Proof.

See [21, Theorem 3.2] and [6, Corollary 1.2]. ∎

Next result will play an important role in next section. It relates the
minimal degree ${\rm mrd}(f)$ of a syzygy of the Jacobian ideal of a reducible
plane curve $C=C_{1}\cup C_{2}=V(f_{1}f_{2})$ with the minimal degrees ${\rm
mrd}(f_{1})$ and ${\rm mrd}(f_{2})$ of a Jacobian syzygy of $C_{1}=V(f_{1})$
and $C_{2}=V(f_{2})$, respectively. Observe that the syzygy module can be
identified with the module of derivations killing the polynomial $g$, that is
the submodule $D_{0}(g)$ of the free $R$-module
$D(g)=\\{\delta=a\partial_{x}+b\partial_{y}+c\partial_{z}\ :\ a,b,c\in R\\}$
of $\mathbb{C}$-derivations of the polynomial ring $R$ annihilated by $g$:

$D_{0}(g)=\\{\delta\in D(g)\ :\ \delta g=0\\}.$

In the case of a smooth curve $V(g)$, the syzygy module is trivial, so we will
indeed consider the module $D_{0}(g)$ instead of ${\rm Syz}(J_{g})$.

######  Theorem 2.5.

Let $C_{i}=V(f_{i})$ for $i=1,2$ be two reduced curves in $\mathbb{P}^{2}$
without common irreducible components. Set $d_{i}=\deg f_{i}$ and $r_{i}={\rm
mrd}(f_{i})$ for $i=1,2$. Let $C=V(f_{1}f_{2})$ be the union of $C_{1}$ and
$C_{2}$, let $d=d_{1}+d_{2}=\deg f$ and $r={\rm mrd}(f)$. Then it holds:

  1. (1)

If $\delta_{1}\in D_{0}(f_{1})$, then

$\delta=f_{2}\delta_{1}-\frac{1}{d}\delta_{1}(f_{2})\ E\in D_{0}(f),$

where $E=x\partial_{x}+y\partial_{y}+z\partial_{z}$ denotes the Euler
derivation;

  2. (2)

$D_{0}(f)\subset D_{0}(f_{1})\cap D_{0}(f_{2})$; more precisely, for
$\delta\neq 0$, one has $\delta\in D_{0}(f)$ if and only if $\delta$ can be
written in a unique way in the form $\delta=h\ E+\delta_{1}=-h\ E+\delta_{2}$,
where $h$ is a suitable homogeneous polynomial and $\delta_{j}\in
D_{0}(f_{j})$ are non-zero for $j=1,2$.

  3. (3)

In particular, we have

$\max(r_{1},r_{2})\leq r\leq\min(r_{1}+d_{2},r_{2}+d_{1}).$

and $r$ is the minimal integer $t$ such that either $D_{0}(f_{1})_{t}\cap
D_{0}(f_{2})_{t}\neq 0$ or $D_{0}(f_{1})_{t}+D_{0}(f_{2})_{t}$ contains a non-
zero multiple of the Euler derivation $E$.

###### Proof.

See [10, Theorem 5.1] and [10, Corollary 5.3]. ∎

###  2.2. Eigenschemes in $\mathbb{P}^{2}$

Since we shall investigate whether a Jacobian scheme is also an eigenscheme of
some tensor, we conclude the preliminary section with the basic definitions
and results concerning tensor eigenschemes.

There are several notions of eigenvectors and eigenvalues for tensors, as
introduced independently in [19] and [24]. Here we focus our attention on the
algebraic-geometric point of view. We choose a basis for $\mathbb{C}^{3}$, we
identify a partially symmetric tensor $T$ with a triple of homogeneous
polynomials of degree $d-2$ and we describe the eigenpoint of a tensor $T$
algebraically by the vanishing of the minors of a homogeneous matrix. More
precisely, we have:

######  Definition 2.6.

Let $T=(g_{1},g_{2},g_{3})\in(Sym^{d-2}\mathbb{C}^{3})^{\oplus 3}$ be a
partially symmetric tensor. The eigenscheme of $T$ is the closed subscheme
$E(T)\subset\mathbb{P}^{2}$ defined by the $2\times 2$ minors of the
homogeneous matrix

(2.3) $M=\begin{pmatrix}x&y&z\\\ g_{1}&g_{2}&g_{3}\end{pmatrix}.$

If $T$ is general, then $E(T)$ is a $0$-dimensional scheme (see, for instance,
[1]). Moreover, by the Hochster-Eagon Theorem [18], the coordinate ring
$R/I(E(T))$ is a Cohen-Macaulay ring, and as a consequence, the homogeneous
ideal $I(E(T))$ is saturated. Hence $E(T)$ is a standard determinantal scheme.
When the tensor $T$ is symmetric, i.e., there is a homogeneous polynomial $f$
and $g_{0}=\partial_{x}f$, $g_{1}=\partial_{y}f$ and $g_{2}=\partial_{z}f$, we
denote its eigenscheme by $E(f)$.

It is worthwhile to point out that in the case of a symmetric tensor
corresponding to some homogeneous polynomial $f$, the eigenpoints are the
fixed points of the polar map

$\nabla
f=(\partial_{x}f,\partial_{y}f,\partial_{z}f):\mathbb{P}^{2}\dasharrow\mathbb{P}^{2}$

of $f$.

######  Example 2.7.

We consider the Fermat cubic $V(f)$ with
$f=x^{3}+y^{3}+z^{3}\in\mathbb{C}[x,y,z]$. The eigenscheme $E(f)$ is the
0-dimensional subscheme of $\mathbb{P}^{2}$ of length 7 defined by the maximal
minors of

$M=\begin{pmatrix}x&y&z\\\ x^{2}&y^{2}&z^{2}\end{pmatrix}.$

Therefore,
$E(f)=\\{(1,0,0,),(0,1,0),(0,0,1),(1,1,0),(1,0,1),(0,1,1),(1,1,1)\\}.$

If we fix an integer $d\geq 2$ and
$T=(g_{1},g_{2},g_{3})\in(Sym^{d-2}\mathbb{C}^{3})^{\oplus 3}$ is a general
partially symmetric tensor, then it holds (see [16, Theorem A2.10]):

  * (1)

$E(T)$ is a reduced 0-dimensional scheme of length $d^{2}-3d+3$.

  * (2)

The homogeneous ideal $I(E(T))\subset R$ has a minimal free $R$-resolution

$0\longrightarrow R(-2d+3)\oplus R(-d)\longrightarrow
R(-d+1)^{3}\longrightarrow I(E(T))\longrightarrow 0.$

The two conditions above are not sufficient for a planar $0$-dimensional
subscheme to be an eigenscheme, and a characterization is given by the
following result (see [2, Proposition 5.2].

