# Addition-deletion results for plus-one generated curves

Anca Măcinic∗ and Piotr Pokora∗∗

###### Abstract.

In the recent paper A. Dimca proves that when one adds to or deletes a line
from a free curve the resulting curve is either free or plus-one generated. We
prove two converse statements for plus-one generated curves and we give an
additional insight into the original deletion result, and derive a
characterisation of free curves in terms of behaviour to addition/deletion of
lines. We describe the possible splitting types of the bundle of logarithmic
vector fields associated to a plus-one generated curve.

∗ Partially supported by a grant of the Romanian Ministry of Education and
Research, CNCS - UEFISCDI, project number PN-III-P4-ID-PCE-2020-2798, within
PNCDI III

∗∗ Partially supported by The Excellent Small Working Groups Programme
DNWZ.711/IDUB/ESWG/2023/01/00002 at the University of the National Education
Commission Krakow.

##  1\. Introduction

Let $\mathcal{C}$ be a reduced curve in
$\mathbb{P}^{2}\coloneqq\mathbb{P}^{2}_{\mathbb{C}}$ defined by the equation
$f_{\mathcal{C}}=0$, where $f_{\mathcal{C}}\in S\coloneqq\mathbb{C}[x,y,z]$.
The logarithmic derivation module associated to the curve $\mathcal{C}$ is the
submodule of ${\rm Der}(S)$ defined by

$D(\mathcal{C})\coloneqq\\{\theta\in{\rm
Der}(S)\;|\;\theta(f_{\mathcal{C}})\in Sf_{\mathcal{C}}\\}.$

The Euler derivation $\theta_{E}=x\partial x+y\partial y+z\partial z$ is an
element of $D(\mathcal{C})$ for any curve $\mathcal{C}$. Furthermore, the
module $D(\mathcal{C})$ admits a decomposition as the direct sum

$D(\mathcal{C})=D_{0}(\mathcal{C})\oplus S\theta_{E},$

where

$D_{0}(\mathcal{C})\coloneqq\\{\theta\in{\rm
Der}(S)\;|\;\theta(f_{\mathcal{C}})=0\\}.$

The curve $\mathcal{C}$ is called free if $D_{0}(\mathcal{C})$ is free as an
S-module. In this case, $D_{0}(\mathcal{C})$ is a rank $2$ graded module and
we call exponents of $\mathcal{C}$ the pair of degrees of the elements in a
basis of homogeneous derivations of $D_{0}(\mathcal{C})$. In other words, if
$\\{\theta_{2},\theta_{3}\\}$ is a basis of homogeneous derivations of
$D_{0}(\mathcal{C})$, then
$\exp(\mathcal{C})\coloneqq(\deg(\theta_{2}),\deg(\theta_{3}))$.

We recall the definition of a recent notion introduced by Abe in [1] for
hyperplane arrangements and extended to curves by Dimca and Sticlaru in [6].

######  Definition 1.1.

A reduced curve $\mathcal{C}$ in $\mathbb{P}^{2}$ is called plus-one generated
if its associated module of derivations $D(\mathcal{C})$ admits a minimal set
of homogeneous generators $\\{\theta_{E},\theta_{2},\theta_{3},\psi\\}$ that
satisfy a unique relation of the form

(1) $f_{1}\theta_{E}+f_{2}\theta_{2}+f_{3}\theta_{3}+\alpha\psi=0,$

where $f_{i}\in S$ for $i=1,2,3$ and $\alpha\in S$ with $\deg(\alpha)=1$. The
exponents of $\mathcal{C}$ are, by definition, the pair
$(\deg(\theta_{2}),\deg(\theta_{3}))$ and $\deg(\psi)$ is called the level of
$\mathcal{C}$.

In the recent paper [3], Dimca proves the following addition result for free
curves (see Section 2 for the additional notions and notations).

######  Theorem 1.2.

Let $\mathcal{C}$ be a free reduced curve in $\mathbb{P}^{2}$ of exponents
$(d_{2},d_{3})$ and $L\subset\mathbb{P}^{2}$ be a line that is not an
irreducible component of $\mathcal{C}$. Then $\mathcal{C}\cup L$ is either
free with the exponents $(d_{2},d_{3}+1)$ and $d_{2}+1=|\mathcal{C}\cap
L|+\epsilon(\mathcal{C},L)$ or plus-one generated with the exponents
$(d_{2}+1,d_{3}+1)$ and level $d=|\mathcal{C}\cap
L|+\epsilon(\mathcal{C},L)-1$.

In Section 2 we recall some necessary notions and results from the literature
and we prove some new results related to Chern numbers and splitting types for
the vector bundle associated to a plus-one generated curve, defined in 2. In
general, evaluating splitting types onto projective lines for vector bundles
is a difficult endeavor. We describe in Theorem 2.8 the set of possible values
for the splitting types of the vector bundle associated to a plus-one
generated curve, generalizing similar results on plus-one generated line
arrangements, respectively on nearly free curves, from [4], respectively from
[2].

In Section 3 we present our main results. We prove in Theorem 3.1 a converse
of the above result. In the same paper Dimca proved a deletion result for free
reduced curves, see [3, Theorem 1.1]. It states that for a free curve
$\mathcal{C}$ with a line $L$ being an irreducible component, the curve
$\mathcal{C}^{\prime}$ obtained by removing from $\mathcal{C}$ line $L$ is
either free or plus-one generated. The author asks whether some relation
between the exponents of $\mathcal{C}$ and the exponents (and level) of
$\mathcal{C}^{\prime}$ can be given. We answer this question in Theorem 3.2.
We prove in Theorem 3.3 a converse of the deletion result [3, Theorem 1.1].

Theorem 3.4 generalizes a similar result on conic-line arrangements of Schenck
and Tohaneanu, see [9, Theorem 2.5], describing the situation when the
addition or the deletion of a projective line from a free curve results in a
free curve.

Finally, summing up Theorems 3.1, 3.2, 3.3, 3.4 and [3, Theorems 1.1, 1.2], we
give in Theorems 3.6 and 3.5 a freeness criterion for curves in terms of
behaviour to addition or to deletion of projective lines.

## Acknowledgments

The first author is grateful to Jean Vallès for patiently answering many
questions regarding Chern polynomials.

