galois group at each point for some self-dual

GALOIS GROUP AT EACH POINT FOR SOME SELF-DUALCURVES

HIROYUKI HAYASHI

Doctoral Program in Information Science and EngineeringGraduate School of Science and Technology

Niigata University

1

2 GALOIS GROUP AT EACH POINT FOR SOME SELF-DUAL CURVES

The purpose of the present thesis is to determine the Galois groups at eachpoint for some self-dual curves. This thesis is divided into four sections. Sec-tions 1-3 are devoted to preliminaries of Section 4.

In Section 1 we begin by defining an analytic subset in a complex manifold.We introduce affine algebraic sets and projective algebraic sets as examples ofanalytic subsets. Section 2 treats Riemann surfaces and covering spaces. Andwe state the connection between Galois covering maps and Galois extensionsof the fields of meromorphic functions. We also note the Riemann-Hurwitzformula for a compact Riemann surface. Section 3 is concerned with algebraiccurves in a projective plane. For an algebraic curve we define the dual curveand give an example. In Section 4 we proceed to the main subject of the presentthesis. First, for an irreducible algebraic curve and a point on a projectiveplane we define the Galois group at the point and give the definition of self-dualcurves. We rewrite a condition being primitive represented by group theory intheory of covering spaces in our case.

The author wishes to express his sincere thanks to Professor Yoshihara forvaluable suggestions given during the period of the preparation of this thesis.

1. COMPLEX MANIFOLDS

1.1. Analytic Subsets. Let X be an n-dimensional complex manifold.

DEFINITION 1.1. A subset A ⊂ X is said to be analytic if for each point q ∈ Xthere are a (connected) open neighborhood U(q) of q and finitely many holo-morphic functions f1(p), . . . , fν(p) on U(q) such that

A ∩ U(q) = {p ∈ U(q) | f1(p) = · · · = fν(p) = 0}.

We call A an analytic hypersurface if we can always take ν = 1.

Examples. (1) A subset A = {1/m | m ∈ N} of a complex plane C is ananalytic subset of C, but A∪{0} = {1, 1/2, . . . , 1/m, . . . }∪{0} is not an analyticsubset of C.

(2) We consider the domain G = G1 ∪G2 with

G1 = {(z1, z2) ∈ C2 | |z1| <1

2and |z2| < 1},

G2 = {(z1, z2) ∈ C2 | |z1| < 1 and1

2< |z2| < 1}.

For the analytic subset we take A = {(z1, z2) ∈ G2 | z1 = z2}. The sets G1, G2

give an open covering of G with A∩G1 = ∅ and A∩G2 = {(z1, z2) ∈ G2 | z1−z2 =0}. So A is an analytic subset of G.

DEFINITION 1.2. Let A ⊂ X be an analytic subset. Then A is said to bereducible if there exist analytic subsets A1, A2 ⊂ X such that A = A1 ∪A2, A 6=A1, A 6= A2. Otherwise, A is said to be irreducible.

Let f1(p), . . . , fν(p) be holomorphic functions that are defined on an opensubset U ⊂ X. Let q ∈ U be a point and z : p −→ z(p) = (z1(p), . . . , zn(p)) be a

GALOIS GROUP AT EACH POINT FOR SOME SELF-DUAL CURVES 3

complex local coordinates of q. The mapping f : p −→ f(p) = (f1(p), . . . , fν(p))is holomorphic, and we consider

∂(f1(z(p)), . . . , fν(z(p)))

∂(z1(p), . . . , zn(p))=

(∂f j(z(p))

∂zk(p)

)j=1,...,νk=1,...,n

.

This is something like a Jacobian matrix of f at p, but it depends on the localcoordinates (z1, . . . , zn). Let w : p −→ w(p) = (w1(p), . . . , wn(p)) be other localcomplex coordinates of q. Since

∂f j(z(p))

∂zk(p)=

n∑l=1

∂f j(w(p))

∂wl(p)· ∂w

l(p)

∂zk(p), wl(p) = wl(z(p)),

we have∂(f1(z(p)), . . . , fν(z(p)))

∂(z1(p), . . . , zn(p))=∂(f1(w(p)), . . . , fν(w(p)))

∂(w1(p), . . . , wn(p))· ∂(w1(p), . . . , wn(p))

∂(z1(p), . . . , zn(p)).

