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RATIONAL ELLIPTIC SURFACES WITH DIHEDRALGROUP ACTION

SHINZO BANNAI

Abstract. We give an explicit method to construct every ratio-nal elliptic surface that has a four torsion section. We use thisconstruction to study and classify the birational equivalence classof rational elliptic surfaces with a relative dihedral group actionfor dihedral groups of order eight.

1. Introduction

Let X, Y be normal projective varieties defined over the field ofcomplex numbers C. Let π : X → Y be a finite surjective morphism.Then π is said to be Galois if the field extension C(X)/C(Y ) of therational function fields that is induced by π is Galois. For a fixed finitegroup G, we say that π : X → Y is a G-cover if it is Galois andGal(X/Y ) ∼= G. In this situation there is a natural G-action on X andX/G ∼= Y .

Let D2n be the dihedral group of order 2n. Elliptic surfaces withD2n-action appear in the study of D2n-covers of P2 done by H. Toku-naga. An existence theorem for D2n-covers π : X → P2 branched alonga quartic and a line is proved by him in [14]. The close relationshipof G-covers and Galois theory of function fields makes it natural toask questions about birationally invariant properties of G-covers andit is important to classify G-covers up to birational equivalence. T.Yasumura studied the birational equivalence of the D6-covers that ap-pear in [14] in terms of the branch curves in P2 ([15]). In this paper westudy the birational equivalence of G-covers π : X → Y by looking atthe covering surface X, where X is a rational elliptic surface, G-actsrelatively on X and G ∼= D2n (n ≥ 4).

Definition 1.1. Let G be a finite group. Let π : X → Y and π′ : X ′ →Y ′ be G-covers. Then π and π′ are birationally equivalent if there existsa G-equivariant birational map φ : X 99K X ′.

Date: January 20, 2011.1

2 SHINZO BANNAI

The classification of G-covers is equivalent to the classification of G-surfaces (i.e. surfaces with G-action). The case where X is a rationalsurface has been studied by many mathematicians. J. Blanc classifiedrational G-surfaces for finite abelian groups ([1]). I. V. Dolgachev andV. A. Iskovskikh studied the general case and essentially completedthe classification of rational G-surfaces for finite G ([2]).Although theclassification is essentially finished, there still are details to be resolvedas stated in Section 9 of [2]. Many interesting rational G-surfaces arenot minimal and it is not easy to see where the non-minimal surfaces fitin the classification. In this paper we clarify the case of rational ellipticsurfaces with relative D2n-action for n ≥ 4 which are not minimal.

This article is organized as follows. In Section 2, we review basic factson rational elliptic surfaces and consider their relative automorphismgroups. In Section 3 we give an explicit method to construct rationalelliptic surfaces that have a 4 torsion section. This construction allowsus to describe the elliptic surfaces in detail. In Section 4, we use theconstruction in Section 3 to determine the G-minimal models of theelliptic surfaces and determine their birational equivalence classes.

2. Dihedral group actions on elliptic surfaces

In this section, we review the definition and basic facts about ellipticsurfaces and discuss the conjugacy classes of finite subgroups of therelative automorphism group that are isomorphic to dihedral groups.The references of this section are [4], [5], [12] and [13].

Definition 2.1. Let E be a smooth projective surface and C be a smoothprojective curve. We say that S is an elliptic surface if there exists asurjective morphism

f : E → C

such that

(1) general fibres of f are smooth curves of genus 1,(2) f is relatively minimal,(3) f has a distinguished section O : C → E called the zero-section,(4) f has at least one singular fiber.

Since we assume the existence of a section, the generic fibre E off : E → C can be viewed as an elliptic curve over C(C) and there is anatural one-to-one correspondence between sections of f : E → C andC(C) rational points E(C(C)) of E.

RATIONAL ELLIPTIC SURFACES WITH DIHEDRAL GROUP ACTION 3

Definition 2.2. The set of sections of f : E → C endowed with a groupstructure induced by the above correspondence is called the Mordell-Weilgroup of E and is denoted by MW (E).

Note that the group structure on E and on MW (E) depends on thechoice of the zero-section. We often identify sections with the imagecurve and abuse language by referring to the curves as sections. Wedenote elements of MW (S) by lower case alphabets s, t. We must notconfuse the group operation of MW (S) and the group operation asdivisors. We use the notation [s] when we want to emphasize that weare considering the curve corresponding to s. Under this notation, s+tis a sum in MW (S) and [s] + [t] is a sum as divisors.

Definition 2.3. An automorphism φ of E is called an relative auto-morphism of f : E → C if it preserves the fibration, i.e. f ◦φ = f . Wedenote the group of relative automorphisms of E by AutC(E).

Definition 2.4. We say that an elliptic surface E has a relative G-action if G ⊂ AutC(E). We also say that E is an elliptic G-surface.

We define an equivalence relation on elliptic G-surfaces.

Definition 2.5. Let f : E → C, f ′ : E ′ → C be elliptic G-surfacesover C (i.e. elliptic surfaces with relative G-action). We say that E , E ′

are isomorphic as elliptic G-surfaces if and only if there exists a G-equivariant isomorphism φ : E → E ′ that preserves the fibration (i.e.f = f ′ ◦ φ).

The following facts are well known.

Lemma 2.1 ([13], Chapter 3, Proposition 9.1). For any point P ∈MW (E)(= E(C(C))), the translation-by-P map extends to a automor-phism of E. The map MW (E) → AutC(E) is a homomorphism.

In the following we assume that AutO(E) = Z/2Z and denote theinvolution by ι.

Propostition 2.1. Let f : E → C be an elliptic surface. Then AutC(E)is generated by MW (E) and ι. Further more ι ◦ s = (−s) ◦ ι for anys ∈MW (E).

Lemma 2.2. Let g ∈ AutC(S). Then g has finite order if and only ifeither

(1) g ∈MW (E)tor or(2) g = ι ◦ t for some t ∈MW (E).

Lemma 2.3. ι ◦ t and ι ◦ t′ are conjugate in AutC(E) if and only ift− t′ is 2-divisible in MW (E).

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Lemma 2.4. Let t, t′ ∈ MW (E). Then t, t′ are conjugate in AutC(E)if and only if t = t′ or t = −t′.

Lemma 2.5. Let t be an n-torsion element of MW (E) and s be anyelement of MW (E). Then the subgroup generated by ι ◦ s and t isisomorphic to a dihedral group of order 2n.

〈ι ◦ s, t〉 ∼= D2n

Conversely, any subgroup of AutC(E) isomorphic to D2n is generatedby ι ◦ s, t for some s, t ∈MW (E).

Proof. We have relations (ι ◦ s)2 = tn = ((ι ◦ s) ◦ t)2 = 1, henceG = 〈ι ◦ s, t〉 is a quotient of D2n, but G has order 2n which impliesG = D2n.

Since finite elements of AutC(E) are of the form t ∈ MW (E) andι ◦ s for some s ∈ MW (S), and since D2n is non-abelian, we see thatany subgroup isomorphic to D2n is generated by ι ◦ s, t for some s, t ∈MW (E). �

The above lemmas show that it is important to know the structure ofthe Mordell-Weil group and especially the torsion part of the Modell-Weil group in order to study the conjugacy classes of relative dihedralgroup actions on an elliptic surface. For rational elliptic surfaces, thestructure has been calculated by U. Persson [8] and many other mathe-maticians and is well known. We list in the following table the surfaceshaving a torsion section of order n for n ≥ 4. The type of S correspondsto the number in Oguiso-Shioda’s list [7] which gives the structure ofthe Mordell-Weil lattice of S, which will be used later.

n MW (S) ] of iso. class. of E Type of E6 Z/6Z 1 665 Z/5Z 1 67

Z/4Z 2 70, 724 Z/2Z ⊕ Z/4Z 1 74

Z ⊕ Z/4Z ∞ 58

From this table and the above Lemmas, we are able to classify ra-tional elliptic G-surfaces up to isomorphism (as elliptic G-surfaces) forG ∼= D2n(n ≥ 4). If MW (E) ∼= Z/nZ then AutC(E) ∼= D2n and thereis an unique relative D2n-action on the surface. For the other cases, wehave the following lemmas.

Propostition 2.2. Let E be the elliptic surface of type 74 such thatMW(E) ∼= Z/2Z ⊕ Z/4Z. Then there are four isomorphism classes ofrelative D8-actions on E.


Proof. They are given by〈ι, t〉, 〈ι, t′〉, 〈ι◦s, t〉, 〈ι◦s, t′〉 where s = (1, 0), t =(0, 1), t′ = (1, 1) ∈MW (E). �Propostition 2.3. Let E be a rational elliptic surface with MW (E) ∼=Z ⊕ Z/4Z. Then there are two isomorphism classes of relative D8-actions on E.

