A note on the stability of pencils of plane curves

https://doi.org/10.1007/s00209-021-02857-w

Mathematische Zeitschrift

A note on the stability of pencils of plane curves

Aline Zanardini¹

Received: 3 March 2021 / Accepted: 18 July 2021 / Published online: 30 August 2021

© The Author(s) 2021

Abstract

We investigate the problem of classifying pencils of plane curves of degreedup to projective equivalence. We obtain explicit stability criteria in terms of the log canonical threshold by relating the stability of a pencil to the stability of the curves lying on it.

Keywords Pencils of plane curves·GIT stability·Log canonical threshold Mathematics Subject Classification Primary 14L24; Secondary 14E99

1 Introduction

The groupP G L(3)acts naturally on the space of all pencils of curves of degreedinP²and in order to construct the corresponding classification space (via geometric invariant theory), a fundamental problem consists in explicitly describing what the (semi)stable pencils are, with respect to the action. In this paper we obtain explicit stability criteria in terms of some known invariants of singularities. We relate the stability of a pencil_Pto the number

lctp(P²,Cd)=sup{c∈Q; (P²,cCd)is log canonical atp}

which is known as the log canonical threshold of the pair(P²,Cd)atp, whereCdis a curve in Pandpis a base point (see [7, Section 8]). We also relate the stability ofPto the global log canonical threshold of the pair(P²,Cd), i.e. the numberlct(P²,Cd)=min_p∈Cdlct_p(P²,Cd) (cf. Definition2.4); and to the multiplicities of its generators at a base point.

LettingPd denote the space of all pencils of plane curves of degreed, our main results are given by Theorems1.1,1.2and1.3below.

Theorem 1.1 (= Theorem4.7)LetP be a pencil inPd containing a curve C_f such that lct(P²,C_f)=α. IfPis unstable (resp. not stable), thenPcontains a curve C_gsuch that lct(P²,C_g) < _2d^3α_α−3 (resp.≤).

Theorem 1.2 (= Theorem4.10)IfP∈Pdis semi-stable (resp. stable), then lctp(P²,Cf)≥

3

2d (resp.>) for any curve Cf inPand any base point p.

B

a.zanardini@math.leidenuniv.nl

1 Mathematical Institute, Leiden University, Leiden, The Netherlands

Theorem 1.3 (= Corollary3.6)LetPbe a pencil inPd. If we can find two curves C_f and CginPsuch that multp(Cf)+multp(Cg) > ^4d₃ (resp.≥) for some base point p, thenPis unstable (resp. not stable).

In particular, we extend and idea of Hacking [4] and Kim-Lee [5] who observed the following connection between two notions of stability, one coming from geometric invariant theory and the other coming from the Minimal Model Program: ifH ⊂Pⁿis a hypersurface of degreedand the pair

Pⁿ,ⁿ⁺_d¹H

is log canonical, thenH is semi-stable for the natural action ofP G L(n+1). And if

Pⁿ, (ⁿ⁺¹_d +ε)H

is log canonical for some 0< ε1, then His stable. Moreover, ford=3 andd=4, we recover some of the results from [9] and [1], respectively.

One of the key ingredients in our approach consists in observing that we can sometimes determine whether a pencilP ∈Pd is (semi)stable or not by looking at the stability of its generators. We also prove Theorems1.4,1.5and1.6below:

Theorem 1.4 (= Corollary3.12)If a pencil _P ∈ Pd has only semi-stable (resp. stable) members, thenPis semi-stable (resp. stable).

Theorem 1.5 (= Theorem3.14)IfP ∈Pd contains at most one strictly semi-stable curve (and all other curves inPare stable), thenPis stable.

Theorem 1.6 (= Theorem3.15)IfP∈Pdcontains at most two semi-stable curves C_f and C_g(and all other curves inPare stable), thenPis strictly semi-stable if and only if there exists a one-parameter subgroupλsuch that C_f and C_gare both non-stable with respect to thisλ.