##  2\. Preliminaries

Unless otherwise stated, $\mathcal{C}$ is a reduced curve in $\mathbb{P}^{2}$
and $L\subset\mathbb{P}^{2}$ is a line that is an irreducible component of
$\mathcal{C}$. Let $f_{\mathcal{C}}\in S$ be a homogeneous polynomial such
that $f_{\mathcal{C}}=0$ defines $\mathcal{C}$. We denote by
$(\mathcal{C},\mathcal{C}^{\prime},\mathcal{C}^{\prime\prime})$ the triple
obtained by the deletion-restriction with respect to the line $L$, i.e.,
$\mathcal{C}^{\prime}$ is the curve obtained by removing from $\mathcal{C}$
line $L$, and $\mathcal{C}^{\prime\prime}=\mathcal{C}^{\prime}\cap L$. Without
loss of generality, we may take $\alpha_{L}=z$ as the defining linear form of
the line $L$. Then $f_{\mathcal{C}}=zf_{\mathcal{C}^{\prime}}$.

Denote by ${\mathcal{E}}_{\mathcal{C}}$ the locally free sheaf (i.e. vector
bundle) obtained by the sheafification of the graded module
$D_{0}(\mathcal{C})$. Recall that
${\mathcal{E}}_{\mathcal{C}}=\mathcal{T}_{\mathcal{C}}(-1)$, where
$\mathcal{T}_{\mathcal{C}}$ is Saito’s sheaf of logarithmic vector ﬁelds along
$\mathcal{C}$ (see [7]).

######  Remark 2.1.

By [3, Theorem 2.3] (see also [8, Theorem 1.6, Remark 1.8]), we have an exact
sequence of sheaves (actually vector bundles) for the triple
$(\mathcal{C},\mathcal{C}^{\prime},\mathcal{C}^{\prime\prime})$ defined above,
namely

(2)
$0\rightarrow\mathcal{E}_{\mathcal{C}^{\prime}}(-1)\overset{\alpha_{L}}{\longrightarrow}\mathcal{E}_{\mathcal{C}}\rightarrow\mathcal{O}_{L}(1-|\mathcal{C}^{\prime\prime}|-\epsilon(\mathcal{C}^{\prime},L))\rightarrow
0,$

where $\epsilon(\mathcal{C}^{\prime},L)\in\mathbb{Z}$ is a constant defined by
Dimca in [3, page 2]. This constant measures the defect from the property of
being a quasi-homogeneous triple, namely if all singularities of the triple
$(\mathcal{C},\mathcal{C}^{\prime},\mathcal{C}^{\prime\prime})$ are quasi-
homogeneous, then the above exact sequence has becomes

(3)
$0\rightarrow\mathcal{E}_{\mathcal{C}^{\prime}}(-1)\overset{\alpha_{L}}{\longrightarrow}\mathcal{E}_{\mathcal{C}}\rightarrow\mathcal{O}_{L}(1-|\mathcal{C}^{\prime\prime}|)\rightarrow
0.$

Note also that by [3, Remark 2.4] we only need to assume that the
singularities of $\mathcal{C}$ on $L$ are quasi-homogeneous in order for the
exact sequence (3) to hold.

Denote by $c(\mathcal{E})$ the Chern polynomial of a complex vector bundle
$\mathcal{E}$. It is well-known that, for an exact sequence of vector bundles

$0\rightarrow\mathcal{E}\rightarrow\mathcal{F}\rightarrow\mathcal{G}\rightarrow
0,$

we have the relation

(4) $c(\mathcal{F})=c(\mathcal{E})c(\mathcal{G}).$

In particular, if $c_{i}(\mathcal{E})$ is the $i$-th Chern number of
$\mathcal{E}$, then we have the equality

(5)
$c_{2}(\mathcal{F})=c_{2}(\mathcal{E})+c_{1}(\mathcal{E})c_{1}(\mathcal{G})+c_{2}(\mathcal{G}).$

######  Lemma 2.2.

Let $\mathcal{E}$ be a rank $2$ vector bundle and
$c(\mathcal{E})=1+c_{1}(\mathcal{E})t+c_{2}(\mathcal{E})t^{2}$ its Chern
polynomial. Then the Chern polynomial of $\,\mathcal{E}(-1)$ has the following
form

$c(\mathcal{E}(-1))=1+(-2+c_{1}(\mathcal{E}))t+(c_{2}(\mathcal{E})-c_{1}(\mathcal{E})+1)t^{2}.$

###### Proof.

Immediate follows from the equality
$\mathcal{E}(-1)=\mathcal{E}\otimes_{\mathcal{O}_{\mathbb{P}^{2}}}\mathcal{O}_{\mathbb{P}^{2}}(-1)$
and the formula for the Chern classes of a tensor product of a rank $2$ vector
bundle and a line bundle, taking into account that
$c(\mathcal{O}_{\mathbb{P}^{2}}(-1))=1-t$. ∎

Let us recall a result due to Yoshinaga devoted to the splitting type along a
line of rank $2$ vector bundles on $\mathbb{P}^{2}$, which we will apply to
sheaves of logarithmic vector fields along a curve, see Remark 2.4.

######  Theorem 2.3 ([10]).

Let $\mathcal{E}$ be a rank $2$ vector bundle on $\mathbb{P}^{2}$, $L$ a line
in $\mathbb{P}^{2}$ and $\mathcal{E}|_{L}={\pazocal O}_{L}(-a)\oplus{\pazocal
O}_{L}(-b)$. Then

$c_{2}(\mathcal{E})-ab={\rm
dim}_{\mathbb{C}}Coker(\Gamma_{*}(\mathcal{E})\overset{\pi_{L}}{\longrightarrow}(\Gamma_{*}(\mathcal{E}|_{L})),$

where $c_{2}(\mathcal{E})$ is the second Chern number of $\mathcal{E}$ and
$\pi_{L}$ is the morphism of graded modules of sections induced by the
restriction to $L$ of the vector bundle $\mathcal{E}$. Moreover, $\mathcal{E}$
is splitting if and only if $\pi_{L}$ is surjective.

The pair $(a,b)$ is called the splitting type onto $L$ of the bundle
$\mathcal{E}$.

######  Remark 2.4.

For an arbitrary reduced curve $\mathcal{C}$ we have an isomorphism
$\Gamma_{*}(\mathcal{E}_{\mathcal{C}})\cong D_{0}(\mathcal{C})$, see [2,
Proposition 2.1] for details. Then $\mathcal{C}$ is free if and only if
$\mathcal{E}_{\mathcal{C}}$ is splitting.