Proof. There are two isomorphism classes of relative D8 actions on Srepresented by 〈ι, t〉 and 〈ι ◦ s, t〉 where s = (1, 0) and t = (0, 1). �Definition 2.6. Let E be an rational elliptic surface of type 58 or 74.Then we call D8-actions on E that are isomorphic to 〈ι, t〉 for somefour torsion section actions of type 1. We call D8-actions that are notisomorphic to 〈ι, t〉 actions of type 2.

3. Rational Elliptic surfaces with four torsion section

In this section we give a method to explicitly construct every rationalelliptic surface that has a four torsion section. We do this by blowingup certain pencils of curves of genus 1 of P1 × P1. We use Kodaira’snotation for singular fibers of elliptic surfaces.

Let ([y0, y1], [z0, z1]) be homogeneous coordinates of P1 × P1. Wedefine a D8

∼= 〈σ, τ |σ2 = τ 4 = (στ)2 = 1〉 action on P1 × P1 as follows:

σ([y0, y1], [z0, z1]) = ([z0, z1], [y0, y1]),

τ([y0, y1], [z0, z1]) = ([z0, z1], [y1, y0]).

Let π : P1 × P1 → P1 × P1/D8 be the quotient morphism.

Lemma 3.1. The quotient of P1 × P1 by the D8 action given above isisomorphic to P2. Furthermore π can be factored into a bi-double coverπ2 : P1 × P1 → P1 × P1 and a double cover π1 : P1 × P1 → P2.

The following corollaries follow immediately from Lemma 3.1.

Corollary 3.1. π is defined by a D8-invariant linear subsystem |Λ| ofH0(O(2, 2)) on P1 × P1.

Corollary 3.2. The branch divisor of π consists of a nonsingular conicB and two lines L1, L2 each of which are tangent to B.

Let

f0 = y0y1z0z1,

f1 = y20z

20 + y2

0z21 + y2

1z20 + y2

1z21 ,

f2 = y0y1z20 + y0y1z

21 + y2

0z0z1 + y21z0z1.

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Then f0, f1, f2 is a basis for |Λ|. Let [x0, x1, x2] be homogeneous coor-dinates of P2. If we fix coordinates so that π : P1 × P1 → P2 is givenby

π([y0, y1], [z0, z1]) = (f0, f1, f2) ∈ P2

the defining equations of B, L1 and L2 are given by

B : 4x0x1 − x22 = 0,

L1 : 4x0 + x1 + 2x2 = 0,

L2 : 4x0 + x1 − 2x2 = 0.

There is an one to one correspondence between members of Λ andlines of P2. For a member C ∈ Λ, we denote the corresponding lineby lC . In general C is a smooth curve of genus 1. C becomes singu-lar, reducible, non-reduced depending on how lC intersects the branchdivisor of π. Let q0 = L1 ∩ L2 and qi = B ∩ Li (i = 1, 2).

Lemma 3.2. The structure of C is given by the following table:

lC C = π∗(lC)q1 ∈ lC, q0, q2 6∈ l I1q2 ∈ lC, q0, q1 6∈ l I1q0 ∈ lC, q1, q2 6∈ l I2

q1, q2 ∈ lC I2q0, q1 ∈ lC 2h0 + 2v0 (non-reduced)q0, q2 ∈ lC 2h1 + 2v1 (non-reduced)

qi 6∈ lC, lC is tangent to B I4lC: general smooth of genus one

I1 is a nodal rational curve. In (n ≥ 2) is a cycle of n smooth rationalcurves. hi, vi (i = 0, 1) are horizontal and vertical fibres of P1 × P1.

Since general members of Λ are smooth curves of genus 1, we canconstruct elliptic surfaces by blowing-up the base points of sub-pencilsof Λ. Let λ be a sub-pencil of Λ. Then λ corresponds to a pencil oflines in P2. Let qλ be the base point of this pencil of lines. We canidentify the dual line lqλ

of qλ with the base curve of the elliptic surface.

Lemma 3.3. The surface Eλ obtained by blowing up the base pointsof a sub-pencil λ of Λ is an rational elliptic surface if and only ifqλ 6= p0, p1, p2. Furthermore the number of the base points of λ, theconfiguration of singular fibers and the Mordell-Weil group of Eλ isgiven by the following table:


qλ |π−1(qλ)| singular fibres MW (Eλ)qλ ∈ B \ {q1, q2} 4 I8, I2, I1, I1 Z/4Z

qλ ∈ Li \ {q0, qi}, i = 1, 2 4 I∗1 , I4, I1 Z/4Zqλ ∈ q1q2 \ {q1, q2} 8 I4, I4, I2, I2 Z/4Z ⊕ Z/2Z

qλ: general 8 I4, I4, I2, I1, I1 Z ⊕ Z/4Z

Proof. If qλ = pi for some i, then every curve in λ becomes singular byLemma 3.2. If qλ 6= q0, q1, q2, then there are only a finite number ofsingular members of λ and general members of λ are smooth curves ofgenus 1. Hence Eλ is an elliptic surface if and only if qλ 6= q0, q1, q2. Theconfiguration of singular fibers can be computed easily and is left tothe reader. The Mordell-Weil group for each type of surfaces is givenin [7]. �

Let lC be defined by αx0 + βx1 + γx2 = 0. Then the j-invariant jCof C can be computed directly by using Maple, and is given by

jC =(α4 − 16α2β2 − 16α2γ2 + 96αβγ2 + 256β4 − 256β2γ2 + 16γ4)

3

(α− 4β)2 (α+ 4β + 4γ) (α+ 4β − 4γ) (αβ − γ2)4 .

We can view this as a rational function on the dual projective space P2

with homogeneous coordinates [α, β, γ]. The j-function of the ellipticsurfaces in Lemma 3.3 can be obtained by restricting this function tothe dual line lqλ

of qλ which can be identified with the base curve of Eλ.Let X be the extremal rational elliptic surface of type 72 whose

singular fibers are I∗1 , I4, I1. A detailed discription of X can be foundin [6]. Let [u, v] be homogeneous coordinates of the base curve P1

X of Xand suppose that the I∗1 , I4, I1 lie over [2, 1], [−2, 1], [1, 0] respectively.Then the j-function of X is given by

jX =256(u2 − 3v2)3

v4(u2 − 4v2).

If we consider a base change P1 → P1X of degree 2 branched over [2, 1]

and another point q 6= [−2, 1], [1, 0] then we obtain an elliptic surfaceof type 58. It is well known (see [6], [10]) that any E with a 4 torsionsection can be obtained by such base change.

On the other hand, if we define ψ : P2 99K P1X by

[α, β, γ] 99K [α2 + 16β2 − 8γ2,−4αβ + 4γ2],

then jC = jX ◦ψ. Hence if we restrict ψ to lqλwe find the base change

morphism ψ|lqλ: lqλ

→ P1X that gives Eλ. Since

(α2 + 16β2 − 8γ2) − 2(−4αβ + 4γ2) = (α− 4β)2

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ψ|lqλis branched over [2, 1] for all λ. The other branch point is given

by [r0, r1] where lqλis tangent to

Cr : r1(α2 + 16β2 − 8γ2) − r0(4αβ − 4γ2) = 0.

Conversely for any [r0, r1] ∈ PX , we can find a line l tangent to Cr andψ|l is branched over [2, 1] and [r0, r1]. Hence every base change mor-phism that gives a surface of type 58 can be found in our constructionand this implies that every surface of type 58 can be constructed usingour method.

We can also see when we obtain isomorphic surfaces. For Eλ andEλ′ of type 58, they are isomorphic if and only if lqλ

and lqλ′ are bothtangent to Cr for some [r0, r1], or equivalently if qλ and q′λ both are onthe dual curve of Cr. In fact the pencil of dual conics of Cr is exactlythe pencil of conics given by {r0(L1 + L2) + r1B}. Summing up thearguments above, we obtain the following theorems.

Theorem 3.1. Every rational elliptic surface that has a four torsionsection can be obtained by the above method by choosing a suitable sub-pencil λ.

Theorem 3.2. Any two surfaces Eλ and Eλ′ obtained in Lemma 3.3are isomorphic as elliptic surfaces if and only if qλ and q′λ lie on thesame member of {r0(L1 + L2) + r1B}.

4. Birational classification of rational elliptic surfaces

First we find the corresponding minimal D8-surfaces.

Propostition 4.1. Let E be a rational elliptic surface of type 70, 72or 74 with relative D8-action. Then E can be D8-equivariantly blowndown to P1 × P1.

To determine the (relatively) minimal models of surfaces of type 58,we need to study how the sections intersect each other in detail. Byutilizing the theory of Mordell-Weil lattices we obtain the followingLemmas .

Lemma 4.1. The sections in the set {[O], [t], [2t], [3t], [s], [s + t], [s +2t], [s+ 3t]} are mutually disjoint.

Lemma 4.2. The sections in the set {[±ks], [±ks+t], [±ks+2t], [±ks+3t]} are not mutually disjoint for any k > 0.