The paper is organized as follows: Sect.2contains the relevant background material on log canonical pairs and on geometric invariant theory. In Sect.3we describe the stability criterion of Hilbert-Mumford for pencils of plane curves and we relate the stability of a pencil to the stability of its generators. Then, in Sect.4we use the notations and results from Sect.

3to relate the stability of a pencil to the log canonical threshold.

We work overCthroughout.

2 Background

For the convenience of the reader, we begin presenting some basic notions concerning log canonical pairs and the relevant background on geometric invariant theory.

2.1 The log canonical threshold

We first introduce the log canonical threshold, which will play an important role in our analysis of the stability of pencils of plane curves (Sect.4). We refer to [7, Section 8] for a more detailed exposition.

LetXbe a normal algebraic variety of dimensionnand letΔ=

diDi be an effective Q-divisor inX, i.e. aQ-linear combination of prime divisors with non-negative coefficients.

Definition 2.1 Given any birational morphismμ: ˜X→X, withX˜ normal, we writeK_X˜ ≡ μ^∗(KX+Δ)+

aEE, whereE⊂ ˜Xare distinct prime divisors,aE =. a(E,X, Δ)are the discrepanciesofEwith respect to(X, Δ)and a non-exceptional divisorEappears in the sum if and only ifE=μ⁻_∗¹D_ifor somei(in that case with coefficienta(E,X, Δ)= −di).

Definition 2.2 Alog resolutionof the pair(X, Δ)consists of a proper birational morphism μ: ˜X→Xsuch thatX˜is smooth andμ⁻¹_∗ (Δ)∪E xc(μ)is a simple normal crossings divisor (that is, each component is smooth and each point étale locally looks like the intersection of r≤ncoordinate hyperplanes).

Definition 2.3 We say(X, Δ)islog canonical(lc) ifKX+ΔisQ-Cartier and given any log resolutionμ: ˜X →Xwe haveK_X˜ ≡μ^∗(KX+Δ)+

aEEwith all the discrepancies satisfyinga_E≥ −1. In particular, ifXis smooth andΔ=d_iD_iis simple normal crossings, then(X, Δ)is log canonical if and only ifd_i ≤1 for alli.

Definition 2.4 The numberlct(X, Δ) .=sup{t ; (X,tΔ)is log canonical}is called thelog canonical thresholdof(X, Δ).

Remark 2.5 We can also consider a local version,lct_p(X, Δ), taking the supremum over all tsuch that(X,tΔ)is log canonical in an open neighborhood ofp, wherep∈Xis a closed point.

2.2 Geometric invariant theory

We now recall the relevant definitions and basic results from geometric invariant theory, and we point the reader to [3] for more details. The general setup consists of a reductive group Gacting on an algebraic varietyX, and we first consider the simple case whenXCⁿ⁺¹. Definition 2.6 A point x ∈ X is said to be semi-stable for the G−action if and only if 0∈/G·x.

Definition 2.7 A pointx ∈Xis said to bestablefor theG−action if and only if the following two conditions hold:

(i) The orbitG·x ⊂Xis closed and (ii) The stabilizerG_x ≤Gis finite

WhenX →Pⁿ is a projective variety, a pointx ∈ X will be called semi-stable (resp.

stable) if any pointx˜ ∈Cⁿ⁺¹ lying overx is semi-stable (resp. stable). Moreover, a point x ∈Xwill be called unstable ifxis not semi-stable.

From now on we assume thatXis a projective variety, embedded in some projective space.

Given a one-parameter subgroupλ:C^× →Gwe may regardCⁿ⁺¹as a representation of C^×. Since any representation ofC^×is completely reducible and every irreducible represen- tation is one dimensional, we can choose a basise₀, . . . ,e_nofCⁿ⁺¹so thatλ(t)·e_i =t^rie_i, for somer_i ∈Z. Then, givenx ∈X→Pⁿwe can pickx˜∈Cone(X)⊂Cⁿ⁺¹lying above xand writex˜=

xieiwith respect to this basis so thatλ(t)·x =. λ(t)· ˜x =

t^rixiei. The weights ofxare the set of integersrifor whichxi is not zero.