Another important instrument to test the freeness property for hyperplane
arrangements is Saito’s criterion. A version of this result holds for reduced
hypersurfaces in $\mathbb{P}^{n}_{\mathbb{C}}$, in particular for reduced
curves in $\mathbb{P}^{2}$, see [9].

######  Theorem 2.5 (Saito’s criterion for curves).

Let $\mathcal{C}$ be a reduced curve in $\mathbb{P}^{2}$ defined by
$f_{\mathcal{C}}=0$ with $f_{\mathcal{C}}\in S$. Given a set of homogeneous
derivations $\\{\theta_{1},\theta_{2},\theta_{3}\\}\subset D(\mathcal{C})$,
the following are equivalent:

  1. (1)

$\\{\theta_{1},\theta_{2},\theta_{3}\\}$ is a basis in $D(\mathcal{C})$.

  2. (2)

The determinant of the matrix $M$ is equal to $f_{\mathcal{C}}$ (modulo a non-
zero constant), where $M$ is the $3\times 3$-matrix with lines
$[\theta_{i}(x)\;\theta_{i}(y)\;\theta_{i}(z)],\;i=1,2,3$.

We will denote by $mdr(\mathcal{C})=mdr(f_{\mathcal{C}})$ the minimal degree
of relations for $f_{\mathcal{C}}$, which is by definition the smallest degree
of a non-zero derivation from $D_{0}(\mathcal{C})$. It is known that for free
or for plus-one generated curves $\mathcal{C}$ one has $mdr(\mathcal{C})={\rm
min}\exp(\mathcal{C})={\rm min}\\{d_{2},d_{3}\\}$.

######  Proposition 2.6.

Let $\mathcal{C}$ be a plus-one generated curve with the exponents
$(d_{2},d_{3})$ and level $d$. Then the second Chern number of
${\mathcal{E}}_{\mathcal{C}}$ is given by the formula:

(6) $c_{2}({\mathcal{E}}_{\mathcal{C}})=d_{2}(d_{3}-1)+d-d_{3}+1$

###### Proof.

Assume without loss of generality that $d_{2}\leq d_{3}$. Since
$d_{2}+d_{3}=\deg(\mathcal{C})$, then
$d_{2}=mdr(\mathcal{C})\leq\deg(\mathcal{C})/2$. Let $L_{0}$ be a generic line
in $\mathbb{P}^{2}$ and denote by
$(d_{2}^{L_{0}},d_{3}^{L_{0}}),\;d_{2}^{L_{0}}\leq d_{3}^{L_{0}},$ the
splitting type onto $L_{0}$ of the vector bundle
${\mathcal{E}}_{\mathcal{C}}$. Recall that
$d_{2}^{L_{0}}+d_{3}^{L_{0}}=\deg(\mathcal{C})-1$. From [2], we know that

(7)
$c_{2}({\mathcal{E}}_{\mathcal{C}})=d_{2}^{L_{0}}d_{3}^{L_{0}}+\nu(\mathcal{C}),$

where $\nu(\mathcal{C})$ is the defect of the curve $\mathcal{C}$, which is
equal to $(d-d_{3}+1)$ by [6, Corollary 3.10]. In order to finish our proof,
it remains to connect the generic splitting type
$(d_{2}^{L_{0}},d_{3}^{L_{0}})$ with the exponents of $\mathcal{C}$. By [2,
Proposition 3.2 (1)], $d_{2}^{L_{0}}\leq mdr(\mathcal{C})$, i.e.,
$d_{2}^{L_{0}}\leq d_{2}$. There are, according to [2, Proposition 3.2
(2)-(4)], three cases to consider, namely:

Case 1: $d_{2}^{L_{0}}<(\deg(\mathcal{C})-2)/2$, and then
$d_{2}^{L_{0}}=mdr(\mathcal{C})$.

Case 2: $d_{2}^{L_{0}}\geq(\deg(\mathcal{C})-2)/2$ and $\deg(\mathcal{C})=2m$.
Then $d_{2}^{L_{0}}=m-1$, which implies that $mdr(\mathcal{C})\in\\{m-1,m\\}$.

Case 3: $d_{2}^{L_{0}}\geq(\deg(\mathcal{C})-2)/2$ and
$\deg(\mathcal{C})=2m+1$. Then $d_{2}^{L_{0}}=m$, so $m\leq
mdr(\mathcal{C})\leq(2m+1)/2$, i.e., $mdr(\mathcal{C})=m$.

In all the above cases we get a formula for $d_{2}^{L_{0}}$ in terms of
$mdr(\mathcal{C})$. In Case 1 and Case 3 we get
$d_{2}^{L_{0}}=mdr(\mathcal{C})$. In Case 2 we get either
$d_{2}^{L_{0}}=mdr(\mathcal{C})=m-1$ or
$d_{2}^{L_{0}}=mdr(\mathcal{C})-1=m-1$. Since
$mdr(\mathcal{C})=d_{2},\;d_{2}+d_{3}=\deg(\mathcal{C})$, and
$d_{2}^{L_{0}}+d_{3}^{L_{0}}=\deg(\mathcal{C})-1$, in all the above cases
formula (6) follows from (7). ∎

######  Corollary 2.7.

Let $\mathcal{C}$ be a plus-one generated curve with exponents
$(d_{2},d_{3}),\;d_{2}\leq d_{3},$ and level $d$. Let
$(d_{2}^{L_{0}},d_{3}^{L_{0}}),d_{2}^{L_{0}}\leq d_{3}^{L_{0}},$ be the
generic splitting type of ${\mathcal{E}}_{\mathcal{C}}$. Then
$d^{L_{0}}_{2}=d_{2}$, except possibly when $\deg(\mathcal{C})=2m$, when
$d_{2}^{L_{0}}\in\\{d_{2}-1,d_{2}\\}$.

###### Proof.

Immediate from the proof of Proposition 2.6. ∎

In fact, we are able to describe all possible splitting types of
${\mathcal{E}}_{\mathcal{C}}$ onto an arbitrary line $L\subset\mathbb{P}^{2}$,
if $\mathcal{C}$ is a plus-one generated curve.