These two lemmas enable us to prove the following proposition.


Propostition 4.2. Let E be a rational elliptic surface of type 58.

(1) A D8-action of Type 1 on E cannot be equivariantly blown downto P1 × P1. Furthermore, the D8-surface Σ obtained by equiv-ariantly blowing down OrbD8([O]) = {[O], [t], [2t], 3t]} is a DelPezzo surface of degree 4 with minimal D8-action.

(2) A D8-action of Type 2 on E can be equivariantly blown down toP1 × P1.

Propositions 4.1 and 4.2 tell us about the underlying surface of theD8-minimal models of E , but they do not tell us about the D8 action onthe surface. We use the explicit description of E as a blow-up of P1×P1

given in Section 3 to determine the group action on the D8-minimalmodels.

Propostition 4.3. Let E be a rational elliptic D8-surface of type 70,72 with the unique D8-action or, 58, 74 with D8-action of type 2. ThenE is birationally equivalent as a D8-surface to P1×P1 with the followingD8 = 〈σ, τ〉 action.

σ([y0, y1], [z0, z1]) = ([z0, z1], [y0, y1]),

τ([y0, y1], [z0, z1]) = ([z0, z1], [y1, y0]).

Proof. By Theorem 3.1 we know that every underlying E can be repre-sented as a (D8-equivariant) blow up of P1 ×P1 with the above action.We need to check whether the relative D8-action on E that is inducedby the D8-action on P1 × P1 coincides with the desired action.

If E is of type 70 or 72 there is nothing to prove since then E has aunique relative D8-action. If E is of type 74 we see that the inducedaction is of type 2 because the D8-action on P1×P1 acts transitively onthe base points of the pencil corresponding to E and hence the inducedaction on E must be transitive on MW (E). If E is of type 58 then wesee that the action must be of type 2 because actions of type 1 cannotbe equivariantly blown down to P1 × P1 by Proposition 4.2. �

Propostition 4.4. Let E be a rational elliptic D8-surface of type 74with action of type 1. Then E is birationally equivalent to P1 ×P1 withthe following D8 = 〈σ, τ〉 action.

σ([y0, y1],[z0, z1])

= ([z0 −√−1z1,

√−1z0 − z1], [y0 −

√−1y1,

√−1y0 − y1]),

τ([y0, y1],[z0, z1])

= ([z0, z1], [y1, y0]).

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Proof. Let C0 and C1 be defined by

C0 : 4y0y1z0z1 − y20z

21 − y2

0z20 − y2

1z20 − y2

1z21 = 0,

C1 : y0y1z20 + y0y1z

21 + z0z1y

20 + z0z1y

21 = 0.

Then the pencil λ = {λ0C0 + λ1C1} is an sub pencil of Λ which wasdefined in Section 3. It can be easily checked that λ is invariant underthe above D8-action and that the D8-action is not transitive on thebase points of λ and that the elliptic surface obtained by blowing upthe base points is of type 74. �

Now that we have found the corresponding minimal models of each E ,we can see where rational elliptic surfaces fit in Dolgachev-Iskovskikh’sclassification. According to the classification of elementary links givenin [2], the two actions on P1 × P1 are birationally distinct. We lookinto the case of minimal Del Pezzo surfaces of degree 4 in detail.

Let E be a rational elliptic surface of type 58. Then we know fromSection 3 that it can be presented as a blow up of P1 × P1 at eightpoints. Let ν : E → P1 × P1 be the blow-up morphism. We label theeight points by

p1 = ([a, b], [c, d]), p2 = τ(p1), p3 = τ 2(p1), p4 = τ 3(p1),p5 = σ(p1), p6 = τσ(p1), p7 = τ 2σ(p1), p8 = τ 3σ(p1).

Where σ, τ are defined as in Section 3. We can suppose that ν([s]) =p1,ν([s + t]) = p2, ν([s + 2t]) = p3, ν([s + 3]) = p4, ν([o]) = p5,ν([t]) = p6, ν([2t]) = p7 and ν([3t]) = p8. Let Σ be the minimal D8

surface corresponding to E . Then Σ is the blow-up of P1 ×P1 centeredat p1, p2, p3, p4. The inverse images of ([1, 1], [1, 1]) and ([1,−1], [1,−1])give D8-fixed points which correspond to the singular points of thesingular fibers of type I1. Let Σ′ be the blow-up of Σ centered at theinverse image of ([1, 1], [1, 1]). The proper transform of a singular fiberof type I1 on Σ′ is an AutC(E)-invariant smooth rational curve with selfintersection number 0. Hence there exists an AutC(E)-invariant conicbundle structure on Σ′. Let ν ′ : Σ′ → P1 × P1 be the blow-up centeredat p1, p2, p3, p4, q = ([1, 1], [1, 1]). Then the conic bundle structure onΣ′ is induced by the proper transform of curves of bi-degree (2, 2) onP1 × P1 passing through p1, p2, p3, p4 with multiplicity one and q withmultiplicity two.

Let Cij be the curve of bi-degree (1, 1) passing through pi, pjand([1, 1], [1, 1]). Let C1 (resp. C2) be the curve of bi-degree (2, 1) (resp.(1, 2)) passing through p1, p2, p3, p4, q. Let L1 be z0 − z1 = 0 and L2

be y0 − y1 = 0. Then the singular fibers of the conic bundle on Σ′

are given by the proper transforms of C1 + L1, C2 + L2, C12 + C34,


C13 +C24 and C14 +C23. By looking at the intersection with the curves{[±s], [±s + t], [±s + 2t], [±s + 3t]} one finds that ι and t act on theset {Ci, Lj, Ckl} as follows:

ι(C1) = L2, ι(C2) = L1, ι(C12) = C23, ι(C34) = C14, ι(C13) = C24,

t(C1) = C2, t(C2) = C1, t(L1) = L2, t(L2) = L1,

t(C12) = C23, t(C23) = C34, t(C34) = C13, t(C14) = C12,

t(C13) = C24, t(C24) = C13.

Lemma 4.3. Let E be a rational elliptic D8-surface of type 58 with D8-action of type 1 obtained by blowing up P1 × P1 at ([a, b], [c, d]) = p1,p2, . . . , p8. Then the involutions ι ◦ t and ι ◦ t3 of E respectively fix acurve C, C ′ of genus 1 with the same j-invariant given by

j(C) = j(C ′) =256n3

1 n32

d41 d

42 d

23

where

n1 = a2c2 + a2d2 + b2c2 + b2d2 + a2cd+ b2cd− abc2 − abd2 − 4abcd,

n2 = a2c2 + a2d2 + b2c2 + b2d2 − a2cd− b2cd+ abc2 + abd2 − 4abcd,

d1 = ac− bd,

d2 = ad− bc,

d3 = a2d2 + a2c2 + b2c2 + b2d2 − 4 abcd.

Proof. Let F1 = C1 +L1, F2 = C2 +L2, F3 = C12 +C34, F4 = C13 +C24

and F5 = C14 + C23 be the five singular fibers of the conic bundlestructure on µ : Σ′ → P1.

By using Maple we can show that it is possible to choose coordinatesso that µ(F1) = [1, 0], µ(F2) = [0, 1] µ(F3) = [(ac − db)2, (da − cb)2],F4 = [1,−1] and F5 = [(da− cb)2, (ac− db)2].

Since ι ◦ t fixes two points on a general fiber and since it inter-changes the components of F1, F2, F3, and F4, the restriction of thestructure map to the fixed curve C is a double cover branched at[1, 0], [1,−1], [0, 1] and [(ac − db)2, (da − cb)2]. Hence C is of genus1 with the desired j-invariant. The case of ι ◦ t3 can be calculated in asimilar way. �

The two involutions ι ◦ t and ι ◦ t3 are cubic involutions in J. Blanc’sterminology [1]. Since the fixed curves of ι ◦ t and ι ◦ t3 are of positivegenus the D8-minimal Del Pezzo surfaces of degree 4 obtained aboveare birationally distinct to the two D8-actions on P1 × P1 which onlyhave rational curves as fixed curves.

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Propostition 4.5. Let E, E ′ be rational elliptic D8-surfaces of type 58with D8-action of type 1. Then E and E ′ are birationally equivalent asD8-surfaces if and only if E and E ′ are isomorphic as elliptic surfaces.

Proof. This is due to Proposition 8.1.8 of [1], which states that two DelPezzo surfaces of degree 4 with Z/2Z ⊕ Z/2Z action generated by twocubic involutions are birationally equivalent as Z/2Z ⊕ Z/2Z-surfacesif and only if there exists a Z/2Z ⊕ Z/2Z-equivariant isomorphismbetween them. In our case, if E , E ′ are birationally equivalent as D8-surfaces, the Del Pezzo surfaces of degree 4 corresponding to thembecome isomorphic as D8-surfaces by Proposition 8.1.8 of [1] and henceE , E ′ are isomorphic as D8-surfaces. �

The arguments in this section add up to the following Theorem:

Theorem 4.1. Let E be a rational elliptic D8-surface. Then E is bi-rationally equivalent to one and only one of the following types of sur-faces.