######  Proposition 2.8.

Let Z be a $0$-dimensional subscheme of $\mathbb{P}^{2}$ of degree
$d^{2}-3d+3$. Then $Z$ is the eigenscheme of a tensor if and only if its
Hilbert-Burch matrix has the form

$\begin{pmatrix}L_{1}&G_{1}\\\ L_{2}&G_{2}\\\ L_{3}&G_{3}\\\ \end{pmatrix},$

where $L_{1},L_{2},L_{3}$ are linearly independent linear forms.

##  3\.  Conic-line arrangements with a linear Jacobian syzygy

We start this section with two series of examples of reduced conic-line
arrangements. All these examples will play an important role since, as we will
see, they are the only examples of reduced conic-line arrangements $C$ in
$\mathbb{P}^{2}$, whose Jacobian ideal has a linear syzygy.

In what follows we shall use the result [5, Theorem 3.5], due to R. O.
Buchweitz and A. Conca, which determines the Hilbert - Burch matrix of
Jacobian schemes admitting a linear syzygy of the type $(ax,by,cz)$ for some
coefficients $a,b,c\in\mathbb{C}$. For completeness, we recall its statement:

######  Theorem 3.1 (Buchweitz - Conca).

Let $K$ be a field of characteristic zero and $f\in K[x,y,z]$ a reduced
polynomial of degree $d$ in three variables such that $f$ is contained in the
ideal of its partial derivatives. Assume further that there is a triple
$(a,b,c)$ of elements of $K$ that are not all zero such that $ax\
\partial_{x}f+by\ \partial_{y}f+cz\ \partial_{z}f=0$.

We then have the following possibilities, up to renaming the variables:

  1. (1)

If $abc\neq 0$, then $f$ is a free divisor with Hilbert-Burch matrix

(3.1) $\begin{pmatrix}ax&\qquad\left(\frac{1}{c}-\frac{1}{b}\right)\
(d+2)^{-1}\partial_{yz}f\\\ by&\qquad\left(\frac{1}{a}-\frac{1}{c}\right)\
(d+2)^{-1}\partial_{xz}f\\\ cz&\qquad\left(\frac{1}{b}-\frac{1}{a}\right)\
(d+2)^{-1}\partial_{xy}f\\\ \end{pmatrix},$

where $\partial_{**}f$ denotes the corresponding second order derivative of
$f$.

  2. (2)

If $a=0$, but $bc\neq 0$, then $f$ is a free divisor if, and only if,
$\partial_{x}f\in(y,z)$. If that condition is verified and
$\partial_{x}f=yg+zh$, then $\frac{\partial
f_{y}}{cz}=\frac{-\partial_{z}f}{by}$ is an element of $K[x,y,z]$, and a
Hilbert-Burch matrix is given by

(3.2) $\begin{pmatrix}0&{\partial_{y}f}/{cz}\\\ by&{-h}/{c}\\\ cz&{g}/{b}\\\
\end{pmatrix}.$

  3. (3)

If $a=b=0$, then $f$ is independent of $z$ and, so, being the suspension of a
reduced plane curve, is a free divisor.

We focus now on conic-line arrangements. The first series of examples
corresponds to reduced conic-line arrangements $C=V(f)$ of degree $d$ with
${\rm mrd}(f)=1$ and maximal Tjurina number $\tau(C)=(d-1)(d-2)+1=d^{2}-3d+3$.

######  Example 3.2.

  1. (1)

We fix an integer $d\geq 3$. Let ${\mathcal{L}}$ be a line arrangement with
$d-1$ lines through a point $p$, and one other line in general position.
Without loss of generality we can assume that $p=(0:0:1)$ and that the general
line is $V(x)$, so that the equation of the line arrangement ${\mathcal{L}}$
is given by

${\mathcal{L}}:\ z\prod_{i=1}^{d-1}(a_{i}x+b_{i}y)=0$

with $(a_{i}:b_{i})\neq(a_{j}:b_{j})$ for $i\neq j$. It is simple to determine
a linear syzygy between the three partials of $f$, by observing that
$\partial_{z}f=\prod_{i=1}^{d-1}(a_{i}x+b_{i}y)$. So, we have

$f=z\ \partial_{z}f.$

On the other hand, by Euler formula we also have $f=\frac{1}{d}(x\
\partial_{x}f+y\partial_{y}f+z\ \partial_{x}f)$, hence we get the identity

$x\ \partial_{x}f+y\ \partial_{y}f+(1-d)z\ \partial_{x}f=0.$

Therefore, according to Theorem 3.1, (1), the Hilbert-Burch matrix of $J_{f}$
is given by

(3.3) $\begin{pmatrix}x&\frac{1}{(1-d)}\ \partial_{yz}f\\\
y&\frac{1}{(d-1)}\partial_{xz}f\\\ (1-d)z&0\\\ \end{pmatrix},$

a minimal free $R$-resolution of $J_{f}$ is given by:

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

and $\mathcal{L}$ is free with exponents $(1,d-2)$ and global Tjurina number
$d^{2}-3d+3$.

It is worthwhile to point out that the 3 entries $(x,y,(1-d)z)\in{\rm
Syz}(J_{f})_{1}$ of the linear syzygy are linearly independent.

  2. (2)

We fix an even integer $d=2m\geq 4$. Let ${\mathcal{C}}_{1}$ be a conic
arrangement with $m$ conics $C_{1},\dots,C_{m}$ such that there exists a point
$p\in\mathbb{P}^{2}$, for all $i,j$, $1\leq i<j\leq m$, satisfying $C_{i}\cap
C_{j}=\\{p\\}$, and the intersection point $p$ is a singularity $A_{7}$ for
$C_{i}\cup C_{j}$. In other words, the $m$ conics belong to a hyperosculating
pencil; such a curve is called an even Płoski curve in [25]. Without loss of
generality, we can assume that $p=(0:0:1)$ and the equation of the conic
arrangement ${\mathcal{C}}_{1}$ is given by

${\mathcal{C}}_{1}:\ f=\prod_{i=1}^{m}(x^{2}+a_{i}(xz+y^{2}))=0,$

with $a_{i}\neq 0$ and $a_{i}\neq a_{j}$ for $i\neq j$. The reduced plane
curve $\mathcal{C}_{1}$ has degree $d=2m$. A linear Jacobian syzygy can be
determined by observing that $(0,x,-2y)$ is a linear syzygy of the Jacobian
ideal of $f_{i}:=x^{2}+a_{i}(xz+y^{2})$, for any $i=1,\dots,m$, so we deduce
that ${\rm Syz}(J_{f})_{1}$ is also generated by $(0,x,-2y)$.

Moreover, we claim that $\mathcal{C}_{1}$ is free with exponents $(1,d-2)$, so
that the global Tjurina number is $d^{2}-3d+3$.