######  Theorem 2.8.

Let $\mathcal{C}$ be a plus-one generated curve with exponents $(d_{2},d_{3})$
and level $d$. Then the splitting type of ${\mathcal{E}}_{\mathcal{C}}$ onto
an arbitrary line $L\subset\mathbb{P}^{2}$ is one of the pairs
$\\{(d_{2},d_{3}-1),(d_{2}-1,d_{3}),(d_{2}-(d-d_{3}+1),d)\\}$.

###### Proof.

The proof is essentially the same as the one of [4, Theorem 4.4], which proves
a similar result for plus-one generated projective line arrangements. We
reproduce the arguments here for completion.

Assume without loss of generality that $d_{2}\leq d_{3}$. Denote by
$(d^{L}_{2},d^{L}_{3}),\;d^{L}_{2}\leq d^{L}_{3}$ the the splitting type of
${\mathcal{E}}_{\mathcal{C}}$ onto the line $L$. Then
$d^{L}_{2}+d^{L}_{3}=\deg(\mathcal{C})-1$, see for instance [2, Proposition
3.1].

Since $\nu(\mathcal{C})=d-d_{3}+1$ ([6, Proposition 3.7]), by [2, Corollary
3.4] we have that

$(d^{L}_{2},d^{L}_{3})\in\\{(d_{2},d_{3}-1),(d_{2}-1,d_{3}),\dots,d_{2}-(d-d_{3}+1,d)\\}.$

If $d^{L}_{2}\in\\{d_{2},d_{2}-1\\}$ we are done, since
$d_{2}+d_{3}=\deg(\mathcal{C})$. It remains to consider the case
$d^{L}_{2}<d_{2}-1$. By Remark 2.4, the domain of the map $\pi_{L}$ from
Theorem 2.3 can be identified to the module of derivations
$D_{0}(\mathcal{C})$. One can choose the minimal system of generators
$\\{\theta_{E},\theta_{2},\theta_{3},\psi\\}$ of $D(\mathcal{C})$ from
Definition 1.1 such that $\\{\theta_{2},\theta_{3},\psi\\}$ is a minimal set
of generators for $D_{0}(\mathcal{C})$, with
$\deg(\theta_{2})=d_{2},\;\deg(\theta_{3})=d_{3},\;\deg(\psi)=d$.

Let $\\{\theta_{2}^{L},\theta_{3}^{L}\\}$ be a homogeneous basis of the
codomain of $\pi_{L}$, $\Gamma_{*}(\mathcal{E}_{\mathcal{C}}|_{L})$, which is
a free graded module of rank $2$ over $\overline{S}\coloneqq S/(\alpha_{L})$,
with $\deg(\theta_{2}^{L})=d^{L}_{2}$ and $\deg(\theta_{3}^{L})=d^{L}_{3}$.
Since $d^{L}_{2}<d_{2}-1$, we get $d^{L}_{3}>d_{3}$, hence
$\pi_{L}(\theta_{2}),\pi_{L}(\theta_{3})$ are in the $\overline{S}$, a
submodule of $\Gamma_{*}(\mathcal{E}_{\mathcal{C}}|_{L})$ generated by
$\theta_{2}^{L}$.

Let $\pi_{L}(\psi)=f\theta^{L}_{2}+g\theta^{L}_{3}$, for some
$f,g\in\overline{S}$. Then necessarily $g\neq 0$, otherwise we would have
$\dim_{\mathbb{C}}\operatorname{Coker}(\pi_{L})=\infty$, a contradiction with
Theorem 2.3. Moreover, we must have $\deg(g)=0$, otherwise we can construct an
infinite series of elements from $\Gamma_{*}(\mathcal{E}_{\mathcal{C}}|_{L})$
which are not in the image of $\pi_{L}$, for instance elements of type
$m\theta^{L}_{2}+\theta^{L}_{3}$, with $m\in\overline{S}$ arbitrary monomial,
again a contradiction to the finite dimensionality over $\mathbb{C}$ of
$\operatorname{Coker}(\pi_{L})$, stated in Theorem 2.3.

In conclusion, $g\in\mathbb{C}\setminus\\{0\\}$, i.e., $\deg\theta^{L}_{3}=d$.
∎

######  Remark 2.9.

If $\mathcal{C}$ is a reduced projective curve, then the first Chern number of
${\mathcal{E}}_{\mathcal{C}}$ is given by the formula
$c_{1}({\mathcal{E}}_{\mathcal{C}})=1-\deg(\mathcal{C})$, see for instance
[2].

######  Proposition 2.10.

Let $\mathcal{C},\mathcal{C}^{\prime}$ be reduced curves in $\mathbb{P}^{2}$
such that $\mathcal{C}=\mathcal{C}^{\prime}\cup\\{L\\}$, where
$L\subset\mathbb{P}^{2}$ is a line not included in $\mathcal{C^{\prime}}$
having the equation $\alpha_{L}=0$. Let
$\mathcal{C}^{\prime\prime}=\mathcal{C}^{\prime}\cap L$. Then we have the
exact sequence of derivation modules, where $\rho$ is defined by taking modulo
$\alpha_{L}$:

(8) $0\rightarrow
D(\mathcal{C}^{\prime})\overset{\cdot\alpha_{L}}{\rightarrow}D(\mathcal{C})\overset{\rho}{\rightarrow}D(\mathcal{C}")$

###### Proof.

Let $f_{\mathcal{C}}\in\mathbb{C}[x,y,z]$ be the defining polynomial of
$\mathcal{C}$. We may assume that $\alpha_{L}=z$. Then $f_{\mathcal{C}}=zf$
with $f\in\mathbb{C}[x,y,z]$, where $f$ is the defining polynomial for
$\mathcal{C}^{\prime}$. It is easy to check that if $\theta^{\prime}\in
D(\mathcal{C}^{\prime})$, then $z\theta^{\prime}\in D(\mathcal{C})$. Also,
notice that the multiplication map by $z$ is obviously injective.