(1) A rational elliptic surface of type 58 with D8 action of type 1.Two surfaces of this type are birationally equivalent if and onlyif they are isomorphic as elliptic surfaces.

(2) P1 × P1 with D8-action given by

σ([y0, y1], [z0, z1]) = ([z0, z1], [y0, y1]),

τ([y0, y1], [z0, z1]) = ([z0, z1], [y1, y0]).

(3) P1 × P1 with D8-action given by

σ([y0, y1],[z0, z1])

= ([z0 −√−1z1,

√−1z0 − z1], [y0 −

√−1y1,

√−1y0 − y1]),

τ([y0, y1],[z0, z1])

= ([z0, z1], [y1, y0]).

References

[1] J. Blanc, Finite abelian subgroups of the Cremona group of theplane, Thesis, University of Geneva, 2006. Available onine athttp://www.unige.ch/cyberdocuments/theses2006/BlancJ/meta.htm

[2] I. V. Dolgachev and V. A. Iskovskikh, Finite subgroups of the plane Cremonagroup, “Algebra, arithmetic and geometry: in honor of Yu. I. Manin” Vol. I,pp443-548, Progress in Math., 269, Birkhauser Boston, 2009.

[3] R. Hartshorne, Algebraic Geometry, Graduate Texts in Math., 52, Springer(1977)

[4] K. Kodaira, On compact analytic surfaces II, Ann. of Math., 77 No. 3, 1963,pp.563–626


[5] K. Kodaira, On compact analytic surfaces III, Ann. of Math., 78 No. 1, 1963,pp.1–40

[6] R. Miranda and U. Persson, On extremal rational elliptic surfaces, Math. Z.193, 537–558 (1986)

[7] K. Oguiso and T. Shioda, The Mordell-Weil lattice of a rational elliptic surface,Comment. Math. Univ. St. Pauli, Vol. 40 No.1, 1991, pp.83–99

[8] U. Persson, Configuration of singular fibers on rational elliptic surfaces, Math.Z. 205, 1–47 (1990)

[9] Z. Reichstein and B. Youssin, Equivariant resolution of points of indetermi-nancy, Proc. Amer. Math. Soc. 130(2002), 2183–2187.

[10] M. Schuett and T. Shioda, Elliptic surfaces, Preprint, To appear in Algebraicgeometry in East Asia III-KIAS 2008, arXiv:0907.0298v3 [math.AG]

[11] T. Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Pauli, Vol.39 No.2, 1990, pp.211–240

[12] J. H. Silverman, The arithmetic of elliptic curves. Graduate Texts in Math.106, Springer (1986).

[13] J. H. Silverman, Advanced topics in the arithmetic of elliptic curves. GraduateTexts in Math. 151, Springer (1994).

[14] H. Tokunaga, Dihedral covers and an elementary arithmetic on elliptic surfaces,J. Math. Kyoto Univ., 44-2 (2004), 255-270

[15] T. Yasumura, Triple coverings over the projective plane whose branch loci arequintic curves, Thesis, Tokyo Metropolitan University, 2011

Examples of coexistence of hyperelliptic curves

and trigonal curves in a linear system

Takeshi Harui

This is a brief overview of a joint work with Professor Kato and Profes-

sor Ohbuchi on the constancy of the gonality of smooth curves in a complete

linear system on a surface. This article is based on the author’s talk on Decem-

ber 11, 2010 in the conference “Symposium on Algebraic Curves” at Saitama

University.

1 Introduction and known results on the con-

stancy of the gonality

We work over the complex number field C in this article.

Notation and Conventions

A grd is a linear system of degree d and dimension r on a smooth curve. A

one-dimensional linear system is called pencil. For a linear system Λ, ΦΛ is the

rational map associated to it.

Let C be a smooth irreducible projective curve of genus g. The gonality of

C is defined as follows:

gon(C) := min{degφ | φ : C → P1 is a finite morphism}= min{d | C has a g1d}

A curve is said to be d-gonal if gon(C) = d. A 2-gonal (resp. 3-gonal) curve

is said to be hyperelliptic (resp. trigonal).

In this article a linear system on a surface is said to be trigonal if its general

member is a smooth trigonal curve.

Consider the following problem:

Problem. Let C be a smooth curve on a surface S. Does any smooth member

of the linear system |C| have the same gonality as C?

For example, the answer is yes when S is the projective plane P2 (classical)

or a Hirzebruch surface (see [Ma]). However, the gonality is not constant

in a linear system in general. For instance, several authors have studied this

problem forK3 surfaces. Among others, Donagi and Morrison [DM] discovered

a counterexample to the constancy of the gonality in a linear system on a K3

surface. This is the only example for K3 surfaces if OS(C) is ample, which

is proved by Ciliberto and Pareschi [CP]. On the other hand, for Del Pezzo

surfaces, Pareschi [P] and Knutsen [K] determined the linear systems where

the gonality is not constant. The author studied curves on ruled surfaces over

elliptic curves in [Ha] and discovered a counterexample to the constancy of the

gonality.

2 Coexistence of hyperelliptic curves and trig-

onal curves

In this article we study coexistence of hyperelliptic curves and trigonal curves

which have plane models of the same degree with the prescribed singularity.

Let p1, p2, . . . , pn be points in P2 (possibly infinitely near), d,m1,m2, . . . ,mn

positive integers with mi ≥ 2 (i = 1, 2, . . . , n). Assume that there exists a

hyperelliptic curve C0 of genus g with a plane model Γ of degree d whose

singular points are p1, p2, . . . , pn with multpi(Γ) = mi (i = 1, 2, . . . , n).

Let

S = Snπn−→ Sn−1 −→ · · · −→ S1

π1−→ S0 = P2

be the composition of the seven one-point-blow-ups, where πi : Si → Si−1 is

the blow-up at pi. We denote the pullback to S of a line in P2 (resp. the

exceptional curve of πi) by l (resp. ei). Then C0 is identified with a member of

the linear system |D|, where D := dl −∑n

i=1miei. First we have a restriction

on the degree of the plane model Γ.

Lemma 2.1. Assume that |D| contains a smooth non-hyperelliptic curve.

(1) d ≥ g + 3 holds.

(2) If d = g + 3, then mi = 2 for i = 1, 2, . . . , n. Furthermore, (g, d, n) =

(3, 6, 7), (4, 7, 11) or (5, 8, 16).

Proof. (1) First note that d ≥ g + 2, since a hyperelliptic curve of genus g

has no plane model of degree less than g + 2. Furthermore, any plane model

of degree g + 2 of a hyperelliptic curve of genus g has a singular point with

multiplicity g. Hence any smooth member of |D| is hyperelliptic if d = g + 2.

Thus we have d ≥ g + 3.

(2) Suppose that d = g + 3 and mi ≥ 3 for some i. Let grn be the complete

linear system associated to the restriction of l − ei to C0. Then r ≥ 1 and

n = d − mi ≤ g. Such a linear system on a hyperelliptic curve is a multiple

of the hyperelliptic pencil g12, that is, grn = rg12. In particular we have n = 2r.

Let C be a smooth member of |D|. From the following short exact sequences

0 → OS(l − ei − C0) → OS(l − ei) → OC0(l − ei) → 0,

0 → OS(l − ei − C) → OS(l − ei) → OC(l − ei) → 0

we obtain the following exact sequences:

0 →H0(S,O(l − ei)) → H0(C0,OC0(l − ei)) → H0(S,O(l − ei − C0)) → 0

|| ||0 →H0(S,O(l − ei)) → H0(C,OC(l − ei)) → H0(S,O(l − ei − C)) → 0.

Thus we have h0(C,OC(l − ei)) = h0(C0,OC0(l − ei)) = r + 1. It follows that

the restriction of l − ei to C is associated to an r-dimensional linear system

of degree n = 2r. Then C is hyperelliptic, which contradicts our assumption.

Hence mi = 2 for i = 1, 2, . . . , n if d = g + 3.

We then have

d− 3 = g =1

2(d− 1)(d− 2)− n.

Since |D| contains at least two members, we have

0 ≤ D2 = d2 − 4n.

Then it is easy to obtain the assertion by straight calculation.

Our main result is the following:

Theorem 2.2. Assume that d = g + 3.

(1) If g = 3, 4 and p1, p2, . . . , pn are general points in P2, then |D| is trigonal.

(2) When g = 5, |D| cannot be trigonal.

3 The case where g = 3

In the rest of this article we show our main result. Assume that (g, d, n) =

(3, 6, 7) and p1, p2, . . . , p7 are general points in P2 in this section. Note that

KS ∼ −3l +∑7

i=1 ei and the anticanonical map Φ|−KS | : S → P2 is a double

covering.