To prove the claim, observe that since $r=1$, the conic arrangement
$\mathcal{C}_{1}$ is either free or nearly free, and by (2.1) and (2.2), it is
free if and only if $J_{f}$ admits a sygyzy of degree $d-2$, which is not
proportional to $(0,x,-2y)$. Let us prove the latter fact by induction on the
number $m$ of conics.

If $m=2$, a degree $2$ syzygy, which is not proportional to $(0,x,-2y)$, is
given by

$\qquad\qquad(-(a_{1}+a_{2})x^{2}-2a_{1}a_{2}(y^{2}+xz),a_{1}a_{2}yz,4x^{2}+2(a_{1}+a_{2})y^{2}+3(a_{1}+a_{2})xz+2a_{1}a_{2}z^{2}).$

Now assume that $m\geq 3$, and that any conic arrangement of $m-1$ conics
belonging to a hyperosculating pencil admits a syzygy of degree $2m-4$, not
proportional to $(0,x,-2y)$. Let $f=\prod_{i=1}^{m}(x^{2}+a_{i}(xz+y^{2}))$,
and set

$f_{1}=\prod_{i=1}^{m-1}(x^{2}+a_{i}(xz+y^{2})),\qquad
f_{2}=x^{2}+a_{m}(xz+y^{2}).$

By induction hypothesis $J_{f_{1}}$ admits a syzygy $\delta_{1}\in{\rm
Syz}(J_{f_{1}})_{2m-4}$, with $\delta_{1}\not\in\langle(0,x,-2y)\rangle$, and
by Theorem 2.5, (1), we have

$\delta=f_{2}\delta_{1}-\frac{1}{2m}(\delta_{1}\cdot\nabla f_{2})\ E\in{\rm
Syz}(J_{f})_{2m-2},$

where we set $\delta_{1}\cdot\nabla
f_{2}=h_{1}\partial_{x}f_{2}+h_{2}\partial_{y}f_{2}+h_{3}\partial_{z}f_{2}$,
and $E=(x,y,z)$ is the Euler relation. As observed in the proof of such a
Theorem (see [10, Theorem 5.1]), since $\delta_{1}\neq 0$, it is also
$\delta\neq 0$. Finally, we claim that $\delta\not\in\langle(0,x,-2y)\rangle$.
Indeed, if $\delta_{1}\cdot\nabla f_{2}=0$, we have $\delta=f_{2}\delta_{1}$
and since $\delta_{1}\not\in\langle(0,x,-2y)\rangle$ by induction hypothesis,
the claim follows. If $\delta_{1}\cdot\nabla f_{2}\neq 0$, we see that

$\delta\cdot\nabla f_{1}=f_{2}\delta_{1}\cdot\nabla
f_{1}-\frac{1}{2m}(\delta_{1}\cdot\nabla f_{2})\ E\cdot\nabla
f_{1}=-\frac{2m-2}{2m}(\delta_{1}\cdot\nabla f_{2})f_{1}\neq 0.$

On the other hand, if we had $\delta=h\ (0,x,-2y)$ for some polynomial $h$, we
would have

$\delta\cdot\nabla f_{1}=h\ (0,x,-2y)\cdot\nabla f_{1}=0,$

as $(0,x,-2y)\in{\rm Syz}(J_{f_{1}})$.

It is important to point out that in this case the linear syzygy of $J_{f}$
has only 2 linearly independent entries, and that such an example is not of
the type considered in Buchweitz - Conca Theorem 3.1.

  3. (3)

We fix an odd integer $d=2m+1\geq 5$. Let ${\mathcal{C}}{\mathcal{L}}_{1}$ be
a conic-line arrangement with $m$ conics $C_{1},\dots,C_{m}$ and a line $\ell$
such that there exists a point $p\in\mathbb{P}^{2}$, $\ell$ is a common
tangent line to all $C_{i}$’s, $C_{i}\cap C_{j}=\\{p\\}$ and the intersection
point $p$ is a singularity $A_{7}$ for $C_{i}\cup C_{j}$, for all $i,j$,
$1\leq i<j\leq m$. In other words, the conics belong to a hypersculating
pencil; such a curve is called an odd Płoski curve in [25]. Without loss of
generality we can assume that $p=(0:0:1)$, the line $\ell=V(x)$ and the
equation of the conic-line arrangement ${\mathcal{C}}{\mathcal{L}}_{1}$ is
given by

${\mathcal{C}}{\mathcal{L}}_{1}:\ f=x\prod_{i=1}^{m}(x^{2}+a_{i}(xz+y^{2}))=0$

with $a_{i}\neq 0$ and $a_{i}\neq a_{j}$ for $i\neq j$, and $a_{i}\neq a_{j}$
if $1\neq j$. The reduced plane curve $\mathcal{C}\mathcal{L}_{1}$ has degree
$d=2m+1$; by using the same argument as in the previous example, it is not
difficult to see that $\mathcal{C}\mathcal{L}_{1}$ is free with exponents
$(1,d-2)$ and global Tjurina number $d^{2}-3d+3$. The linear syzygy of ${\rm
Syz}(J_{f})_{1}$ is generated by $(0,x,-2y)$. So, again the linear syzygy of
the Jacobian ideal of $f$ has only two independent entries, and is not of the
type considered in Buchweitz - Conca Theorem 3.1.

  4. (4)

We fix an odd integer $d=2m+1\geq 5$. Let ${\mathcal{C}}{\mathcal{L}}_{2}$ be
a conic-line arrangement with $m$ conics $C_{1},\dots,C_{m}$ and a line $\ell$
such that there exist two points $p,q\in\mathbb{P}^{2}$ such that $C_{i}\cap
C_{j}=\\{p,q\\}$ and the two intersection points $p,q$ are tacnodes for
$C_{i}\cup C_{j}$, for all $i,j$, $1\leq i<j\leq m$, and $\ell$ is a common
tangent line to all $C_{i}$’s at $p$. Without loss of generality we can assume
that $p=(0:0:1)$, $q=(1:0:0)$, $\ell=V(x)$, so that the equation of the conic-
line arrangement ${\mathcal{C}}{\mathcal{L}}_{2}$ is given by

${\mathcal{C}}{\mathcal{L}}_{2}:\ f=x\prod_{i=1}^{m}(xz+a_{i}y^{2})=0$

with $a_{i}\neq 0$ for all $i$, $1\leq i\leq m$, and $a_{i}\neq a_{j}$ if
$1\neq j$. The reduced singular plane curve $\mathcal{C}\mathcal{L}_{2}$ has
degree $d=2m+1$. We claim that
$Syz(J_{f})_{1}=\langle((d-1)x,-y,-(d+1)z)\rangle$. Indeed, set
$q_{i}(x,y,z)=xz+a_{i}y^{2}$. We have

$\partial_{x}f=\prod_{i=1}^{m}q_{i}+xz\left(\sum_{j=1}^{m}\prod_{i=1,i\neq
j}^{m}q_{i}\right),\
\partial_{y}f=2xy\left(\sum_{j=1}^{m}a_{j}\prod_{i=1,i\neq
j}^{m}q_{i}\right),\ \partial_{z}f=x^{2}\left(\sum_{j=1}^{m}\prod_{i=1,i\neq
j}^{m}q_{i}\right),$

which, in particular, gives

$f=x\ \partial_{x}f-z\ \partial_{z}f.$

Thus by the Euler identity we get

$(d-1)x\ \partial_{x}f-y\ \partial_{y}f-(d+1)z\ \partial_{z}f=0.$

Therefore, the linear syzygy of the Jacobian ideal of $f$ has again 3 linear
independent entries.