Let $\theta\in D(\mathcal{C})$. Then $f_{\mathcal{C}}$ divides
$\theta(f_{\mathcal{C}})$. This implies, since ${\rm gcd}(z,f)=1$, that $z$
divides $\theta(z)$ and $f$ divides $\theta(f)$. Let

$\theta=f_{1}\partial_{x}+f_{2}\partial_{y}+f_{3}\partial_{z}\in\operatorname{Ker}(\rho).$

Since $z$ divides $\theta(z)$, we get that $z\,|\,f_{3}$. Now from

$\rho(\theta)=f_{1}(x,y,0)\partial_{x}+f_{2}(x,y,0)\partial_{y}=0$

it follows that $z\,|f_{1}$ and $z\,|f_{2}$. Then $\theta=z\theta^{\prime}$.
Since $f$ divides $\theta(f)=z\theta^{\prime}(f)$ and ${\rm gcd}(z,f)=1$, it
follows that $f$ divides $\theta^{\prime}(f)$, so $\theta^{\prime}\in
D(\mathcal{C}^{\prime})$. In conclusion, we obtain $\operatorname{Im}(\cdot
z)=\operatorname{Im}(\cdot\alpha_{L})\supset\operatorname{Ker}(\rho)$. The
converse inclusion is obvious, so we have
$\operatorname{Im}(\cdot\alpha_{L})=\operatorname{Ker}(\rho)$.

Let us check that the map $\rho$ is well-defined, i.e., for $\theta\in
D(\mathcal{C})$ we have $\rho(\theta)\in D(\mathcal{C}^{\prime\prime})$. For
$X\in\mathcal{C}^{\prime\prime}$ having a defining equation
$\gamma_{X}=0,\;\gamma_{X}\in\mathbb{C}[x,y]$, we have that $\gamma_{X}$
divides $f(x,y,0)$. Let $k\in\mathbb{N}_{\geq 1}$ be maximal such that

$f(x,y,0)=\gamma_{X}^{k}h,\;h\in\mathbb{C}[x,y],\;{\rm gcd}(\gamma_{X},h)=1.$

Evaluating at $z=0$ the expression $f\,|\,\theta(f)$ one gets
$f(x,y,0)\,|\,\rho(\theta)(f(x,y,0))$. Denote furthermore
$\overline{\theta}:=\rho(\theta)$. We have the divisibility

$\gamma_{X}^{k}h\,|\,\overline{\theta}(\gamma_{X}^{k}h)$

and we need to show that

$\gamma_{X}\,|\,\overline{\theta}(\gamma_{X}).$

Observe that
$\overline{\theta}(\gamma_{X}^{k}h)=\gamma_{X}^{k}\overline{\theta}(h)+h\overline{\theta}(\gamma_{X}^{k})$
and since ${\rm gcd}(\gamma_{X},h)=1$, one has
$\gamma_{X}^{k}\,|\,\overline{\theta}(\gamma_{X}^{k})$. If $k=1$, then we are
done. Otherwise, if $k>1$, we compute
$\overline{\theta}(\gamma_{X}^{k})=k\gamma_{X}^{k-1}\overline{\theta}(\gamma_{X})$
to conclude that $\gamma_{X}\,|\,\overline{\theta}(\gamma_{X})$. Since this
holds for an arbitrary $X\in\mathcal{C}^{\prime\prime}$, it follows
$\overline{\theta}\in D(\mathcal{C}^{\prime\prime})$. ∎

######  Remark 2.11.

If one takes $\mathcal{E}$ from Theorem 2.3 to be sheaf
$\mathcal{E}_{\mathcal{C}}$ of logarithmic vector fields along $\mathcal{C}$,
which is a locally free sheaf, i.e., a vector bundle by [7], then $\pi_{L}$ is
actually the restriction to $D_{0}(\mathcal{C})$ of the map $\rho$ from (8)
via the identification of $\Gamma_{*}(\widetilde{D_{0}(\mathcal{C})})$ to
$D_{0}(\mathcal{C})$, recalled in Remark 2.4.

##  3\. Main results

######  Theorem 3.1.

Let $\mathcal{C}$ be a plus-one generated reduced curve in $\mathbb{P}^{2}$
with the exponents $(d_{2},d_{3})$ and level $d$ such that there exists a line
$L\subset\mathbb{P}^{2}$ that is an irreducible component of $\mathcal{C}$,
$\mathcal{C}=\mathcal{C}^{\prime}\cup\\{L\\}$, $L$ is not included in
$\mathcal{C}^{\prime}$, and
$|\mathcal{C}^{\prime\prime}|+\epsilon(\mathcal{C}^{\prime},L)=d+1$, where
$\mathcal{C}^{\prime\prime}\coloneqq\mathcal{C}^{\prime}\cap L$. Then
$\mathcal{C}^{\prime}$ is free with the exponents $(d_{2}-1,d_{3}-1)$.

###### Proof.

A computation of the second Chern number of
$\mathcal{E}_{\mathcal{C}^{\prime}}$, using (5) and (2) and taking into
account that
$d+1=|\mathcal{C}^{\prime\prime}|+\epsilon(\mathcal{C}^{\prime},L)$, shows
that

$c_{2}(\mathcal{E}_{\mathcal{C}^{\prime}})=(d_{2}-1)(d_{3}-1)$

To prove that $\mathcal{C}^{\prime}$ is free, it would be enough to show that
there is a line $K$ in $\mathbb{P}^{2}$ such that the splitting type of
$\mathcal{E}_{\mathcal{C}^{\prime}}$ onto $K$ is equal to $(d_{2}-1,d_{3}-1)$.
We will show that $(d_{2}-1,d_{3}-1)$ is the generic splitting type of
$\mathcal{E}_{\mathcal{C}^{\prime}}$ onto a line. Let $K$ be a generic line,
with corresponding splitting type $(d_{2}^{K},d_{3}^{K}),\;d_{2}^{K}\leq
d_{3}^{K}$.

We may assume without loss of generality that $d_{2}\leq d_{3}$. Since
$d_{2}^{K}+d_{3}^{K}=d_{2}+d_{3}-2$ and $(d_{2}-1)(d_{3}-1)\geq
d_{2}^{K}d_{3}^{K}$ by Theorem 2.3, it follows that

$d_{2}^{K}\leq d_{2}-1$

From [2, Proposition 3.2 (2)-(4)], we know that
$d_{2}^{K}=mdr(\mathcal{C}^{\prime})$ except possibly when
$\deg(\mathcal{C}^{\prime})=2m$ and $d_{2}^{K}=m-1$ or
$\deg(\mathcal{C}^{\prime})=2m+1$ and $d_{2}^{K}=m$. We will treat separately
these cases.

First assume $d_{2}^{K}=mdr(\mathcal{C}^{\prime})$. By [5, Proposition 3.1],
$mdr(\mathcal{C}^{\prime})\in\\{d_{2}-1,d_{2}\\}$. If
$mdr(\mathcal{C}^{\prime})=d_{2}-1$ then $d_{2}^{K}=d_{2}-1$, so
$(d_{2}^{K},d_{3}^{K})=(d_{2}-1,d_{3}-1)$, which proves the claim.