Proposition 3.1. Let C be a smooth member of |D| = |6l− 2∑7

i=1 ei|. Then

(1) If Φ|−KS |(C) is a conic, then C is hyperelliptic.

(2) If Φ|−KS |(C) is a quartic, then C is trigonal.

In particular |D| is a trigonal linear system containing a hyperelliptic curve.

Proof. Note that (C. −KS) = 4. If Φ|−KS |(C) is a conic, then C is a double

covering of a rational curve. Hence it is hyperelliptic. If Φ|−KS |(C) is a quartic,

then we easily show that C is isomorphic to the quartic Φ|−KS |(C). Hence it

is trigonal.


In this section we assume that (g, d, n) = (4, 7, 11) and p1, p2, . . . , p11 are

general points in P2. Let L be a divisor on S linearly equivalent to 4l− 2e1 −e2 − e3 − · · · − e11. Then dim|L| = 1 and |L| has two base points q1, q2.

Proposition 4.1. The linear system |D| is trigonal and contains a hyperel-

liptic curve. To be more precise, a smooth member C of |D| = |7l− 2∑11

i=1 ei|is hyperelliptic if and only if it passes through q1 and q2.

Outline of proof. First we note that dim|D| = 2 and any member of |D| isirreducible since p1, p2, . . . , p11 are general points. Hence |D| contains an irre-

ducible curve passing through q1 and q2. It is not difficult to show that the

curve is in fact smooth and hyperelliptic. On the other hand, a general mem-

ber C of |D| does not pass through both q1 and q2. Then the restriction of L

to C induces a pencil whose moving part is of degree 3 or 4. Thus C has a

complete g13 or g14 without base points, hence non-hyperelliptic.


In this section we consider the case where (g, d, n) = (5, 8, 16). As proved in

the previous sections, there exists a trigonal linear system with a hyperelliptic

member when g = 3, 4. We show there exists no such linear system when

g = 5.

Proposition 5.1. Assume that the complete linear system |D| = |8l−2∑16

i=1 ei|contains a hyperelliptic curve. Then |D| is not trigonal.

We prove this proposition by using the following result.

Theorem 5.2. [BN, Theorem 3.1] Let π : S → B be a fibration such that for

non countably many t ∈ B, the fiber Ft over t is d-gonal.

(a) There exist a base change B′ → B, a ruled surface R′ over B′ and a

rational map Φ : S ′ −− → R′ over B′ such that degΦ = d.

(b) If for non countably many t ∈ B, Ft has a unique g1d (hence complete),

then base change is not needed.

Outline of proof of Proposition 5.1. Suppose that |D| is trigonal, that is, its

general member is a smooth trigonal curve. Note that |D| is free from base

points and dim|D| = 1 since D2 = 0. Then Φ|D| : S → P1 is a trigonal

fibration of genus 5. Its general fiber Ct has a unique g13. It follows from the

above theorem that there exists a rational map Φ : S − − → Σe of degree 3,

where Σe is a Hirzebruch surface of index e). By the elimination of points of

indeterminacy we obtain a morphjsm Φ : S → Σe. Let ∆0 (resp. Γt) denote

the minimal section (resp. the fiber over t ∈ P1) of a ruling Σe → P1. Then,

for a general t ∈ P1, Φ|Ct = Φg13: Ct → Γt. Hence

(Ct.Φ∗∆0) = (Φ∗Ct.∆0) = 3(Γt.∆0) = 3,

which contradicts Ct ∼ 2(4l −∑16

i=1 ei).

Refferences

[BN] M. A. Barja, J. C. Naranjo: Extension of maps defined on many fibres,

Collect. Math. 49, 227-238 (1998).

[CP] C. Ciliberto, G. Pareschi: Pencils of minimal degree on curves on a K3

surface, ] J. Reine Angew. Math. 460, 15-36 (1995).

[DM] R. Donagi, D. R. Morrison: Linear systems on K3 sections, J. Diff.

Geom. 29, 49-64 (1989).

[GL] M. Green, R. Lazarsfeld: Special divisors on curves on a K3 surface,

Invent. Math. 89, 357-370 (1987).

[Ha] T. Harui: The gonality and the Clifford index of curves on an elliptic

ruled surface, Arch. Math. 84 131-147 (2005).

[K] A. L. Knutsen: Exceptional curves on Del Pezzo surfaces, Math. Nachr.

256, 58-81 (2003).

[Ma] G. Martens: The gonality of curves on a Hirzebruch surface, Arch. Math.

67, 349-352 (1996).

[P] G. Pareschi: Exceptional linear systems on curves on Del Pezzo surfaces,

Math. Ann. 291, 17-38 (1991).

Takeshi Harui

Department of Mathematics, Graduate school of Science,

Osaka University, Toyonaka, Osaka, 560-0043, Japan

e-mail address: [email protected]

[email protected]

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Birationally Very Ample Linear Series

of degree g + 1 and dimension 3on Smooth Projective Algebraic Curves

Changho Keem

This article is based on the talk which was delivered by the author in theSymposium on Algebraic Curves held in December 2010 at Saitama Uni-versity. The author is very grateful to Professor F. Sakai as well as to theother organizers Professor J. Komeda and Professor A. Ohbuchi for the kindinvitation and the hospitality.

1 Introduction and Motivations

Let X be a smooth projective algebraic curve of genus g over the field of com-plex numbers. By a well-known theorem of Halphen, X may be embeddedin P3 as a smooth curve of degree g + 3, such that the hyperplane sectionsform a non-special and complete linear series. In other words, every smoothprojective curve of genus g admits a linearly normal space model of degreeg + 3. One then may pose the following natural question:

Can every X has a linearly normal space model of degree lowerthan g + 3 ?

The answer to this seemingly naive question is definitely “No”, as one seeseasily. One first recalls that the existence of such a space model of degreeless than g + 3 is equivalent to the existence of a very ample linear seriesg3d, d ≤ g + 2. For example, if X is a hyperelliptic curve, by two successive

projections from general points on the space model, this in turn implies theexistence of a base-point-free pencil of degree d − 2 ≤ g. However this isimpossible for an hyperelliptic curve; note that a complete pencil of degreee ≤ g on a hyperelliptic curve is necessarily of the form g1

2 + B, where B isthe base locus of the pencil.

In view of the above observation and discussion, one therefore would like todetermine (or classify) those curves which can (or cannot) be embedded inP3 as a curve of degree g+ 2. Fortunately, we have a complete picture whichcan be stated as follows.

1

Theorem A. [Arbarello, Cornalba, Grifiths and Harris] A curve X of genusg ≥ 6 has a very ample and complete linear series of dimension 3 and degreeg + 2 if and only if X is not hyperelliptic, not bi-elliptic, not trigonal andnot a smooth plane quintic.

The above statement appears in the literature in [1, p220, Exercise B].The proof utilizes the standard tools and techniques in the classical theoryof special linear series on algebraic curves, especially a theorem due to D.Mumford concerning the variety of special linear series; cf. [10]. After settingup some basic definitions and notations, we will briefly sketch the proof ofthe above theorem for the convenience of the reader, especially to demon-strate how the classical theory of variety of special linear series plays a ratheressential role in our business.

Let’s recall some basic notations and conventions before proceeding fur-ther. Basically, we will follow those in [1]. As usual, given positive integersr and d,

W rd (X) := {|D|; degD = d, r(D) ≥ r}

is the set effective divisor classes (under linear equivalence) of degree d whichmove in a linear system of (projective) dimension at least r. W r

d (X), whichwe often abbreviate as W r

d if no confusion is likely to occur, is a subvariety ofthe Jacobian variety of X, upon fixing a base point. For other notations andconventions which are not explicitly explained here, the readers may consult[1].

Sketch of a proof of Theorem A: For a smooth curve X of genus g, wenote that a linear series L = g3

g+2 is very ample if and only if

KXL−1 ∈ W := Wg−4\[W 1

g−2 −W2].

Recall that dimW 1g−2 ≤ g − 6 unless X is bi-ellipitc, hyperelliptic, trigonal

or a smooth plane quintic by Mumford’s theorem. If dimW 1g−2 < g− 6, then

W 6= ∅. For the case dimW 1g−2 = g − 6 and assuming the non-existence of

a very ample g3g+2 as well, we have W = ∅ and hence Wg−4 = Z −W2 for

a certain irreducible component of Z of W 1g−2, which is impossible since in

general “no component of W rd is entirely contained in W r+1

d ”. This showsthat the curves X has a very ample g3

g+2 unless X is bi-ellipitc, hyperelliptic,trigonal or a smooth plane quintic. The converse is easier to show which areleft as an exercise to the readers.