These notations allow us to define the so called Hilbert-Mumford weight of a pointx∈X: Definition 2.8 Given x ∈ Xand a one-parameter subgroupλ : C^× → G, we define the Hilbert-Mumford weight ofxatλto beμ(x, λ) .=min{ri : x_i=0}.

Remark 2.9 The Hilbert-Mumford weight satisfies the following properties:

(i) μ(x, λⁿ)=nμ(x, λ)for alln∈N (ii) μ(g·x,gλg⁻¹)=μ(x, λ)for allg∈G

The known numerical criterion for stability can thus be stated:

Theorem 2.10 (Hilbert-Mumford criterion) Let G be a reductive group acting linearly on a projective variety X →Pⁿ. Then for a point x ∈ X we have that x is semi-stable (resp.

stable) if and only ifμ(x, λ)≤0(resp.<) for all one-parameter subgroupsλof G.

That is, a pointx ∈ Xis unstable (resp. not stable) for theG-action if and only if there exists a one-parameter subgroupλ:C^×→Gfor which all the weights ofxare all positive (resp. non-negative).

In this paper we are interested in the case whereGis the groupP G L(3)andXis the space Pd of pencils of plane curves of degreed, embedded via Plücker coordinates in projective space.

3 Stability criterion for pencils of plane curves

As in [9], we view a pencil of plane curves of degreedas a choice of line in the space of all plane curves of degreed. We identify the spacePdof all such pencils with the Grassmannian Gr(2,S^dV^∗), where V =. H⁰(P²,O_P2(1)), which we further embed inP(Λ²S^dV^∗)via the Plücker embedding. The group P G L(V)acts naturally onV, hence on the invariant subvarietyPd, and our goal in this Section is to describe the corresponding (semi)stable points for the action.

It turns out that we are able to partially determine whether a pencilP∈Pdis (semi)stable or not by looking at the stability of its generators. Therefore, from now on we will consider the actions ofP G L(V)on bothPdandS^dV^∗, the space of plane curves of degreed.

The numerical criterion of Hilbert-Mumford (Theorem2.10) tells us we need to know how the diagonal elements ofP G L(V)act, and since both P G L(V)andS L(V)act with the same orbits, we will focus on the action of diagonal elements of the latter.

Note that if we choose a pencilP ∈ Pd and two curvesC_f andC_g as generators, these represented (in some choice of coordinates) by f =

f_{i j}xⁱy^jz^d−i−j = 0 and

g =

g_{i j}xⁱy^jz^d−i−j = 0, respectively; then the Plücker coordinates of_P are given by all the 2×2 minorsmi j kl =.

fi j fkl

g_{i j} g_kl

. Thus, the action of

⎛

⎝α 0 0

0 β 0

0 0 γ

⎞

⎠ ∈S L(V)on the Plücker coordinates is given by

(mi j kl)→(α^i+kβ^j+lγ^{2d−i−j−k−l}mi j kl)

In order to obtain the desired stability criteria from Theorems1.1and1.2, the first step in our approach consists in introducing an “affine" analogue of the Hilbert-Mumford weight (see Definition2.8) and then translating the Hilbert-Mumford criterion in terms of this quantity (Proposition3.2). The definition is as follows:

Definition 3.1 Given_P ∈ Pd and a one-parameter subgroupλofS L(V), we define the affine weight ofPatλto be

ω(P, λ) .=min{(ax−az)(i+k)+(ay−az)(j+l) : mi j kl=0}

where we choose coordinates inP²so thatλ:C^×→S L(V)is given by t →

⎛

⎝t^ax 0 0 0 t^ay 0 0 0 t^az

⎞

⎠ (1)

for some weightsa_x,a_y,a_z ∈Zwitha_x≥a_y≥a_z,a_x>0 anda_x+a_y+a_z=0 Stated in terms ofω(P, λ), the Hilbert-Mumford criterion becomes:

Proposition 3.2 A pencilP ∈ Pd is unstable (resp. not stable) if and only if there exists a one-parameter subgroupλ:C^× → S L(V)and a choice of coordinates inP²such that ω(P, λ) > ^2d₃ (a_x+a_y−2a_z)(resp.≥)

Proof A pencil P ∈ Pd is unstable (resp. not stable) if and only if there exists a one- parameter subgroupλ:C^× →S L(V)and a choice of coordinates inP²satisfying that for anyi,j,kandlsuch thatmi j kl=0 (in those coordinates) we haveax(i+k)+ay(j+l)+ a_z(2d−i−j−k−l) >0(resp.≥) if and only if

(ax−az)(i+k)+(ay−az)(j+l)−2d

3 (ax+ay−2az) >0 (resp. ≥0) Similarly, we define an affine weight for plane curves of degreed:

Definition 3.3 Given a plane curve of degreed Cf and a one-parameter subgroupλ:C^×→ S L(V)we define the affine weight of f atλto be

ω(f, λ) .=min{(ax−az)i+(ay−az)j : fi j =0}

And for curves the Hilbert-Mumford criterion becomes:

Proposition 3.4 A curve C_f is unstable (resp. not stable) if and only if there exists a one- parameter subgroupλ:C^×→S L(V)and a choice of coordinates inP²such thatω(f, λ) >

d

3(ax+ay−2az)(resp.≥).

The inspiration for Definition3.1comes from [6, Definition 2.2] and it is justified by Corollary4.2. Given a pencilP∈Pdand a curveC_f ∈P, the idea will be to use this affine weight to bound the log canonical threshold of the pair(P²,C_f)at a base point of_P, as well as the global log canonical threshold.

3.1 The stability of the generators

Given a pencilP∈Pd and a curveCf ∈P, it is interesting to compare the affine weights ω(f, λ)andω(P, λ)for a fixed one-parameter subgroupλ. We state and prove a series of results in this direction that allow us to relate the stability of a pencil to the stability of its generators (Corollary3.12and Theorems3.14and3.15).

Even when omitted, we will always choose coordinates[x,y,z]in P² so that a one- parameter subgroupλ:C^×→S L(V)is normalized as in (1).

Proposition 3.5 Given a pencilP∈Pd and any two (distinct) curves Cf,Cg∈Pwe have thatω(f, λ)≤ω(f, λ)+ω(g, λ)≤ω(P, λ), for all one-parameter subgroupsλ:C^×→ S L(V).

Proof Given P andλ : C^× → S L(V), choose coordinates in P² that normalizeλand choose any two curvesCf andCgofPso thatPis represented by the Plücker coordinates m_{i j kl}= f_{i j}g_kl−g_{i j}f_kl.

Leti,j,kandlbe such thatm_{i j kl}= f_{i j}g_kl−g_{i j}f_kl =0 and ω(P, λ)=(a_x−a_z)(i+k)+(a_y−a_z)(j+l)

Then eitheri andj are such that f_{i j} =0 orkandlare such that f_kl =0. In the first case there are two possibilities: eitherg_kl =0, which impliesg_{i j} =0 and f_kl =0; org_kl =0.

Similarly, in the second case eitherg_{i j}=0, which impliesg_kl =0 and f_{i j}=0; org_{i j}=0.

In any case we have

(ax−az)(i+k)+(ay−az)(j+l)=

(ax−az)i+(ay−az)j + +

(a_x−a_z)k+(a_y−a_z)l

≥ω(f, λ)+ω(g, λ)

As a consequence, we can relate the stability of a pencilP∈Pd to the multiplicity of its generators at a base point.

Corollary 3.6 LetP be a pencil inPd with generators C_f and C_g. If there exists a base point p ofPsuch that mult_p(Cf)+mult_p(Cg) > ^4d₃ (resp.≥), thenPis unstable (resp. not stable).