If $mdr(\mathcal{C}^{\prime})=d_{2}$, then $d_{2}^{K}=d_{2}$, but this
contradicts the inequality $d_{2}^{K}\leq d_{2}-1$, so the case
$mdr(\mathcal{C}^{\prime})=d_{2}$ cannot happen.

Case $\deg(\mathcal{C}^{\prime})=2m,\;d_{2}^{K}=m-1$. Then $m\leq d_{2}$, but
$\deg(\mathcal{C})=2m+1$, so $d_{2}=m$. It follows
$(d_{2}^{K},d_{3}^{K})=(d_{2}-1,d_{3}-1)$, which proves the claim.

Case $\deg(\mathcal{C}^{\prime})=2m+1,\;d_{2}^{K}=m$. Then $m+1\leq d_{2}$,
but $\deg(\mathcal{C})=2m+2$, so $d_{2}=m+1$. Again it follows that
$(d_{2}^{K},d_{3}^{K})=(d_{2}-1,d_{3}-1)$, which completes our proof.
∎

It has come to our attention that the next result is also proved in a more
recent version of [3], but with a different proof.

######  Theorem 3.2.

Let $\mathcal{C}$ be a reduced free curve with the exponents $(d_{2},d_{3})$,
not necessarily ordered, such that $\mathcal{C}=\mathcal{C}^{\prime}\cup L$
for some line $L\subset\mathbb{P}^{2},\;L\not\subset\mathcal{C}^{\prime}$. Let
$\mathcal{C}^{\prime\prime}\coloneqq\mathcal{C}^{\prime}\cap L$.

  1. 1.

If $\mathcal{C}^{\prime}$ is free, then
$\exp(\mathcal{C}^{\prime})=(d_{2},d_{3}-1)$ and
$|\mathcal{C}^{\prime\prime}|+\epsilon(\mathcal{C}^{\prime},L)=d_{2}+1$.

  2. 2.

If $\mathcal{C}^{\prime}$ is plus-one generated, then
$\exp(\mathcal{C}^{\prime})=(d_{2},d_{3})$ and the level
$d=\deg(\mathcal{C}^{\prime})-|\mathcal{C}^{\prime\prime}|-\epsilon(\mathcal{C}^{\prime},L)$.

###### Proof.

Assume first $\mathcal{C}^{\prime}$ is free with the exponents
$(d^{\prime}_{2},d^{\prime}_{3}),\,d^{\prime}_{2}\leq d^{\prime}_{3}$. Notice
that $d_{2}+d_{3}=d^{\prime}_{2}+d^{\prime}_{3}+1$ and recall that by [5,
Proposition 3.1],
$mdr(\mathcal{C})=\min\\{d_{2},d_{3}\\}\in\\{d^{\prime}_{2},d^{\prime}_{2}+1\\}$.
It follows that

$(d^{\prime}_{2},d^{\prime}_{3})\in\\{(d_{2}-1,d_{3}),\;(d_{2},d_{3}-1)\\}.$

Assume without loss of generality that
$(d^{\prime}_{2},d^{\prime}_{3})=(d_{2},d_{3}-1)$, since $(d_{2},d_{3})$ are
unordered. Then the Chern polynomial has the following form

$c(\mathcal{E}_{\mathcal{C}^{\prime}})=1+(-d^{\prime}_{2}-d^{\prime}_{3})t+(d^{\prime}_{2}d^{\prime}_{3})t^{2}=1+(-d_{2}+1-d_{3})t+d_{2}(d_{3}-1)t^{2}$

and by Lemma 2.2,

$c(\mathcal{E}_{\mathcal{C}^{\prime}}(-1))=1+(-1-d_{2}-d_{3})t+(d_{2}d_{3}+d_{3})t^{2}.$

Since $\mathcal{C}$ is free with the exponents
$(d_{2},d_{3}),\;c(\mathcal{E}_{\mathcal{C}})=1+(-d_{2}-d_{3})t+d_{2}d_{3}t^{2}$.
Compute (5) for the exact sequence (2) to conclude
$|\mathcal{C}^{\prime\prime}|+\epsilon(\mathcal{C}^{\prime},L)=d_{2}+1$.

Assume next that $\mathcal{C}^{\prime}$ is plus-one generated with
$\exp(\mathcal{C}^{\prime})=(d^{\prime}_{2},d^{\prime}_{3}),\,d^{\prime}_{2}\leq
d^{\prime}_{3},$ and level $d$. Choose an order among the exponents of
$\mathcal{C}$, say $d_{2}\leq d_{3}$.

We have $d_{2}+d_{3}=d^{\prime}_{2}+d^{\prime}_{3}$ and, by [5, Proposition
3.1], $mdr(\mathcal{C})=d_{2}\in\\{d^{\prime}_{2},d^{\prime}_{2}+1\\}$. Assume
that $d_{2}=d^{\prime}_{2}+1$. Since
$f_{\mathcal{C}^{\prime}}\;|\;f_{\mathcal{C}}$, we have an inclusion between
modules of derivations:

$D(\mathcal{C})\subset D(\mathcal{C}^{\prime})$

Since $\mathcal{C}$ is free with the exponents $(d_{2},d_{3})$, the module
$D(\mathcal{C})$ is spanned by $\theta_{E}$ and two other homogeneous
derivations $\theta_{2},\theta_{3}$ of degrees $d_{2}=d^{\prime}_{2}+1$,
respectively $d_{3}=d^{\prime}_{3}-1$. The module of derivations
$D(\mathcal{C}^{\prime})$ of the plus-one generated curve
$\mathcal{C}^{\prime}$ is spanned by $\theta_{E}$ and three other homogeneous
derivations $\theta^{\prime}_{2},\theta^{\prime}_{3},\psi$ of degrees
$d^{\prime}_{2},\;d^{\prime}_{3}$, respectively $d$. Due to degree
constraints, both $\theta_{2},\theta_{3}$ are in the submodule of
$D(\mathcal{C}^{\prime})$ spanned by $\theta_{E}$ and $\theta^{\prime}_{2}$,
but this implies that $\theta_{2},\theta_{3},\theta_{E}$ are not
S-independent, contradiction. Then necessarily $d_{2}=d^{\prime}_{2}$, so

$\exp(\mathcal{C}^{\prime})=(d_{2},d_{3})$

It remains to prove the formula for the level $d$.