Therefore, according to Theorem 3.1, (1), the curve
$\mathcal{C}\mathcal{L}_{2}$ is free with global Tjurina number $d^{2}-3d+3$,
and the Hilbert-Burch matrix of $J_{f}$ is

(3.4) $\begin{pmatrix}(d-1)x&\frac{1}{(d+1)}\ \partial_{yz}f\\\
-y&\frac{2}{(d^{2}-1)}\partial_{xz}f\\\
-(d+1)z&-\frac{1}{(d-1)}\partial_{xy}f\\\ \end{pmatrix}.$

  5. (5)

The following three examples are of the type given in [5, Example 3.7], and
they all correspond to free curves of exponents $(1,d-2)$ and ${\rm
Syz}(J_{f})_{1}$ generated by $(x,0,-z)$.

     * •

Let $d=2m+2\geq 6$ and let ${\mathcal{C}}{\mathcal{L}}_{3}$ be a conic-line
arrangement of degree $d$ with $m$ conics $C_{1},\dots,C_{m}$ and two lines
$\ell_{1}$ and $\ell_{2}$ such that there exist two points
$p,q\in\mathbb{P}^{2}$ satisfying $C_{i}\cap C_{j}=\\{p,q\\}$ and the two
intersection points $p,q$ are tacnodes for $C_{i}\cup C_{j}$, for all $i,j$,
$1\leq i<j\leq m$, $\ell_{1}$ is a common tangent line to all $C_{i}$’s at
$p$, $\ell_{2}$ is a common tangent line to all $C_{i}$’s at $q$. Without loss
of generality we can assume that $p=(0:0:1)$, $q=(1:0:0)$, $\ell_{1}=V(x)$,
$\ell_{2}=V(z)$, so that the equation of the conic-line arrangement
${\mathcal{C}}{\mathcal{L}}_{3}$ is given by

${\mathcal{C}}{\mathcal{L}}_{3}:\ f=xz\prod_{i=1}^{m}(xz+a_{i}y^{2}))=0$

with $a_{i}\neq 0$ for all $i$, $1\leq i\leq m$, and $a_{i}\neq a_{j}$ if
$1\neq j$.

     * •

Let $d=2m+2\geq 6$ and let ${\mathcal{C}}{\mathcal{L}}_{4}$ be a conic-line
arrangement of degree $d$ with $m$ conics $C_{1},\dots,C_{m}$ and two lines
$\ell_{1}$ and $\ell_{2}$ such that there exist two points
$p,q\in\mathbb{P}^{2}$ satisfying $C_{i}\cap C_{j}=\\{p,q\\}$ and the two
intersection points $p,q$ are tacnodes for $C_{i}\cup C_{j}$, for all $i,j$,
$1\leq i<j\leq m$, $\ell_{1}$ is a common tangent line to all $C_{i}$’s at
$p$, $\ell_{2}$ is the line joining $p$ and $q$. Without loss of generality we
can assume that $p=(0:0:1)$, $q=(1:0:0)$, $\ell_{1}=V(x)$, $\ell_{2}=V(y)$, so
that the equation of the conic-line arrangement
${\mathcal{C}}{\mathcal{L}}_{4}$ is given by

${\mathcal{C}}{\mathcal{L}}_{4}:\ f=xy\prod_{i=1}^{m}(xz+a_{i}y^{2}))=0$

with $a_{i}\neq 0$ for all $i$, $1\leq i\leq m$, and $a_{i}\neq a_{j}$ if
$1\neq j$.

     * •

Let $d=2m+3\geq 6$ and let ${\mathcal{C}}{\mathcal{L}}_{5}$ be a conic-line
arrangement of degree $d$ with $m$ conics $C_{1},\dots,C_{m}$ and three lines
$\ell_{1}$, $\ell_{2}$ and $\ell_{3}$ such that there exist two points
$p,q\in\mathbb{P}^{2}$ satisfying $C_{i}\cap C_{j}=\\{p,q\\}$ and the two
intersection points $p,q$ are tacnodes for $C_{i}\cup C_{j}$, for all $i,j$,
$1\leq i<j\leq m$, $\ell_{1}$ is a common tangent line to all $C_{i}$’s at
$p$, $\ell_{2}$ is a common tangent line to all $C_{i}$’s at $q$, $ell_{3}$ is
the line joining $p$ and $q$. Without loss of generality we can assume that
$p=(0:0:1)$, $q=(1:0:0)$, $\ell_{1}=V(x)$, $\ell_{2}=V(z)$ and
$\ell_{3}=V(y)$, so that the equation of the conic-line arrangement
${\mathcal{C}}{\mathcal{L}}_{5}$ is given by

${\mathcal{C}}{\mathcal{L}}_{5}:\ f=xyz\prod_{i=1}^{m}(xz+a_{i}y^{2}))=0$

with $a_{i}\neq 0$ for all $i$, $1\leq i\leq m$, and $a_{i}\neq a_{j}$ if
$1\neq j$.

Next series of example corresponds to reduced nearly free plane curves
$C=V(f)$ of degree $d$ with ${\rm mrd}(f)=1$ and $\tau(C)=d^{2}-3d+2$.

######  Example 3.3.

By [5, Example 3.7], next examples have the property that ${\rm
Syz}(J_{f})_{1}$ is generated by $(x,0,-z)$ and they are not free. Since
$r=1$, we have $\tau(C)=(d-1)(d-2)$ and they are nearly free by Lemma 3.4.

  1. (1)

Let ${\mathcal{C}}_{2}$ be a conic arrangement with $m$ conics
$C_{1},\dots,C_{m}$ such that there exist two points $p,q\in\mathbb{P}^{2}$
satisfying $C_{i}\cap C_{j}=\\{p,q\\}$ and the two intersection points $p,q$
are tacnodes for $C_{i}\cup C_{j}$, for all $i,j$, $1\leq i<j\leq m$.

Without loss of generality we can assume that $p=(0:0:1)$, $q=(1:0:0)$, so
that the equation of the conic arrangement is

${\mathcal{C}}_{2}:\ f=\prod_{i=1}^{m}(xz+a_{i}y^{2})=0$

with $a_{i}\neq 0$ for all $i$, $1\leq i\leq m$, and $a_{i}\neq a_{j}$ if
$1\neq j$.