Proof IfPis any base point ofP, we can always choose coordinates so that we havep=(0: 0:1). Lettingax =1,ay=1,az = −2 andλbe the corresponding one-parameter subgroup (which in these coordinates is normalized as in (1)), we have thatω(f, λ)=3·multp(Cf) andω(g, λ)=3·multp(Cg)for any choice of generators ofP, sayCf andCg. These two equalities, together with Proposition3.5, imply

ω(P, λ)

(a_x−a_z)+(a_y−a_z) ≥3·mult_p(C_f)+mult_p(C_g) (a_x−a_z)+(a_y−a_z)

and since(ax−az)+(ay−az)=6, the result follows from the Hilbert-Mumford criterion

(Proposition3.2).

Remark 3.7 In [2, Lemma 3.3] it is proved that a plane curve of degreed, sayCd, satisfying mult_pCd) > ^2d₃ is unstable. Our proof of Corollary3.6above is an adaptation of the argument therein.

Example 3.8 Assumed<6. Let_P∈Pd be a pencil which contains a curve_Cd consisting ofdlines through a common pointp, and which contains another curve with a double point atp. ThenPis unstable.

Note that the content of Proposition3.5gives a lower bound for the affine weightω(P, λ).

We can also find an upper bound:

Proposition 3.9 GivenP ∈ Pd, a one-parameter subgroupλ : C^× → S L(V) and any curve C_f ∈Pthere exists a curve C_gin_Psuch that

ω(P, λ)≤ω(f, λ)+ω(g, λ)

Proof Fix a one-parameter subgroupλ:C^×→S L(V)and coordinates inP²that normalize λ. Choose any two curvesCf andCg ofP. Letiandjbe such that fi j=0 andω(f, λ)= (a_x−a_z)i+(a_y−a_z)j. Replacinggbyg=g−^g_f^{i j}_{i j} f we haveg_{i j} =0, hencem_{i j kl} =0 for allkandlsuch thatg_kl =0 and it follows thatω(P, λ)≤ω(f, λ)+ω(g, λ).

Corollary 3.10 GivenP∈Pd, a one-parameter subgroupλ:C^× →S L(V)and any curve C_f ∈Pthere exists a curve C_ginPsuch that

ω(P, λ)≤2 max{ω(f, λ), ω(g, λ)}

Corollary 3.11 GivenP∈Pd, a one-parameter subgroupλ:C^× →S L(V)and any curve C_f ∈Pthere exists a curve C_gin_Psuch that

ω(P, λ)=ω(f, λ)+ω(g, λ)

Corollary 3.12 If a pencil_P ∈Pd has only semi-stable (resp. stable) members, then_Pis semi-stable (resp. stable).

Corollary 3.13 If a pencil _P ∈ Pd contains only plane curves C_d such that the pairs P²,3/dCd

(resp.

P², (3/d+ε)Cd

,0 < ε << 1) are log canonical, thenPis semi- stable (resp. stable).

Proof As observed in [4] and [5], in this case all members ofPare semi-stable (resp. stable).

As a result of our comparison betweenω(f, λ)andω(P, λ)we can now prove Theorems 3.14and3.15below:

Theorem 3.14 IfP ∈ Pd contains at most one strictly semi-stable curve (and all other curves inPare stable), thenPis stable.

Proof GivenPas above, if all curves inPare stable, thenP is stable by Corollary3.12.

Otherwise, letCf be the unique strictly semi-stable curve inP. Given any one-parameter subgroupλ, by Proposition3.9there exists a curveC_g such that

ω(P, λ)

(ax−az)+(ay−az) ≤ ω(f, λ)

(ax−az)+(ay−az) + ω(g, λ) (ax−az)+(ay−az) And becauseCf (resp.Cg) is strictly semi-stable (resp. stable) it follows that

ω(f, λ)

(a_x−a_z)+(a_y−a_z) ≤d

3 and ω(h, λ)

(a_x−a_z)+(a_y−a_z) < d 3 and hence_(a ^ω(P,λ)

x−az)+(ay−az) < ^2d₃. That is,Pis stable.

