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 pencilPto 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)=minp∈Cdlctp(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 Cf such that lct(P2,Cf)=α. IfPis unstable (resp. not stable), thenPcontains a curve Cgsuch that lct(P2,Cg) < 2d3αα−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
Aline Zanardinia.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 Cf 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+1d +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 Cf and Cg(and all other curves inPare stable), thenPis strictly semi-stable if and only if there exists a one-parameter subgroupλsuch that Cf and Cgare 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 writeKX˜ ≡ 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−∗1Difor 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 haveKX˜ ≡μ∗(KX+Δ)+
aEEwith all the discrepancies satisfyingaE≥ −1. In particular, ifXis smooth andΔ=diDiis simple normal crossings, then(X, Δ)is log canonical if and only ifdi ≤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,lctp(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 stabilizerGx ≤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, . . . ,enofCn+1so thatλ(t)·ei =triei, for someri ∈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 =
trixiei. 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 : xi=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,SdV∗), where V =. H0(P2,OP2(1)), which we further embed inP(Λ2SdV∗)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 bothPdandSdV∗, 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 curvesCf andCg as generators, these represented (in some choice of coordinates) by f =
fi jxiyjzd−i−j = 0 and
g =
gi jxiyjzd−i−j = 0, respectively; then the Plücker coordinates ofP are given by all the 2×2 minorsmi j kl =.
fi j fkl
gi j gkl
. 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)→(αi+kβj+lγ2d−i−j−k−lmi 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 GivenP ∈ 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 →
⎛
⎝tax 0 0 0 tay 0 0 0 taz
⎞
⎠ (1)
for some weightsax,ay,az ∈Zwithax≥ay≥az,ax>0 andax+ay+az=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 (ax+ay−2az)(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)+ az(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 Cf 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 curveCf ∈P, the idea will be to use this affine weight to bound the log canonical threshold of the pair(P2,Cf)at a base point ofP, 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 mi j kl= fi jgkl−gi jfkl.
Leti,j,kandlbe such thatmi j kl= fi jgkl−gi jfkl =0 and ω(P, λ)=(ax−az)(i+k)+(ay−az)(j+l)
Then eitheri andj are such that fi j =0 orkandlare such that fkl =0. In the first case there are two possibilities: eithergkl =0, which impliesgi j =0 and fkl =0; orgkl =0.
Similarly, in the second case eithergi j=0, which impliesgkl =0 and fi j=0; orgi j=0.
In any case we have
(ax−az)(i+k)+(ay−az)(j+l)=
(ax−az)i+(ay−az)j + +
(ax−az)k+(ay−az)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 Cf and Cg. If there exists a base point p ofPsuch that multp(Cf)+multp(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)
(ax−az)+(ay−az) ≥3·multp(Cf)+multp(Cg) (ax−az)+(ay−az)
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 multpCd) > 2d3 is unstable. Our proof of Corollary3.6above is an adaptation of the argument therein.
Example 3.8 Assumed<6. LetP∈Pd be a pencil which contains a curveCd 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 Cf ∈Pthere exists a curve CginPsuch 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)= (ax−az)i+(ay−az)j. Replacinggbyg=g−gfi ji j f we havegi j =0, hencemi j kl =0 for allkandlsuch thatgkl =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 Cf ∈Pthere exists a curve CginPsuch 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 Cf ∈Pthere exists a curve CginPsuch that
ω(P, λ)=ω(f, λ)+ω(g, λ)
Corollary 3.12 If a pencilP ∈Pd has only semi-stable (resp. stable) members, thenPis semi-stable (resp. stable).
Corollary 3.13 If a pencil P ∈ Pd contains only plane curves Cd 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 curveCg 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)
(ax−az)+(ay−az) ≤d
3 and ω(h, λ)
(ax−az)+(ay−az) < 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 Cf and Cg(and all other curves inPare stable), thenPis 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−az)+(ay−az), ω(h, λ) (ax−az)+(ay−az)
In particular, eitherCf orChis non-stable with respect to thisλ. ButCf andCgare the only potentially non-stable curves inP. Therefore, either
ω(f, λ)
(ax−az)+(ay−az) ≥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)
(ax−az)+(ay−az) = ω(g, λ)
(ax−az)+(ay−az) = d 3 In fact, ifCh =Cg and (3) holds, then (a ω(g,λ)
x−az)+(ay−az) = d3 becauseCg 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 becauseCf 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
Ch=Cg(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(xi) ω(f) Corollary 4.2 Let Cf be any plane curve. Then
ω(f, λ)
(ax−az)+(ay−az) ≤ 1
lct(P2,Cf) (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 Cf ∈Psuch that
ω(P, λ)
(ax−az)+(ay−az) ≤ 2
lct(P2,Cf) (5)
And, as a consequence, we recover the statement from Corollary3.13:
Corollary 4.5 IfP∈Pdis a pencil such that lct(P2,Cf)≥3/d (resp.>) for any curve Cf inP, thenPis semi-stable (resp. stable).
Next, we prove the following:
Proposition 4.6 GivenP ∈ Pd and any base point p ofP, there exists a one-parameter subgroupλ: C× →S L(V)(and coordinates inP2) such that for any curve Cf inPwe have that (ax−aω(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 integersax,ayandaz, withay−az =0. Then c0=. (ax−aω(Pz)+(a,λ)y−az) ≤1, becausef00=0 for any curveCf inPso that we havem00kl=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)→(uω(u), vω(v))and let
Δ .=(1−ω(u))Hu+(1−ω(v))Hv+c· ˜f(uω(u), vω(v)) whereHu(resp.Hv) is the divisor ofu=0 (resp.v=0) andc∈Q∩ [0,1].
Thenϕ∗(KC2+c· ˜f(u, v))=KC2+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>lctp(P2,Cf)=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 Cf such that lct(P2,Cf)=α. IfPis unstable (resp. not stable), thenPcontains a curve Cgsuch that lct(P2,Cg) < 2d3αα−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)
(ax−az)+(ay−az) ≤ ω(f, λ)
(ax−az)+(ay−az) + ω(g, λ) (ax−az)+(ay−az) 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,Cg) < 2dα−33α (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 curveCdinPsatisfiesm1
p ≤lct(P2,Cd)(see [7, Lemma 8.10]) and since
2d−33 ≤m1p (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 ofPwe have2d3 ≤lctp(P2,Cf)(resp.<).
Proof FixP∈Pd and a base point pas above. GivenCf 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 Cd−m is a curve of degree d−m. ThenPis unstable (resp. not stable). In particular, ifPcontains a multiple line with multiplicity d, thenPis unstable.
Note that Theorems4.7and4.10provide explicit stability criteria for a pencilP∈Pdin terms log canonical thresholds of pairs(P2,Cf), whereCfis 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 byCgen. By [7, Theorem 4.8] we have thatlct(P2,P) =lct(P2,Cgen). By Bertini’s theorem,Cgenis smooth away from the base points ofP, hencelct(P2,P)=lctp(P2,Cgen)for some base pointp. Now, it follows from [7, Lemma 8.6] thatlctp(P2,Cf)≤lctp(P2,Cgen)for any curveCfinP. As a consequence, Theorem4.10implies that wheneverPis semi-stable (resp. stable), then2d3 ≤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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