  2. (2)

Let ${\mathcal{C}}{\mathcal{L}}_{6}$ be a conic-line arrangement with $m$
conics $C_{1},\dots,C_{m}$ and a line $\ell$ such that there exist two points
$p,q\in\mathbb{P}^{2}$ such that $C_{i}\cap C_{j}=\\{p,q\\}$ and the two
intersection points $p,q$ are tacnodes for $C_{i}\cup C_{j}$, for all $i,j$,
$1\leq i<j\leq m$, and $\ell$ is the line through $p$ and $q$. We can assume
that $p=(0:0:1)$, $q=(1:0:0)$, $\ell=V(y)$ and the equation of the conic-line
arrangement is

${\mathcal{C}}{\mathcal{L}}_{6}:\ f=y\prod_{i=1}^{m}(xz+a_{i}y^{2})=0$

with $a_{i}\neq 0$ for all $i$, $1\leq i\leq m$, and $a_{i}\neq a_{j}$ if
$i\neq j$.

Our next goal is to establish a geometric characterization of all reduced
conic-line arrangements $C=V(f)$ in $\mathbb{P}^{2}$ of degree $d\geq 3$,
whose Jacobian ideal $J_{f}$ has a linear syzygy, i.e., ${\rm mrd}(f)=1$.

######  Lemma 3.4.

Let $C=V(f)$ be a reduced singular curve in $\mathbb{P}^{2}$ of degree $d\geq
3$ and let $J_{f}=(f_{x},f_{y},f_{z})$ be its Jacobian ideal. Assume that
${\rm mrd}(f)=1$. Then $d^{2}-3d+2\leq\tau(C)\leq d^{2}-3d+3$. Moreover, if
$\tau(C)=d^{2}-3d+3$ (resp. $\tau(C)=d^{2}-3d+2$) then $C$ is free (nearly
free).

###### Proof.

By Proposition 2.4 we have $d^{2}-3d+2\leq\tau(C)\leq d^{2}-3d+3$. By [6,
Theorem 1.2] (see also [17, Proposition 26]), if $\tau(C)=d^{2}-3d+3$
(respectively $d^{2}-3d+2$) then $C$ is free (nearly free). ∎

###  3.1. Free conic-line arrangements with a linear Jacobian syzygy

Our first goal is to classify free conic-line arrangements $C=V(f)$ of degree
$d$ in $\mathbb{P}^{2}$ with $r={\rm mrd}(f)=1$; such curves have maximum
Tjurina number $\tau(C)=d^{2}-3d+3$. We have

######  Theorem 3.5.

Let $C=V(f)$ be a conic-line arrangement in $\mathbb{P}^{2}$ of degree $d\geq
5$. Then, $\tau(C)=d^{2}-3d+3$ if and only if $C$ is either a line arrangement
${\mathcal{L}}$ as in example 3.2(1), or a conic arrangement
${\mathcal{C}}_{1}$ as in example 3.2(2), or a conic-line arrangement
${\mathcal{C}}{\mathcal{L}}_{1}$, ${\mathcal{C}}{\mathcal{L}}_{2}$,
${\mathcal{C}}{\mathcal{L}}_{3}$, ${\mathcal{C}}{\mathcal{L}}_{4}$ or
${\mathcal{C}}{\mathcal{L}}_{5}$ as in examples 3.2(3)-(7).

###### Proof.

All conic-line arrangements $C\subset\mathbb{P}^{2}$ described in examples
3.2(1)-(7) have total Tjurina number $\tau(C)=d^{2}-3d+3$ and ${\rm
mdr}(f)=1$. Let us prove the converse. The hypothesis $\tau(C)=d^{2}-3d+3$,
$d\geq 5$ and Proposition 2.4 imply that ${\rm mrd}(f)=1$. We distinguish
several cases:

Case 1: $C$ is a line arrangement. By [9, Proposition 4.7(5)], $C$ is the
union of $d-1$ lines through a point $p$, and one other line in general
position.

Case 2: Let $C=\cup_{i=1}^{m}C_{i}:\ f=\prod_{i=1}^{m}f_{i}=0$ be a conic
arrangement. By Theorem 2.5, (3), whenever we extract the union
$C^{\prime}=C_{1}\cup C_{2}$ of two conics, the relative $r^{\prime}={\rm
mrd}(C^{\prime})=1$, so by the classification given in [10, Proposition 5.5],
the only possible cases are: either $|C_{1}\cap C_{2}|=2$ and the two
intersection points are two tacnodes for $C^{\prime}$, or $|C_{1}\cap
C_{2}|=1$ and the singular point is an $A_{7}$ singularity (the two conics are
hyperosculating). Since in the first case the total Tjurina number is
$\tau=6$, it does not occur. This settles the case $d=4$. Assume now $d\geq
6$.

Claim:  Whenever we extract the union $C_{i_{1}}\cup C_{i_{2}}\cup C_{i_{3}}$
of three conics, they belong to the same pencil, so they are either bitangent,
or they are hyperosculating.

Proof of the Claim. Indeed, assume first that $C_{i_{1}}\cap
C_{i_{2}}=\\{p,q\\}$ and $p,q$ are two tacnodes for $C_{i_{1}}\cup C_{i_{2}}$.
Without loss of generality we can assume that

$C_{i_{1}}=V(f_{i_{1}})=V(xz+a_{i_{1}}y^{2}),\qquad
C_{i_{2}}=V(f_{i_{2}})=V(xz+a_{i_{2}}y^{2})$

with $a_{i_{1}},a_{i_{2}}\in\mathbb{C}$. Therefore,
$D_{0}(f_{i_{1}})_{1}=D_{0}(f_{i_{2}})_{1}\cong{\rm
Syz}(f_{i_{1}}f_{i_{2}})_{1}=\langle(x,0,-z)\rangle$. By hypothesis ${\rm
mrd}(f_{i_{1}}f_{i_{2}}f_{i_{3}})=1$. So, applying Theorem 2.5, we have

$D_{0}(f_{i_{3}})_{1}\cap\langle(x,0,-z)\rangle\neq{0},\ {\rm or}\
\alpha(x,0,-z)+\beta(x,y,z)\in D_{0}(f_{i_{3}})_{1}.$

A straightforward computation shows that necessarily
$C_{i_{3}}=V(f_{i_{3}})=V(\lambda xz+\mu y^{2})$.