By Proposition 2.6,

$c(\mathcal{E}_{\mathcal{C}^{\prime}})=1+(1-d_{2}-d_{3})t+(d_{2}(d_{3}-1)+d-d_{3}+1)t^{2}.$

By Lemma 2.2,

$c(\mathcal{E}_{\mathcal{C}^{\prime}}(-1))=1+(-1-d_{2}-d_{3})t+(d_{2}d_{3}+d+1)t^{2}.$

Compute again (5) for the exact sequence (2), and the equality
$d=\deg(\mathcal{C}^{\prime})-|\mathcal{C}^{\prime\prime}|-\epsilon(\mathcal{C}^{\prime},L)$
follows.
∎

######  Theorem 3.3.

Let $\mathcal{C}^{\prime}$ be a plus-one generated curve with the exponents
$(d_{2},d_{3})$ and level
$d=\deg(\mathcal{C}^{\prime})-|\mathcal{C}^{\prime\prime}|-\epsilon(\mathcal{C}^{\prime},L),\mathcal{C}^{\prime\prime}\coloneqq\mathcal{C}^{\prime}\cap
L$ for some line $L\subset\mathbb{P}^{2}$ such that $L$ is not included in
$\mathcal{C}^{\prime}$. Then $\mathcal{C}:=\mathcal{C}^{\prime}\cup L$ is free
with the exponents $(d_{2},d_{3})$.

###### Proof.

Using Proposition 2.6 and Lemma 2.2, we get that

$c(\mathcal{E}_{\mathcal{C}^{\prime}}(-1))=1+(-1-d_{2}-d_{3})t+(d_{2}d_{3}+d+1)t^{2}.$

A computation of the second Chern number of $\mathcal{E}_{\mathcal{C}}$, using
(5) and (2) and taking into account that
$d=\deg(\mathcal{C}^{\prime})-|\mathcal{C}^{\prime\prime}|-\epsilon(\mathcal{C}^{\prime},L)$,
shows that

$c_{2}(\mathcal{E}_{\mathcal{C}})=d_{2}d_{3}$

By Theorem 2.3, to prove that $\mathcal{C}$ is free with the exponents
$(d_{2},d_{3})$, it would be enough to show that the generic splitting type of
$\mathcal{E}_{\mathcal{C}}$ onto a projective line is $(d_{2},d_{3})$.

Assume without loss of generality $d_{2}\leq d_{3}$. Let
$(d_{2}^{K},d_{3}^{K}),\,d_{2}^{K}\leq d_{3}^{K},$ be the splitting type of
$\mathcal{E}_{\mathcal{C}}$ onto some generic line $K\subset\mathbb{P}^{2}$.
In any case, from Theorem 2.3 it follows that

$d_{2}d_{3}\geq d_{2}^{K}d_{3}^{K}$

Since $d_{2}+d_{3}=d_{2}^{K}+d_{3}^{K}$, it follows that $d_{2}^{K}\leq
d_{2}$. From [2, Proposition 3.2 (2)-(4)] (see the proof of Proposition 2.6
for details) we know that $d_{2}^{K}=mdr(\mathcal{C})$ except possibly when
$\deg(\mathcal{C})=2m$ and $d_{2}^{K}=m-1$ or $\deg(\mathcal{C})=2m+1$ and
$d_{2}^{K}=m$. We will treat separately these cases.

For now, assume $d_{2}^{K}=mdr(\mathcal{C})$. Since
$mdr(\mathcal{C}^{\prime})=d_{2}$, then
$mdr(\mathcal{C})\in\\{d_{2},d_{2}+1\\}$. But $mdr(\mathcal{C})=d_{2}+1$ is
not possible, since $mdr(\mathcal{C})=d_{2}^{K}\leq d_{2}$. So
$mdr(\mathcal{C})=d_{2}=d_{2}^{K}$. But this means that the generic splitting
type of $\mathcal{E}_{\mathcal{C}}$ onto a line is $(d_{2},d_{3})$, and we are
done.

Case $\deg(\mathcal{C})=2m,\;d_{2}^{K}=m-1$. Notice that $d_{2}+d_{3}=2m-1$,
so $d_{2}\leq m-1$. But $d_{2}\geq d_{2}^{K}=m-1$, so $d_{2}=m-1$. It follows
$(d_{2}^{K},d_{3}^{K})=(d_{2},d_{3})$, and we are done.

Case $\deg(\mathcal{C})=2m+1,\;d_{2}^{K}=m$. Notice that $d_{2}+d_{3}=2m$, so
$d_{2}\leq m$. In fact, $d_{2}=m$ since $d_{2}\geq d_{2}^{K}$. Then
$(d_{2}^{K},d_{3}^{K})=(d_{2},d_{3})$, and this concludes the proof. ∎

######  Theorem 3.4.

Let $(\mathcal{C},\mathcal{C}^{\prime},\mathcal{C}^{\prime\prime})$ be a
triple with respect to a line $L\subset\mathbb{P}^{2}$, such that
$|\mathcal{C}^{\prime\prime}|+\epsilon(\mathcal{C}^{\prime},L)=d_{2}+1$. Then
the following are equivalent:

  1. (1)

$\mathcal{C}^{\prime}$ is free of exponents $(d_{2},d_{3})$.

  2. (2)

$\mathcal{C}$ is free of exponents $(d_{2},d_{3}+1)$.

###### Proof.

$(1)\Rightarrow(2)$ Assume $\mathcal{C}^{\prime}$ is free with the exponents
$(d_{2},d_{3})$. We already know that $\mathcal{C}=\mathcal{C}^{\prime}\cup L$
must be either free or plus-one generated. If it is free, then, since
$|\mathcal{C}^{\prime\prime}|+\epsilon(\mathcal{C}^{\prime},L)=d_{2}+1$, its
exponents are $(d_{2},d_{3}+1)$ by [3, Theorem 1.3].

If $\mathcal{C}$ would be plus-one generated, then its exponents would be
$(d_{2}+1,d_{3}+1)$ and level $d$ such that
$|\mathcal{C}^{\prime\prime}|+\epsilon(\mathcal{C}^{\prime},L)=d+1$. But then
$d=d_{2}$, so $d<d_{2}+1$, i.e., the level is strictly smaller than one of the
exponents of the plus-one generated $\mathcal{C}$, a contradiction.