Theorem 3.15 IfP∈Pdcontains at most two semi-stable curves C_f and C_g(and all other curves in_Pare stable), then_Pis strictly semi-stable if and only if there exists a one-parameter subgroupλ(and coordinates inP²) such that Cf and Cgare both non-stable with respect to thisλthat is,

ω(f, λ)

(ax−az)+(ay−az) = d

3 and ω(g, λ)

(ax−az)+(ay−az) = d 3

Proof FixPas above and note thatP is semi-stable by Corollary3.12. Next, note that if the two inequalities above hold for someλ, thenPis strictly semi-stable by Proposition3.5.

Thus, assumePis strictly semi-stable. Then there exists a one-parameter subgroupλ(and

coordinates inP²) such that _(a ^ω(P,λ)

x−az)+(ay−az) = ^2d₃ and, by Corollary3.10, it must exist a curveChinPsuch that

d 3 ≤max

ω(f, λ)

(ax−a_z)+(ay−a_z), ω(h, λ) (ax−a_z)+(ay−a_z)

In particular, eitherC_f orC_his non-stable with respect to thisλ. ButC_f andC_gare the only potentially non-stable curves inP. Therefore, either

ω(f, λ)

(ax−a_z)+(ay−a_z) ≥d

3 (2)

or

Ch =Cg and ω(g, λ)

(ax−az)+(ay−az) ≥d

3 (3)

In any case, we claim that the following equalities hold ω(f, λ)

(a_x−a_z)+(a_y−a_z) = ω(g, λ)

(a_x−a_z)+(a_y−a_z) = d 3 In fact, ifC_h =C_g and (3) holds, then _(a ^ω(g,λ)

x−az)+(ay−az) = ^d₃ becauseC_g is semi-stable.

Thus, by Proposition3.9, inequality (2) must be true also.

Now, if (2) holds, then _(a ^ω(f,λ)

x−az)+(ay−az) = ^d₃ becauseC_f is semi-stable. Thus, by Propo- sition3.9, we have that _(a ^ω(h,λ)

x−az)+(ay−az) ≥ ^d₃ and, by assumption, it must be the case that

C_h=C_g(and (3) holds).

4 Stability and the log canonical threshold

We are now ready to describe howω(P, λ)andω(f, λ)are related to the log canonical threshold of the pair(P²,Cf). We first recall the following known result and its corollary:

Proposition 4.1 (=[7, Proposition 8.13]) Let f be a holomorphic function near0 ∈ Cⁿ. Assign rational weights ω(xi) to the variables x1, . . . ,xn and letω(f) be the weighted multiplicity of f (i.e. the lowest weight of the monomials appearing in f ). Then

lct₀(Cⁿ,f)≤

ω(x_i) ω(f) Corollary 4.2 Let Cf be any plane curve. Then

ω(f, λ)

(a_x−a_z)+(a_y−a_z) ≤ 1

lct(P²,C_f) (4)

for any one-parameter subgroupλ:C^×→S L(V).

Remark 4.3 The main idea behind Corollary4.2is that in order to check whether the pair (P²,tCf)is log canonical, it suffices to check that for each weighted blowup of a point p in the plane we havea(E,P²,tCf)≥ −1, whereEdenotes the corresponding exceptional divisor (Cf. [4, Section 10]).

Corollary4.2together with Corollary3.10allow us to conclude that:

Proposition 4.4 Given a pencilP ∈ Pd we have that for any one-parameter subgroup λ:C^×→S L(V)there exists C_f ∈Psuch that

ω(P, λ)

(a_x−a_z)+(a_y−a_z) ≤ 2

lct(P²,C_f) (5)

And, as a consequence, we recover the statement from Corollary3.13:

Corollary 4.5 IfP∈Pdis a pencil such that lct(P²,C_f)≥3/d (resp.>) for any curve C_f in_P, then_Pis semi-stable (resp. stable).

Next, we prove the following:

Proposition 4.6 Given_P ∈ Pd and any base point p of_P, there exists a one-parameter subgroupλ: C^× →S L(V)(and coordinates inP²) such that for any curve Cf inPwe have that ^(ax−a_ω(P,λ)^z)+(ay−az) ≤lctp(P²,Cf).