Assume now that $C_{i_{1}}\cap C_{i_{2}}=\\{p\\}$ and $p$ is a singularity
$A_{7}$ for $C_{i_{1}}\cup C_{i_{2}}$. Without loss of generality we can
assume

$C_{i_{1}}=V(f_{i_{1}})=V(x^{2}+a_{i_{1}}(xz+y^{2})),\qquad
C_{i_{2}}=V(f_{i_{2}})=V(x^{2}+a_{i_{2}}(xz+y^{2}))$

with $a_{i_{1}},a_{i_{2}}\in\mathbb{C}$. Therefore,
$D_{0}(f_{i_{1}})_{1}=D_{0}(f_{i_{2}})_{1}\cong{\rm
Syz}(f_{i_{1}}f_{i_{2}})_{1}=\langle(0,x,-2y)\rangle$. By hypothesis ${\rm
mrd}(f_{i_{1}}f_{i_{2}}f_{i_{3}})=1$, hence by Theorem 2.5, we have in this
case

$D_{0}(f_{i_{3}})_{1}\cap\langle(0,x,-2y))\rangle\neq{0},\ {\rm or}\
\alpha(0,x,-2y)+\beta(x,y,z)\in D_{0}(f_{i_{3}})_{1}.$

A direct computation shows that in this case necessarily
$C_{i_{3}}=V(f_{i_{3}})=V(\lambda x^{2}+\mu(xz+y^{2}))$.

It follows from the claim that the irreducible components of
$C=\cup_{i=1}^{m}C_{i}$ belong either to a bitangent pencil of conics or to a
hyperosculating pencil of conics. In the first case we have
$\tau(C)=(2m)^{2}-3(2m)+2$ (see Example 3.3(1)) and in the second case it is
$\tau(C)=(2m)^{2}-3(2m)+3$ (see Example 3.2(2)), which proves what we want.

Case 3: : Let $C=\cup_{i=1}^{m}C_{i}\bigcup\cup_{j=1}^{s}L_{j}:\
f=\prod_{i=1}^{m}f_{i}\cdot\prod_{j=1}^{s}\ell_{j}=0$ be a conic-line
arrangement. By Theorem 2.5, the conic arrangement $\cup_{i=1}^{m}C_{i}$
satisfies ${\rm mrd}(\prod_{i=1}^{m}f_{i})=1$. By the above discussion, the
components of $\cup_{i=1}^{m}C_{i}$ belong either to a hyperosculating pencil
or to a bitangent pencil of conics. We analyze this two cases separately.

Let us first assume that
$\cup_{i=1}^{m}C_{i}=V(\prod_{i=1}^{m}(x^{2}+a_{i}(xz+y^{2})))$ belong to a
hyperosculating pencil of conics with hyperosculating point $p=(0:0:1)$.
Therefore, ${\rm Syz}(\prod_{i=1}^{m}f_{i})_{1}=\langle(0,x,-2y)\rangle$. We
look for a line $L=V(\ell)=V(ax+by+cz)$ such that ${\rm
mrd}((ax+by+cz)\prod_{i=1}^{m}(x^{2}+a_{i}(xz+y^{2}))=1$. By Theorem 2.5, (3),
if we set

$f_{1}=ax+by+cz,\quad f_{2}=\prod_{i=1}^{m}(x^{2}+a_{i}(xz+y^{2})),$

we have that either $D_{0}(f_{1})_{1}\cap D_{0}(f_{2})_{1}\neq 0$, or
$\alpha(x\partial_{x}+y\partial_{y}+z\partial_{z})\in
D_{0}(f_{1})_{1}+D_{0}(f_{2})_{1}$ for some nonzero constant $\alpha$. As to
the line $V(f_{1})$, if $a\neq 0$, we have

$D_{0}(f_{1})_{1}=\\{L_{1}(-b\partial_{x}+a\partial_{y})+L_{2}(-c\partial_{x}+a\partial_{z})\
:\ L_{1},L_{2}\in R_{1}\\},$

while $D_{0}(f_{2})=\langle x\partial_{y}-2y\partial_{z}\rangle$.

The condition $D_{0}(f_{1})_{1}\cap D_{0}(f_{2})_{1}\neq 0$ is never
satisfied, while the condition
$\alpha(x\partial_{x}+y\partial_{y}+z\partial_{z})=L_{1}(-b\partial_{x}+a\partial_{y})+L_{2}(-c\partial_{x}+a\partial_{z})+\beta
x\partial_{y}-2\beta y\partial_{z}$ for $\alpha\neq 0$ has as a unique
solution $b=c=0$. It follows that

$\ell=x.$

The case $a=0$ can be treated similarly and it never occurs.

Next it is possible to check in a similar way that
$h=x(ax+by+cz)\prod_{i=1}^{m}(x^{2}+a_{i}(xz+y^{2}))$ has a linear Jacobian
syzygy if and only if $b=c=0$, but this would give rise to a non reduced
polynomial.

So, $C$ is as in example 3.2.

Let us now assume that
$\cup_{i=1}^{m}C_{i}=V(\prod_{i=1}^{m}(xz+a_{i_{1}}y^{2}))$ belongs to a
pencil of conics, all of them bitangent at $\\{p=(1:0:0),q=(0:0:1)\\}$, so
that we have the linear syzygy $(x,0,-z)$. Arguing as above we can determine
the lines that we can add to this conic arrangement in such a way that the new
conic-line arrangement has Jacobian ideal with a linear syzygy. It turns out
that we have only six possibilities:

$x\prod_{i=1}^{m}(xz+a_{i_{1}}y^{2}),\quad
y\prod_{i=1}^{m}(xz+a_{i_{1}}y^{2}),\quad
z\prod_{i=1}^{m}(xz+a_{i_{1}}y^{2}),$
$xz\prod_{i=1}^{m}(xz+a_{i_{1}}y^{2}),\quad
yz\prod_{i=1}^{m}(xz+a_{i_{1}}y^{2}),\quad
xyz\prod_{i=1}^{m}(xz+a_{i_{1}}y^{2}).$

The case $y\prod_{i=1}^{m}(xz+a_{i_{1}}y^{2})=0$ is not free by Theorem 3.1
(2). By observing that the first and the third case are projectively
equivalent, this concludes the proof. ∎

###  3.2. Nearly free conic-line arrangements with a linear Jacobian syzygy

In this subsection, we classify conic-line arrangements $C=V(f)$ of degree $d$
in $\mathbb{P}^{2}$ with $r={\rm mrd}(f)=1$ and minimal Tjurina number
$\tau(C)=d^{2}-3d+2$.

######  Theorem 3.6.

Let $C=V(f)$ be a conic-line arrangement in $\mathbb{P}^{2}$ of degree $d\geq
6$. Then, $\tau(C)=d^{2}-3d+2$ if and only if $C$ is either a conic
arrangement ${\mathcal{C}}_{2}$ as in example 3.3(1), or a conic-line
arrangement ${\mathcal{C}}{\mathcal{L}}_{6}$ as in example 3.3(2).

###### Proof.

All conic-line arrangements $C\subset\mathbb{P}^{2}$ described in examples
3.3(1)-(2) have total Tjurina number $\tau(C)=d^{2}-3d+2$.