$(2)\Rightarrow(1)$ Assume $\mathcal{C}$ is free of exponents
$(d_{2},d_{3}+1)$. We already know that $\mathcal{C}^{\prime}$ must be either
free or plus-one generated. If it is free, then, since
$|\mathcal{C}^{\prime\prime}|+\epsilon(\mathcal{C}^{\prime},L)=d_{2}+1$, by
Theorem 3.2, it must be free with the exponents $(d_{2},d_{3})$.

If $\mathcal{C}^{\prime}$ would be plus-one generated plus-one generated,
then, again by Theorem 3.2, its exponents would be $(d_{2},d_{3})$ and its
level
$d=\deg(\mathcal{C}^{\prime})-|\mathcal{C}^{\prime\prime}|-\epsilon(\mathcal{C}^{\prime},L)$.
Then $d=d_{3}-1<d_{3}$, contradiction. ∎

######  Theorem 3.5.

Let $\mathcal{C}$ be a reduced curve such that
$\mathcal{C}=\mathcal{C}^{\prime}\cup L$ for some line
$L\subset\mathbb{P}^{2},\;L\not\subset\mathcal{C}^{\prime}$.The following are
equivalent:

  1. 1.

$\mathcal{C}$ is free with exponents $(d_{2},d_{3})$.

  2. 2.

$\mathcal{C}^{\prime}$ is either

    1. (i)

free with $\exp(\mathcal{C}^{\prime})=(d_{2},d_{3}-1)$ and
$|\mathcal{C}^{\prime}\cap L|+\epsilon(\mathcal{C}^{\prime},L)=d_{2}+1$ or

    2. (ii)

plus-one generated with $\exp(\mathcal{C}^{\prime})=(d_{2},d_{3})$ and level
$d=\deg(\mathcal{C})-1-|\mathcal{C}^{\prime}\cap
L|-\epsilon(\mathcal{C}^{\prime},L)$.

###### Proof.

$1.\Rightarrow 2.$ By [3, Theorem 1.1] and Theorem 3.2.
$2.\Rightarrow 1.$ If (i) holds, then the claim follows from Theorem 3.4. If
(ii) holds, then the claim follows from Theorem 3.3. ∎

######  Theorem 3.6.

Let $\mathcal{C}$ be a reduced curve. The following are equivalent:

  1. 1.

$\mathcal{C}$ is free with exponents $(d_{2},d_{3})$.

  2. 2.

For some line $L\subset\mathbb{P}^{2}$ that is not an irreducible component of
$\mathcal{C}$, $\mathcal{C}\cup L$ is either

    1. (i)

free with exponents $(d_{2},d_{3}+1)$ and $|\mathcal{C}\cap
L|+\epsilon(\mathcal{C},L)=d_{2}+1$ or

    2. (ii)

plus-one generated with exponents $(d_{2}+1,d_{3}+1)$ and level
$d=\epsilon(\mathcal{C},L)+|\mathcal{C}\cap L|-1$.

  3. 3.

For any line $L\subset\mathbb{P}^{2}$ that is not an irreducible component of
$\mathcal{C}$, $\mathcal{C}\cup L$ is either

    1. (i)

free with exponents free with exponents $(d_{2},d_{3}+1)$ and
$|\mathcal{C}\cap L|+\epsilon(\mathcal{C},L)=d_{2}+1$ or

    2. (ii)

plus-one generated with exponents $(d_{2}+1,d_{3}+1)$ and level
$d=\epsilon(\mathcal{C},L)+|\mathcal{C}\cap L|-1$.

###### Proof.

$1.\Rightarrow 3.$ follows from Theorem 1.2. $3.\Rightarrow 2.$ is obvious.
Let us prove $2.\Rightarrow 1.$ If (i) holds, then the claim follows from
Theorem 3.4. If (ii) holds, then the claim follows from Theorem 3.1. ∎

######  Remark 3.7.

The above Theorems 3.5 and 3.6 generalize [1, Theorem 1.11], where the case of
projective line arrangements is treated.

## References

  * [1] T. Abe, Plus-one generated and next to free arrangements of hyperplanes, Int. Math. Res. Not. (2021), Vol. 2021, Issue 12, 9233–9261.
  * [2] T. Abe, A. Dimca, Splitting types of bundles of logarithmic vector fields along plane curves, Internat. J. Math. 29 (2018), no. 8, 1850055.
  * [3] A. Dimca, On plus-one generated curves arising from free curves, arXiv:2310.08972.
  * [4] T. Abe, D. Ibadula, A. Macinic, On some freeness-type properties for line arrangements, preprint arXiv:2101.02150, accepted for publication (2022) in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, DOI: $10.2422/2036-2145.202105\\_038$
  * [5] A. Dimca, G. Ilardi, G. Sticlaru, Addition-deletion results for the minimal degree of a Jacobian syzygy of a union of two curves, Journal of Algebra, Vol. 615 (2023), 77–102.
  * [6] A. Dimca, G. Sticlaru, Plane curves with three syzygies, minimal Tjurina curves curves, and nearly cuspidal curves, Geom. Dedicata 207 (2020), 29–49.
  * [7] K. Saito, Theory of logarithmic diﬀerential forms and logarithmic vector ﬁelds, J. Fac.Sci. Univ. Tokyo 27 (1980), 265–291.
  * [8] H. Schenck, H. Terao, M. Yoshinaga, Logarithmic vector ﬁelds for curve conﬁgurations in $\mathbb{P}^{2}$ with quasihomogeneous singularities, Math. Res. Lett. 25 (2018), 1977–1992.
  * [9] H. Schenck, S. Tohaneanu, Freeness of conic-line arrangements in $\mathbb{P}^{2}$, Comment. Math. Helv. 84(2) (2009), 235–258.
  * [10] M. Yoshinaga, Freeness of hyperplane arrangements and related topics, Annales de la Faculté des Sciences de Toulouse, Vol. XXIII (2014), 483–512.

Anca Măcinic, Simion Stoilow Institute of Mathematics of the Romanian Academy,
P.O. Box 1-764, RO-014700 Bucharest, Romania.
E-mail address<EMAIL_ADDRESS>

Piotr Pokora, Department of Mathematics, University of the National Education
Commission Krakow, Podchora̧żych 2, PL-30-084 Kraków, Poland.
E-mail address<EMAIL_ADDRESS>