The necessary and sufficient condition for the existence of a very ampleweb of degree g+2 suggests us that a further extension of this type of resultsmay become possible. To be more specific, we may throw way several further

2

“very special” types of curves (whatever these would be) so that the remain-ing curves still admits a very ample web of degree g + 1. As we have seen inthe proof (or sketch) of the Theorem A, some extensions of H. Martens’ or D.Mumford’s type regarding the variety of special linear series on curves wouldbe desirable. Indeed, a full extension immediately right after Mumford’s the-orem exists; cf. [2] and references therein. However, as the readers would beable to figure out easily, we not only need the extension immediately afterthe theorem of Mumford, but also need several steps beyond this point withfullest possible generality, which would be hopeless to achieve optimal andfull version of this kind at this moment.

Therefore, not compromising too much (and as an intermediate step to-ward our original goal as well), we would like to throw away or give up therequirement being very ample of degree g+ 1 and thus allowing singulari-ties on our space model. Therefore, we may pose a modified question by onlyasking:

What curves do (or do not) have a base-point-free andbirationally very ample linear series of degree g + 1 ?

The following theorem, which is quite recent, is the full answer to the abovemodified question.

Theorem B. Let X be a smooth projective algebraic curve of genus g ≥ 8.X does not have a birationally very ample web of degree g + 1 if and only ifX is either a smooth plane sextic, bielliptic or hyperelliptic.

The proof utilizes the notion of the so-called primitive linear series andof course usual and standard methods in the theory of special linear series. Infact, Theorem B is an outcome of very recent work jointly with G. Martensand M. Coppens; cf. [5]. Because the proof of the above result would appearsomewhere else (hopefully), instead of providing a detailed and full proofof the theorem in this article, we will merely mention the basic and keyingredients of the idea leading toward the proof of the theorem in the nextsection. This will be done in a series of remarks. Also, the author would liketo add some examples and other related facts which may help (or hamper)the readers to understand our current problem better.

3

2 Basic ideas and remarks toward a proof of

Theorem B

This section is mainly devoted to convey the ideas for the proof of Theo-rem B. Several important ideas and methods toward the proof of the theoremwill be presented in the following series of remarks. Some typical or patho-logical examples will also be presented.

Remarks 1. Recall that a complete linear series is called primitive if it hasno base locus and the residual series also has no base locus.

2. It is relatively easy to show that a on k-gonal curve X, the pencil com-puting the gonality of X is always primitive unless X is a smooth planecurve.

3. Also, it is very easy to see that on a smooth plane curve, a pencil com-puting the gonality is never primitive; note that the pencil computing thegonality on a smooth plane curve is cut out by lines through a point on thecurve.

4. It is worthwhile to remark that the genus assumption g ≥ 8 in our Theo-rem B is quite a natural one. Indeed, the inequality g ≥ 8 is equivalent to thecondition that the Brill-Noether number ρ(g+1, g, 3) is non-negative. There-fore, the condition g ≥ 8 guarantees that all the curves in our considerationhave a g3

g+1.

5. The first step toward the proof of the theorem is to show if X a k-gonalcurve which is not hyperelliptic, not bi-elliptic and not a smooth plane curve,then the series |KX−g1

k| is not only base-point-free (because g1k is primitive by

above), but also simple; for this, one only needs conventional and elementarymethods, even though this process is rather tedious. After showing that|KX − g1

k| is simple, one take off appropriate number of general points fromthis linear series |KX − g1

k| and we get a simple g3g+1 on X.

6. The curves which were left out in the above step need to be handledseparately; these are hyperelliptic, bi-elliptic or smooth plane curves.

7. The hyperelliptic curve case is trivial; as we saw earlier in the previoussection, the same argument applies. So, let’s assume that X is a bi-ellipticcurve. The residual series of a g3

g+1 is a g1g−3. By the Castelnuovo-Severi

inequality [11] or [8, p368, Ex. 1.9] any such pencil of degree g − 3 is of theform,

g1g−3 = |π∗(g1

2) + p1 + · · ·+ pg−7|,

where π is the bi-ellpitic covering map from X onto an elliptic curve E.

4

Hence it follows that any given g3g+1 is of the form

g3g+1 = |KX − π∗(g1

2)− p1 − · · · − pg−7|,

which can never be simple; note that

|KX−π∗(g12)−p1−· · ·−pg−7−π∗(q)| = |KX−π∗(g1

2+q)−p1−· · ·−pg−7| = g2g−1,

for every q ∈ E.

8. Handling the smooth plane curves may not be so trivial. Note that weonly need to consider smooth plane curves of degree at least six, since we areassuming g ≥ 8.

9. If X is a smooth plane sextic, then one can show without any difficultythat X does not have a simple g3

11. Aa a matter of fact, a smooth planesextic does not even have a base-point-free g3

11 which can be argued in anelementary manner as follows.

Assume that there exists a base-point-free g311, which is automatically simple

since 11 is prime. This in turn implies the existence of a base-point-free andsimple g2

10. However, everybody knows that any g210 on a smooth plane sextic

is of the form

|L+ p+ q + r + s| or |2L− p− q − r + s|,

where L is a hyperplane divisor; one may use Max-Noether’s fundamentaltheorem, or one may use super abundance of points on a projective plane.Specifically, one recalls that eight points on a projective plane fails to imposeindependent conditions on the linear system of cubics if and only if all eightpoints line on a conic or at least five among eight points are collinear; notethat the residual series of a g2

10 is g18. However, both |L+ p+ q + r + s| and

|2L− p− q − r + s| have non-empty base locus.

10. It can be shown that a smooth plane curve of degree d ≥ 7 always havea simple g3

g+1. Moreover, if d ≥ 10 then a smooth plane curve of degree dalways carries a very ample g3

g+1. The proof for smooth plane curve caseis quite subtle, which does not seem to be adequate to be presented here,for otherwise this article would become too lengthy. The whole proof is in[5] which hopefully would appear somewhere in the literature in the nearfuture. The method is utilizing those techniques which have been used byM. Coppens to determine the Luroth’s semi-group of smooth plane curves;cf. [3] and [4].

11. Here we only look at the case of smooth plane septic, which is the borderline case, just to give some flavor to the interested readers. Let X be a smooth

5

plane septic. First we will show that there exists a primitive complete pencilof degree 12; the residual series of such a pencil would be a base-point-freeg3

16. A divisor D of degree 12 on X moves in a pencil if and only if D failsto impose independent conditions on the canonical series, which is cut outby quartics in our situation. Some possible configurations of 12 points Don a projective plane which may fail to impose independent conditions onquartics includes:

(i) six among D are collinear(ii) ten among D lie on a conic or(iii) D is a complete intersection of a cubic and a quartic.

We remark that for example, in the case of (i) above, we get a pencil ofdegree 12 which has six base points and the moving part of the pencil is ofthe form |L− p| where L is a hyperplane divisor. In the case (ii), we have apencil of degree 12 with two base points and the moving part of the pencilis just the pencil of degree 10 cut out by conics through 4 prescribed pointson the curve. Therefore, in the case of (iii), we have a chance to produce abase-point-free pencil of degree 12.

Of course, it is possible to produce a smooth plane septic containing a com-plete intersection of a quartic and a cubic as a subscheme, by using Bertini’stype theorems etc. However, the issue here is that given a smooth plane curveof degree 7, whether there always exists a subscheme of the curve which is acomplete intersection of a cubic and a quartic. Indeed, this is true not onlyfor our current case of smooth plane septic but also true for other smoothplane curves in a more general context; for more details, the readers areadvised to look at [6] or [3].

At any rate, the key ingredient of the proof at this stage is that on a smoothplane curve of degree 7, there always exists 12 points (as a subscheme of X)which is a complete intersection of a cubic and a quartic; call this Γ := C3∩C4.Γ certainly fails to impose independent conditions on the linear system ofquartics, which cut out the canonical linear series. By considering the linearsystem of quartics through Γ, we have

|IP2,Γ(4)||X = |KX − Γ| = g316.

We need to make sure that this g316 is base-point-free. Look at a member

C ∈ |IP2,Γ(4)|. Note that C should be of the form

C = λC3 ·M + µC4,

where M is a linear form. By varying M , there is no common locus otherthan Γ, which implies that our g3

16 constructed in this way is indeed base-point-free.