Proof GivenP and a base pointp, we can always choose coordinates inP² so that p = (0:0:1). We can then consider any one-parameter subgroupλ, which in these coordinates is normalized as in (1) for some choice of integersa_x,a_yanda_z, witha_y−a_z =0. Then c₀=. ^(ax−a_ω(P^z)+(a_,λ)^y−az) ≤1, becausef₀₀=0 for any curveC_f inPso that we havem_00kl=0 for all 0≤k,l≤d.

We claim that any choice ofλas above is such that for any curve Cf inP we have c0≤lctp(P²,Cf).

By contradiction, assume there exists Cf in P such that lctp(P²,Cf) < c0. Write f˜(u, v)= f(x,y,1)and assign weightsω(u) .=ax−azto the variableuandω(v) .=ay−az

to the variablevso that the weighted multiplicity of f˜is preciselyω(f, λ). Now, consider the finite morphismϕ:C²→C²given by(u, v)→(u^ω(u), v^ω(v))and let

Δ .=(1−ω(u))Hu+(1−ω(v))H_v+c· ˜f(u^ω(u), v^ω(v)) whereHu(resp.H_v) is the divisor ofu=0 (resp.v=0) andc∈Q∩ [0,1].

Thenϕ^∗(K_C2+c· ˜f(u, v))=K_C2+Δand by Proposition 5.20 (4) in [8] we know that the pair(C²,c· ˜f)is log canonical at(0,0)if and only if the pair(C², Δ)is log canonical at(0,0). In particular, takingc=c₀>lct_p(P²,C_f)=lct₀(C²,f˜)it follows that

a(E;C², Δ)= −1+ω(u)+ω(v)−c·ω(f, λ) <−1

whereEis the exceptional divisor of the blow-up ofC²at the origin anda(E;C², Δ)is the corresponding discrepancy. But the last inequality is equivalent to the inequalityω(P, λ) <

ω(f, λ), contradicting Proposition3.5.

Finally, Proposition4.6above and Corollary4.2together with the other results obtained in this paper, allow us to prove Theorems4.7and4.10below, thus providing explicit stability criteria.

Theorem 4.7 LetPbe a pencil inPdwhich contains a curve C_f such that lct(P²,C_f)=α. IfPis unstable (resp. not stable), thenPcontains a curve Cgsuch that lct(P²,Cg) < _2d^3α_α−3 (resp.≤).

Proof If P is unstable (resp. not stable), then by Proposition3.2we can choose a one- parameter subgroupλ(and coordinates inP²) so that

2d

3 < ω(P, λ)

(ax−az)+(ay−az) (resp. ≤) By Proposition3.9, we can find a a curveCginPsuch that

ω(P, λ)

(a_x−a_z)+(a_y−a_z) ≤ ω(f, λ)

(a_x−a_z)+(a_y−a_z) + ω(g, λ) (a_x−a_z)+(a_y−a_z) Moreover, by Corollary4.2we have that

ω(f, λ)

(ax−az)+(ay−az) ≤ 1

lct(P²,Cf) and ω(g, λ)

(ax−az)+(ay−az) ≤ 1 lct(P²,Cg) Now, because lct(P²,Cf) = α, combining the above inequalities we conclude that

lct(P²,C_g) < _2dα−3^3α (resp.≤).

Corollary 4.8 Assume d ≥5and letP∈Pd be a pencil which contains a smooth member.

If any curveCdinPis reduced and any point inCd has multiplicity m<d, thenPis stable.

Proof It was proved in [2] that under such conditons any such curveCdsatisfieslct(P²,Cd)≥

(d−1)2d−3². Now, becaused ≥5 we have that _(d−1)^2d−3₂ > _2d³₋₃ and the conclusion follows from

Theorem4.7withα=1.

Corollary 4.9 LetP∈Pdbe a pencil which contains a smooth member. If any curveCd ∈P contains only points p with multiplicity mp ≤ ^2d−3₃ (resp.<), thenPis semi-stable (resp.

stable).