Let us prove the converse. The hypothesis $\tau(C)=d^{2}-3d+2$, $d\geq 6$ and
Proposition 2.4 imply that ${\rm mrd}(f)=1$. By [14, Proposition 4.3], there
are no line arrangements with ${\rm mrd}(f)=1$ and $\tau(C)=d^{2}-3d+2$.
Therefore, we only have two possibilities: either $C$ is a conic arrangement,
or $C$ is a conic-line arrangement. Arguing as in the proof of Theorem 3.5 is
it possible to conclude. ∎

##  4\. Applications

As application of the previous results we obtain the main result of this
paper, namely, we determine when the Jacobian ideal of a conic-line
arrangement is the ideal of an eigenscheme. More precisely, we have:

######  Theorem 4.1.

Let $C=V(f)$ be a conic-line arrangement in $\mathbb{P}^{2}$ of degree $d\geq
4$. The Jacobian ideal $J_{f}$ of $f$ is the ideal of an eigenscheme $E(T)$ if
and only if $C$ is either a line arrangement ${\mathcal{L}}$, or a conic-line
arrangement ${\mathcal{C}}{\mathcal{L}}_{2}$.

###### Proof.

Let $C=V(f)$ be a line arrangement (resp. conic-line arrangement) as described
in the statement of the theorem. We have seen in example 3.2(1) (resp. example
3.2(4)) that the Jacobian ideal $J_{f}$ of $C$ has is defined by the maximal
minors of the matrix

$\begin{pmatrix}(d-1)x&g_{0}\\\ -y&g_{1}\\\ -(1+d)z&g_{2}\\\
\end{pmatrix},\quad\text{resp. }\begin{pmatrix}x&h_{0}\\\ y&h_{1}\\\
(1-d)z&h_{2}\\\ \end{pmatrix}.$

Equivalently, the Jacobian ideal is generated by the minors of the matrix

$\begin{pmatrix}x&\frac{1}{d-1}g_{0}\\\ y&-g_{1}\\\ z&-\frac{1}{d+1}g_{2}\\\
\end{pmatrix},\quad\text{resp. }\begin{pmatrix}x&h_{0}\\\ y&h_{1}\\\
z&\frac{1}{d-1}h_{2}\\\ \end{pmatrix}.$

By definition we have $J_{f}=I(E(T))$, where
$T=(\frac{1}{d-1}g_{0},-g_{1},-\frac{1}{d+1}g_{2})\in(Sym^{d-1}\mathbb{C}^{3})^{\oplus(3)}$,
resp.
$T=(h_{0},h_{1},\frac{1}{d-1}h_{2})\in(Sym^{d-1}\mathbb{C}^{3})^{\oplus(3)}$
are partially symmetric tensors.

Let us prove the converse. Assume that there is a partially symmetric tensor
$T=(g_{1},g_{2},g_{3})\in(Sym^{d-1}\mathbb{C}^{3})^{\oplus(3)}$ such that
$I(E(T))=J_{f}$. This implies that $C$ is free, $\tau(C)=d^{2}-3d+3$, ${\rm
mrd}(f)=1$ and that ${\rm Syz}(J_{f})_{1}$ is generated by three linearly
independent linear forms. Example 3.2 together with Proposition 3.5 proves
what we want. ∎

######  Remark 4.2.

There are examples of reduced plane curves $C=V(f)\subset\mathbb{P}^{2}$ whose
Jacobian ideal $J_{f}$ is the ideal of an eigenscheme $E(T)$ and they are not
conic-line arrangements. For instance, $f=y(x^{3}-y^{2}z)$.

######  Remark 4.3.

In particular, the geometry of the Jacobian scheme of a line arrangement of
type ${\mathcal{L}}$ or a conic-line arrangement
${\mathcal{C}}{\mathcal{L}}_{2}$ is completely described by [4, Theorem 5.5
and Remark 5.8]. We observe that the cited result concerns only reduced
eigenschemes, but it is not difficult to see, that it can be extended to all
non reduced zero-dimensional eigenschemes.

Specifically, we have that if $k\in\\{2,\dots,d-1\\}$ then no subscheme of
degree $kd$ of $\Sigma_{f}$ lies on a curve of degree $k$. Moreover, the class
of $S={\rm Bl}_{\Sigma_{f}}\mathbb{P}^{2}$ in the Chow ring
$A(\mathbb{P}^{2}\times\mathbb{P}^{2})$ can be determined. By choosing $L_{1}$
and $L_{2}$ as generators of the Picard groups of the two factors, and by
setting $p_{i}:\mathbb{P}^{2}\times\mathbb{P}^{2}\to\mathbb{P}^{2}$ to be the
two projections, we have that the two divisors $h_{1}=p_{1}^{\star}L_{1}$ and
$h_{2}=p_{2}^{\star}L_{2}$ are generators for
$A(\mathbb{P}^{2}\times\mathbb{P}^{2})$. Then it is simple to check that the
class of $S$ in $A(\mathbb{P}^{2}\times\mathbb{P}^{2})$ is given by

$[S]=(d-1)h_{1}^{2}+dh_{1}h_{2}+h_{2}^{2},$

and, by taking into account the Hilbert-Burch matrices given in (3.3) and
(3.4), the surface $S$ turns out to be the complete intersection of the two
divisors $T\sim h_{1}+h_{1}$ and $D\sim(d-2)h_{1}+h_{2}$ given by

$T=V(p_{0}x+p_{1}y+(1-d)p_{2}z),\quad
D=V(p_{0}\partial_{yz}f+p_{1}\partial_{xz}f)$

in case $\mathcal{L}$, respectively

$T=V((d-1)p_{0}x-p_{1}y-(d+1)p_{2}z),\quad D=V((d-1)\
p_{0}\partial_{yz}f+2p_{1}\partial_{xz}f-(d+1)p_{2}\partial_{xy}f),$

in case $\mathcal{C}\mathcal{L}_{2}$, where
$((x:y:z),(p_{0}:p_{1}:p_{2}))\in\mathbb{P}^{2}\times\mathbb{P}^{2}$.

Finally, we observe that by [20] or [1, Lemma 5.6], every planar eigenscheme
is the zero locus of section $s\in
H^{0}({\mathcal{T}}_{\mathbb{P}^{2}}(d-2))$, where
${\mathcal{T}}_{\mathbb{P}^{2}}$ denotes the tangent bundle of
$\mathbb{P}^{2}$.

Next we shall study the polar map associated with ${\mathcal{L}}$ and
${\mathcal{C}}{\mathcal{L}}_{2}$ arrangements. Observe that since we are
concerned with curves of maximal total Tjurina number and quasihomogeneous
singularities, the degree of the generically finite polar map is
$(d-1)^{2}-\mu(C)=d-2$.

######  Remark 4.4.

We can apply the argument used in the proof of [4, Theorem 5.5] and we see
that the possible contracted curves by the polar map associated with line
arrangements of type ${\mathcal{L}}$ or conic-line arrangements
${\mathcal{C}}{\mathcal{L}}_{2}$ are only lines.