6

12. The last thing which needs to be verified for the smooth plane septiccase is that this g3

16 is simple. If it were not simple, g316 induces either a 4-

sheeted map onto an elliptic curve or a 2-sheeted map onto a curve of genush ≤ 9, by Castelnuovo genus bound. Recall that on a smooth plane curve ofdegree d, there is no complete base-point-free pencil of degree n < 3d − 9,for n 6= d− 1 or n 6= 2d− 4; this fact is classically known as a folklore or onecan refer [13] and [3]. Suppose that our g3

16 induces a 4-sheeted map onto anelliptic curve. Then X would have a base-point-free g1

8 by pulling back g12 on

the elliptic curve, which is not the case for a smooth plane septic. If our g316

induces a double covering onto a curve X ′ of genus h ≤ 9. One may argueas follows. If h = 9, then X ′ is an extremal curve in P3 of degree 8, lying ona quadric surface. X ′ certainly has a g1

4 which is cut out by rulings of thequadric surface, hence X has a base-point-free g1

8 which is the impossibilityas before. If the base curve X ′ has genus 8, then X ′ has gonality k ≤ 5 bythe Meis’ bound; a special case of a more general existence theorem of thespecial linear series on smooth curves by S. Kleimann and D. Laksov, whichwas proved long after Meis. By pulling this g1

k back, we get a g12k on X, where

2k ≤ 10. For k = 1, · · · 4, we get a contradiction as before. We now claimthat k = 5 does not occur either. In fact the our base curve X ′ of genus 8and of degree 8 in P3 still lie on a quadric surface, hence X ′ has a g1

e wheree ≤ 4. The cases for h ≤ 7 are left to the reader as an exercise. Or onecan directly argue that the linear series D we have chosen explicitly has theproperty that |D − p| is has no base locus for a general p ∈ X, which wouldnot be so painful to see.

13. It is worthwhile to remark that on a smooth plane septic X, there is novery ample complete linear series of dimension 3 and of degree 16. Therefore,being birationally very ample is the best what one can hope for a linear seriesg3

16. To see this, one may argue as follows.

14. On any curve X, the non-existence of a very ample grd is equivalent to

W rd ⊂ W r−1

d−2 +W2.

For a smooth plane septic X, we need to show

W 316 ⊂ W 2

14 +W2,

and by taking the residual series we will show that

(†) W 112 ⊂ W 2

14 −W2.

According to the previous remark (11), a pencil of degree 12 has either 6base points, two base points or is base-point-free, corresponding to the three

7

possibilities for the configurations of 12 points which fail to impose indepen-dent conditions on quartics. Let L be the hyperplane series of X. In the case(11)-(i), we have that such a g1

12 is in the locus L −W1 + W6. In the case(11)-(ii), such a g1

12 is in the locus

2L−W4 +W2 ⊂ 2L−W4 + L−W5 = 3L−W9 ⊂ W12.

In the last case, a base-point-free g112 is in the locus

3L−W9 ⊂ W12.

Therefore it follows that

(††) W 112 = (L−W1 +W6) ∪ ((3L−W9) ∩W 1

12).

Trivially, the first locus is contained in the locus W 214 −W2;

(† † †) (L−W1 +W6) ⊂ L−W2 +W7 ⊂ W 214 −W2.

The second locus ((3L −W9) ∩W 112) is also contained in W 2

14 −W2; indeedwe have

(† † ††) 3L−W9 = 3L−W7 −W2 ⊂ W 214 −W2.

Therefore, by (††), († † †) and († † ††), (†) holds.

15. To show the existence of a birationally very ample linear series of degreeg and of dimension 3 on a smooth plane curve of degree d ≥ 6 is far easierthan to show the existence of a birationally very ample g3

g+1.

In fact, the series |(d−4)·L| is certainly very ample, where L is the hyperplaneseries. Therefore, by taking off suitable number of points from this series,one eventually get a birationally very ample linear series of degree g and ofdimension 3.

3 Digressions and Examples

In this final section, we would like to add some more phenomena and remarksas well as an example of sporadic kind which may help the reader (or mayprevent them from) enlarging their overview related to our business.

Remark. 1. Remember that a general curve of genus g possesses a veryample web of degree g + 1.

8

2. But even for a general element in a proper closed sub-locus of the wholemoduli space, this statement is still true. For example, on a general k-gonalcurve of genus g ≥ 8, k ≥ 4, there exists a very ample web of degree g + 1.This result is due to G. Martens and S. Park; cf. [12].

3. However, in the other extreme, there exists a very special k-gonal curvewithout a very ample web of degree g+ 1, only possessing a birationally veryample ones. The following is one of such a typical example.

Example 1. 1. Let X be a complete intersection of two cubic surfaces inP3.

2. In fact this is a very peculiar kind of curve in the sense that this is theonly curve with the so-called Clifford dimension 3.

3. We know that X has gonality 6, cut out by hyperplanes through trisecantlines. The genus of X is 10. Morevoer, X is not isomorphic to a smoothplane sextic.

4. By Theorem B, X has birationally very ample g3g+1 = g3

11.

5. We now claim that X dose not have a very ample g311. Recall that there

is a base-point-free g17 which is cut out by a secant line on X. Choose any

base-point-free g17 and call it L. Let H be the hyperplane series which is

certainly semi-canonical. By considering the natural map

H0(X,H)⊕H0(X,H)ν−→ H0(X,H ⊕ L),

we see that h0(X,H ⊕ L−1) = 1; otherwise if h0(X,H ⊕ L−1) = 0, then bythe base-point-free pencil trick, we have ker ν = H0(X,H ⊕ L−1 ≥ 1, whichis an obvious contradiction.

Recall that any g1d is either base-point-free or with non-empty base locus.

The fact that h0(X,H ⊕ L−1) = 1 implies

(∗∗) W 17 = (W 1

6 +W1) ∪ (H −W2).

The non-existence of a very ample g311 is equivalent to the condition

W 311 ⊂ W 2

9 +W2,

which is also equivalent to the condition

W 17 ⊂ W 2

9 −W2,

by taking the residual series and this is what we want to show.

9

Note that the unique g39, which we call H is in W 2

9 . Hence

H −W2 ⊂ W 29 −W2.

Hence by (**) the component of W 17 whose general element is base-point-free,

which is denote by BFW 17 , we have

BFW 17 ⊂ W 2

9 −W2.

By a similar way which we derived the inclusion (**), we also have

BFW 16 = W 1

6 ⊂ W 39 −W3.

Combining all these, we conclude that the other component of W 17 , which is

W 16 +W1, also satisfies

W 16 +W1 ⊂ (W 3

9 −W3) +W1 = (W 39 −W1 +W1)−W2 ⊂ W 2

9 −W2,

which finishes the verification of our example.

Example 2. It is possible to produce a curve of genus g = 9 or 8 with abirationally very ample web of degree g + 1, but no very ample g3

g+1.The readers may check that, in case g = 9, a curve of degree six with onenode would be an example of such kind.

The following are the last words which the author would like to spell out:

Remark. 1. As a special case of a celebrated theorem of Eisenbud and Har-ris, a general curve of genus g ≥ 8 always have a very ample and completelinear series g3

g+1. Therefore, we stress again that it makes full sense to askwhat special curves do not have such a linear series, and this is the originalmain concern of ours which we still do not have a satisfactory answer yet.

2. A linear series of dimension 3 and of degree g+ 1 is very special, i.e. itsindex of specialty is at least two. Therefore, those curves without complete,very special and very ample linear series are the curves we are looking for.However, our knowledge regarding this matter is relatively poor in general.

References

[1] E. Arbarrello, M. Cornalba, P. Griffiths and J. Harris, Geometry of al-gebraic curves, Grundlehren Math. Wiss. 267, Springer-Verlag, Berlin(1985).

10

[2] E. Ballico, C. Keem, G. Martens, and A. Ohbuchi, On curves of genuseight, Math. Z. 227 (1998), no. 3, 543–554.

[3] M. Coppens, The existence of base point free linear systems on smoothplane curves, J. Alg. Geometry 4 (1995), 1-15.

[4] M. Coppens, Embeddings of general blowing-ups at points, J. reineAngew. Math. 469 (1995), 179-198.

[5] M. Coppens, C. Keem and G. Martens, Birational space models of degreeg + 1 on curves, preprint.

[6] M. Coppens and T. Kato, Non-trivial linear systems on smooth planecurves, Math. Nachr. 166 (1994), 71–82.

[7] M. Coppens and G. Martens, Linear series on 4-gonal curves, Math.Nachr. 213 (2000), 35–55.

[8] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977,Graduate Texts in Mathematics, No. 52.

[9] G. Martens and F.-O. Schreyer, Line bundles and syzygies of trigonalcurves, Abh. Math. Sem. Univ. Hamburg 56 (1986), 169–189.

[10] D. Mumford, Prym varieties I, Contribution to Analysis, Acad. Press.(1974), 325–355.

[11] F. Severi, Vorlesungen uber algebraische Geometrie: Geometrie auf einerKurve, Riemannsche Flachen, Abelsche Integrale, Johnson Reprint Corp.,New York, 1968.

[12] G. Martens and S. Park, A remark on very ample linear series, Archivder Mathematik, 80 (2003), 611-614.

[13] M. Namba, Geometry of projective algebraic curves, Monographs andtextbooks in pure and applied math. 88, Marcel Dekker, Bazel (1984).