Proof In fact, any curveCdinPsatisfies_m¹

p ≤lct(P²,Cd)(see [7, Lemma 8.10]) and since

2d−33 ≤_m¹_p (r esp. <), the conclusion follows from Theorem4.7withα=1.

Theorem 4.10 IfP∈Pd is semi-stable (resp. stable), then for any curve Cf inPand any base point p of_Pwe have_2d³ ≤lct_p(P²,C_f)(resp.<).

Proof FixP∈Pd and a base point pas above. GivenC_f we can always find coordinates inP² so that p = (0 : 0 : 1)and we can chooseλas in Proposition4.6. BecauseP is semi-stable (resp. stable) for thisλwe have that

3

2d ≤(ax−az)+(ay−az)

ω(P, λ) (resp. <)

and the result follows from Proposition4.6.

Corollary 4.11 LetP∈Pd be a pencil which contains a curveCd of the form m L+Cd−m, where m L is a multiple line with multiplicity m ≥2d/3(resp.>) and C_d−m is a curve of degree d−m. ThenPis unstable (resp. not stable). In particular, ifPcontains a multiple line with multiplicity d, then_Pis unstable.

Note that Theorems4.7and4.10provide explicit stability criteria for a pencilP∈Pdin terms log canonical thresholds of pairs(P²,Cf), whereC_fis a curve lying inP. As suggested by the referee, it is interesting to observe that the stability ofPseems to also be related to the log canonical threshold of the pair(P²,P)– where we extend the notions introduced in Sect.

2to linear systems as in [7, Definition 4.6]. We prove the following Corollary to Theorem 4.10and we hope to further investigate this relation in a future project.

Corollary 4.12 IfP∈Pdis semi-stable (resp. stable), then _2d³ ≤lct(P²,P)(resp.<).

Proof GivenP∈Pd, let us denote its general member byC^gen. By [7, Theorem 4.8] we have thatlct(P²,P) =lct(P²,C^gen). By Bertini’s theorem,C^genis smooth away from the base points ofP, hencelct(P²,P)=lct_p(P²,C^gen)for some base pointp. Now, it follows from [7, Lemma 8.6] thatlct_p(P²,C_f)≤lct_p(P²,C^gen)for any curveC_finP. As a consequence, Theorem4.10implies that wheneverPis semi-stable (resp. stable), then_2d³ ≤lct(P²,P)

(resp.<).

Acknowledgements I am grateful to my advisor, Antonella Grassi, for her constant guidance, the many conversations and the numerous suggestions on earlier versions of this paper. I am also thankful to the referee for providing insightful comments. This work is part of my PhD thesis and it was partially supported by a Dissertation Completion Fellowship at the University of Pennsylvania.

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References

1. Ballico, E., Oliverio, P.: Stability of pencils of plane quartic curves. Atti Accad. Naz. Lincei Rend. Cl. Sci.

Fis. Mat. Nat.74(4), 234–241 (1983)

2. Cheltsov, I.: Worst singularities of plane curves of given degree. J. Geom. Anal.27(3), 2302–2338 (2017) 3. Dolgachev, I.: Lectures on Invariant Theory: London Mathematical Society Lecture Note Series. Cambridge

University Press, Cambridge (2003)

4. Hacking, P.: Compact moduli of plane curves. Duke Math. J.124(2), 213–257 (2004)

5. Kim, H., Lee, Y.: Log canonical thresholds of semistable plane curves. Math. Proc. Camb. Philos. Soc.

137(2), 273–280 (2004)

6. Kollár, J.: Polynomials with integral coefficients, equivalent to a given polynomial. Electron. Res. Announc.

Am. Math. Soc.3(3), 17–27 (1997). (Electronic only)

7. Kollár, J.: Singularities of pairs. In: Algebraic geometry—Santa Cruz 1995,Proceedings of symposium on Pure Mathematics, vol. 62, pp. 221–287. American Mathematical Society, Providence, RI (1997) 8. Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, vol.

134. Cambridge University Press, Cambridge (2008)

9. Miranda, R.: On the stability of pencils of cubic curves. Am. J. Math.102(6), 1177–1202 (1980)

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