Indeed, we observe that for any $p=(p_{0}:p_{1}:p_{2})\notin\Sigma_{f}$, the
point $\nabla f(p)$ is the intersection point of the two distinct lines

$\nabla f(p):\ \left\\{\begin{array}[]{l}p_{0}x+p_{1}y+(1-d)p_{2}z=0\\\
\partial_{yz}f(p)x+\partial_{xz}f(p)y=0.\end{array}\right.$

As a consequence, the fiber of $\nabla f$ over any point
$q=(q_{0}:q_{1}:q_{2})\in\mathbb{P}^{2}$ is given by the zero locus of

$\left\\{\begin{array}[]{l}q_{0}x+q_{1}y+(1-d)q_{2}z=0\\\
q_{0}\partial_{yz}f+q_{1}\partial_{xz}f=0.\end{array}\right.$

Since the first equations represents a line for any choice of
$q\in\mathbb{P}^{2}$, the claim follows.

We shall see in the next result that the presence of contracted lines is
indeed always confirmed for ${\mathcal{L}}$ and
${\mathcal{C}}{\mathcal{L}}_{2}$ arrangements. Recall that the critical locus
of the polar map is given by the hessian curve, and it consists of the
contracted curves and the ramification points for the polar map.

######  Proposition 4.5.

Let $C=V(f)$ be a conic-line arrangement in $\mathbb{P}^{2}$ of degree $d$
such that the Jacobian ideal $J_{f}$ of $f$ is the ideal of an eigenscheme
$E(T)$.

Then, in case $\mathcal{L}$, the critical locus of $\nabla f$ is given by an
arrangement of $3(d-2)$ lines of the same type of $\mathcal{L}$, it contains
$\mathcal{L}$ and the contracted lines by $\nabla f$ are precisely the lines
of $\mathcal{L}$.

In case ${\mathcal{C}}{\mathcal{L}}_{2}$, the critical locus contains the
tangent line $\ell$, and it is the only contracted line.

###### Proof.

It is classically known that all the lines of a line arrangement are contained
in the hessian curve.

Now we verify that the residual curve to ${\mathcal{L}}$ in $\text{Hess}(f)$
consists of $3(d-2)-d=2d-6$ concurrent lines through $O=(0:0:1)$ and that such
residual lines are not contracted by $\nabla f$.

The first claim follows by writing the hessian matrix explicitly:

$\text{Hess}(f)=\begin{pmatrix}\partial_{xx}f&\partial_{xy}f&\partial_{xz}f\\\
\partial_{xy}f&\partial_{yy}f&\partial_{yz}f\\\
\partial_{xz}f&\partial_{yz}f&0\\\ \end{pmatrix}.$

Since both $\partial_{xz}f$ and $\partial_{yz}f$ are polynomials in $x$ and
$y$ only, by developing the determinant $h(f)=\det\text{Hess}(f)$ with respect
to the last row we see that $\frac{h(f)}{f}$ is a polynomial in $x$ and $y$.

Moreover, as the polar map is given by

$\nabla f=\left(x_{2}\left(\sum_{i=1}^{d-1}a_{i}\prod_{j\neq
i,j=1}^{d-1}(a_{j}x+b_{j}y)\right),z\left(\sum_{i=1}^{d-1}b_{i}\prod_{j\neq
i,j=1}^{d-1}(a_{j}x+b_{j}y)\right),\prod_{i=1}^{d-1}(a_{i}x+b_{i}y)\right),$

we see that the line $z=0$ is contracted to the point $(0:0:1)$ and the lines
$a_{i}x+b_{i}y=0$ to the points $(a_{i}:b_{i}:0)$.

Finally, to prove that there are no other contracted lines, we recall that the
Hilbert-Burch matrix of $J_{f}$ is given by (3.1), and that $\nabla f$ is
given by its $2\times 2$ minors. It follows that for any
$p=(p_{0}:p_{1}:p_{2})\notin\Sigma_{f}$, the point $\nabla f(p)$ is the
intersection point of the two distinct lines

$\nabla f(p):\ \left\\{\begin{array}[]{l}p_{0}x+p_{1}y+(1-d)p_{2}z=0\\\
\partial_{yz}f(p)x+\partial_{xz}f(p)y=0.\end{array}\right.$

As a consequence, the fiber of $\nabla f$ over a point
$q=(q_{0}:q_{1}:q_{2})\in\mathbb{P}^{2}$ is given by the zero locus of

$\left\\{\begin{array}[]{l}q_{0}x+q_{1}y+(1-d)q_{2}z=0\\\
q_{0}\partial_{yz}f+q_{1}\partial_{xz}f=0.\end{array}\right.$

In particular, a ramification point appears in a fiber if and only if the set
of $d-2$ concurrent lines through $(0:0:1)$ given by the equation
$q_{0}\partial_{yz}f+q_{1}\partial_{xz}f=0$ contains a (non reduced) double
line, so the question is to determine the non reduced elements of the pencil
$q_{0}\partial_{yz}f+q_{1}\partial_{xz}f$. But the latter can be seen as a
pencil of divisors in $\mathbb{P}^{1}$, and precisely the Jacobian pencil of
the polynomial $\partial_{2}f$. If the factors of $f$ are general, the
polynomial $\partial_{2}f\in\mathbb{C}[x,y]_{d-1}$ is general too. Therefore,
the ramification points of the polar map $\nabla\partial_{2}f$ are given by
its hessian.

We finally treat the case of an ${\mathcal{C}}{\mathcal{L}}_{2}$ arrangement.
It is well known that any linear component of a plane curve is contained in
the Hessian curve. Moreover, if $f=x\prod_{i=1}^{m}(xz+a_{i}y^{2})$, the polar
map is given by

$\nabla f=\left(\prod_{i=1}^{m}q_{i}+xz\left(\sum_{j=1}^{m}\prod_{i=1,i\neq
j}^{m}q_{i}\right),2xy\left(\sum_{j=1}^{m}a_{j}\prod_{i=1,i\neq
j}^{m}q_{i}\right),x^{2}\left(\sum_{j=1}^{m}\prod_{i=1,i\neq
j}^{m}q_{i}\right)\right),$

where we set $q_{i}=xz+a_{i}y^{2}$; we see that the line $V(x)$ is contracted
to a point. To see that there are no other contracted lines, we observe that
such a line should contain a subscheme of degree at least $d-1$ in the
Jacobian scheme; the only possible candidates are the tangent line in the
second osculating point of the conics, that is the line $V(z)$, or the line
$V(y)$ connecting the two singular points; but we can directly check that
these cases do not occur.

∎

We conclude by observing that the geometry of the polar map seems to encode
some information concerning the topological type of the singularities of a
given curve, so we believe that it deserves further investigations.

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