Department of MathermaticsCollege of Natural ScienceSeoul National UniversitySeoul, 151-742 Korea

e-mail: [email protected]@gmail.com

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The projective characterization of plane curves of

genus two with one place at infinity

Keita Tono

1 Introduction

Let C be a plane curve on P2 = P2(C). A singular point of C is said to bea cusp if it is a locally irreducible singular point. We say that C is cuspidal(resp. unicuspidal) if C has only cusps (resp. one cusp) as its singular points.For a cuspidal plane curve C we denote by C ′ its strict transform via theminimal embedded resolution of its cusps. We say that a unicuspidal planecurve C is of Abhyankar-Moh-Suzuki type (AMS type, for short) if thereexists a line L such that C ∩ L = {the cusp}. By regarding L as the line atinfinity, we identify such curves with smooth affine plane curves having oneplace at infinity. Rational unicuspidal plane curves of AMS type have thefollowing property.

Theorem ([AM, Su]). Let C be a rational unicuspidal plane curve of AMStype and L a line such that C∩L = {the cusp}. Then there exists a birationalmap f : P2 99K P2 such that f |P2\L ∈ Aut(P2 \ L) and f(C) is a line.

In [Y], rational unicuspidal plane curves C of AMS type were character-ized by (C ′)2 in the following way.

Theorem ([Y]). Let C be a rational unicuspidal plane curve. Then C is ofAMS type if and only if (C ′)2 ≥ 2.

We next consider elliptic unicuspidal plane curve of AMS type. Theyhave the following property.

Theorem ([AO, M]). Let C be an elliptic unicuspidal plane curve of AMStype and L a line such that C∩L = {the cusp}. Then there exists a birationalmap f : P2 99K P2 such that f |P2\L ∈ Aut(P2 \ L) and f(C) is a smoothcubic curve.

The next theorem characterizes elliptic unicuspidal plane curves C ofAMS type by (C ′)2.

Theorem ([T, Theorem 1]). Let C be an elliptic unicuspidal plane curve.Then (C ′)2 ≤ 6. The equality holds if and only if C is of AMS type.

1

Elliptic unicuspidal plane curves of non-AMS type have the followingproperty.

Theorem ([T, Theorem 2]). If C is an elliptic unicuspidal plane curve ofnon-AMS type, then (C ′)2 ≤ 3. The equality holds if and only if there existan irreducible conic C2 ⊂ P2 and a birational map f : P2 99K P2 such thatC ∩ C2 = {the cusp}, f |P2\C2

∈ Aut(P2 \ C2) and f(C) is a smooth cubiccurve.

We next consider unicuspidal plane curves of genus two.

Theorem ([AO, M]). Let C be a unicuspidal plane curve of genus two.Suppose that C is of AMS type. Let L be the line such that C ∩ L ={the cusp}. Then there exists a birational map f : P2 99K P2 such thatf |P2\L ∈ Aut(P2 \L) and f(C) is a unicuspidal quintic curve. There existsno such f with deg f(C) < 5.

In my talk I claimed, as Theorem 2, that if C is a unicuspidal planecurve of genus two then (C ′)2 ≤ 10 and the equality holds if and only if C isof AMS type. After my talk I found a counterexample to these assertions.So I would like to withdraw Theorem 2. I apologize for this error.

For unicuspidal plane curves of genus g ≥ 2, we have the following esti-mation of (C ′)2, which was claimed for g ≥ 3 as Theorem 3 in my talk.

Theorem 1. Let C be a unicuspidal plane curve of genus g ≥ 2. Then(C ′)2 ≤ 5g + 3.

2 Proof of Theorem 1

Let C be a unicuspidal plane curve of genus g ≥ 2 and P the cusp of C. Letσ : V → P2 denote the minimal embedded resolution of P . That is, σ is thecomposite of the shortest sequence of blow-ups such that the strict transformC ′ of C intersects σ−1(P ) transversally. The dual graph of D := σ−1(C)has the following shape, where h ≥ 1 and all Ai, Bi are not empty.

◦ ◦︸︷︷︸A1

◦

◦

◦B1

◦︸︷︷︸A2

◦

◦

◦B2

◦ ◦

◦

◦Bh−1

◦︸︷︷︸Ah

◦

D0

◦

◦Bh

◦ C′

Here D0 is the exceptional curve of the last blow-up and A1 contains that ofthe first one. The morphism σ contracts Ah +D0 + Bh to a (−1)-curve E,Ah−1+E+Bh−1 to a (−1)-curve and so on. Every irreducible component Eof Ai and Bi is a smooth rational curve with E2 < −1. Each Ai contains anirreducible component E such that E2 < −2. Cf. [BK, MS]. We give weights

2

to A1, . . . , Ah, B1, . . . , Bh in the usual way. Let D1 (resp. D2) denote theirreducible component of Bh (resp. Ah) meeting with D0.

Put n := (C ′)2. We may assume that n ≥ g + 1. Perform (n− g)-timesof blow-ups τ0 : W0 → V over C ′∩D0 in the following way, where ∗ (resp. •)denotes a (−1)-curve (resp. (−2)-curve) and Ei is the exceptional curve ofthe i-th blow-up. We use the same symbols to denote the strict transformsvia τ0 of C ′, Ei, etc.

◦

D2

∗

D0

◦ D1

◦

◦

C′

n

τ0

◦

D2

•

D0

◦ D1

◦

•

E1

•

En−g−1

∗

En−g

◦

C′

g

On W0, there exists an exact sequence:

0 −→ H0(OW0) −→ H0(OW0(C′)) −→ H0(OC′(C ′)) −→ H1(OW0) = 0.

Because (C ′)2 = g on W0, we have h0(OC′(C ′)) ≥ 1. Let Λ0 ⊂ |C ′| bea pencil such that C ′ ∈ Λ0. The base points of Λ can be resolved by g-times of blow-ups τ1 : W → W0. Put Λ = τ∗1Λ0 − the fixed part of τ∗1Λ0.Set τ = τ0 ◦ τ1. We may assume that the first n′ ≥ 0 blow-ups of τ1 aredone over En−g ∩ C ′ and the remaining ones are not. Let Ei denote theexceptional curve of the i-th blow-up of τ . Put m = n+ n′ − g.

The morphism ΦΛ : W → P1 is a fibration, whose nonsingular fibers aresmooth curves of genus two. Let φ : W → X be successive contractions of(−1)-curves in the singular fibers of ΦΛ such that the fibration p := ΦΛ◦φ−1 :X → P1 is relatively minimal. The fibration has the following properties.

Lemma 2. The morphism φ does not contract D0, D1, D2, E1, . . . , Em−1.The divisor φ(D0 +D1 +D2 + E1 + · · ·+ Em−1) is contained in a fiber F0

of p.

In order to prove Theorem 1 we use the following fact, which follows from[Sh, Theorem 3] and the fact that the Picard number ρ(X) of X satisfiesthe inequality ρ(X) ≤ 4g + 6. See [SS, Theorem 2.8] and its proof.

Lemma 3.∑F

(r(F ) − 1) ≤ 4g + 4, where F runs over all fibers of p and

r(F ) denotes the number of irreducible components of F .

By Lemma 2, r(F0) ≥ m + 2 ≥ n − g + 2. By Lemma 3, we have4g + 4 ≥ r(F0)− 1 ≥ n− g + 1, which proves Theorem 1.

3

References

[AM] Abhyankar, S. S. and Moh, T. T.: Embeddings of lines in the plane,J. Reine Angew. Math. 276 (1975), 148–166.

[AO] A’Campo, N. and Oka, M.: Geometry of plane curves via Tschirn-hausen resolution tower, Osaka J. Math. 33 (1996), 1003–1033.

[BK] Brieskorn, E. and Knorrer, H.: Plane algebraic curves. Basel, Boston,Stuttgart: Birkhauser 1986.

[MS] Matsuoka, T. and Sakai, F.: The degree of rational cuspidal curves,Math. Ann. 285 (1989), 233–247.

[M] Miyanishi. M.: Minimization of the embeddings of the curves into theaffine plane, J. Math. Kyoto Univ. 36 (1996), 311–329.

[SS] Saito, M.-H. and Sakakibara, K.: On Mordell-Weil lattices of highergenus fibrations on rational surfaces, J. Math. Kyoto Univ. 34, 859–871 (1994)

[Sh] Shioda, T.: Mordell-Weil lattices for higher genus fibration over acurve. New trends in algebraic geometry (Warwick, 1996), 359–373,London Math. Soc. Lecture Note Ser., 264, Cambridge Univ. Press,Cambridge, 1999

[Su] Suzuki, M.: Proprietes topologiques des polynomes de deux variablescomplexes, et automorphismes algebriques de l’espace C2, J. Math.Soc. Japan 26 (1974), 241–257.

[T] Tono, K.: The projective characterization of elliptic plane curveswhich have one place at infinity, Saitama Math. J. 25 (2008), 35–46.

[Y] Yoshihara, H.: Rational curve with one cusp. II, Proc. Amer. Math.Soc. 100 (1987), 405–406.

Department of Mathematics, Faculty of Science, Saitama University,

Shimo-Okubo 255, Urawa Saitama 338–8570, Japan.

E-mail address: [email protected]

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