This shows that

rank∂(f1(z(p)), . . . , fν(z(p)))

∂(z1(p), . . . , zn(p))

is independent of the chosen local complex coordinates (z1, . . . , zn).

DEFINITION 1.3. An analytic subset A ⊂ X is said to be regular (or smooth ornonsingular) of codimension ν at a point q ∈ A if there are open neighborhoodU(q) ⊂ X of q and holomorphic functions f1(p), . . . , fν(p), ν = ν(q), on U(q)such that:

(i) A ∩ U(q) = {p ∈ U(q) | f1(p) = · · · = fν(p) = 0}.

(ii) rank∂(f1(z(q)), . . . , fν(z(q)))

∂(z1(q), . . . , zn(q))= ν.

The number n− ν is called the dimension of A at q.

1.2. Projective Algebraic Manifolds. Denote by P n a complex projectivespace of dimension n.

DEFINITION 1.4. An analytic subset X ⊂ P n that is the zero set of finitelymany homogeneous polynomials is called a (projective) algebraic set. The sub-sets

X ∩ {(ζ0, . . . , ζn) ∈ P n | ζj 6= 0}are called (affine) algebraic set.

A complex manifold X is called projective algebraic manifold if there are apositive integer N and a holomorphic embedding j : X −→ PN such that j(X)is a regular algebraic set.

Example. We consider C − 0. Using a mapping t −→ (z1, z2) = (t, 1/t), t ∈C − 0, we find that C − 0 is mapped biholomorphically onto an affine algebraichypersurface {z ∈ C2 | P (z1, z2) = 0}, where P (z1, z2) = z1z2 − 1. Then

P (ζ0, ζ1, ζ2) = ζ1ζ2 − ζ20


is a homogeneous polynomial of degree 2. The projective algebraic subset {ζ ∈P 2 | P (ζ0, ζ1, ζ2) = 0} is such that

{ζ ∈ P 2 | P (ζ0, ζ1, ζ2) = 0} − {ζ ∈ P 2 | ζ0 = 0} ∼= {z ∈ C2 | P (z1, z2) = 0}.

Let π : Cn+1 − 0 −→ P n be the canonical projection.

THEOREM 1.1 (Chow). Every analytic subset X in the projective space P n isthe zero set of finitely many homogeneous polynomials F1, . . . , Fs such that ifx ∈ X is a regular point of codimension ν, then

rank∂(F1(z), . . . , Fs(z))

∂(z0, . . . , zn)= ν

for every z ∈ π−1(x).

Let V ⊂ Cn+1 be a complex linear subspace of codimension ν. Then thereare linear forms f1, . . . , fν on Cn+1 such that

V = {z ∈ Cn+1 | fj(z) = 0, j = 1, . . . , ν}.

Since the linear forms are homogeneous polynomials of degree 1,

P (V ) = {(ζ0, . . . , ζn) ∈ P n | fj(ζ0, . . . , ζn) = 0, j = 1, . . . , ν}

is a regular algebraic set. We call P (V ) a (projective) linear subspace. It iscomplex analytically homeomorphic to P n−ν .

The dimension of a linear subspace P (V ) ⊂ P n is one less than the dimen-sion of the vector subspace V :

dimP (V ) = dimV − 1.

By definition, the empty set has dimension −1.Linear subspaces of dimension zero are the points; a linear subspace of di-

mension one is called a line. In general, a linear subspace of dimension k iscalled a k-plane.

Suppose that Z ⊂ P n is any subset. We define the span of Z, denoted by〈Z〉, to be the intersection of all linear subspaces containing Z. If P (V ) = 〈Z〉,we might also say that Z spans P (V ). We say that Z is nondegenerate if Zspans all of P n.

We have the following dimension formula, which follows easily from thecorresponding formula for vector subspaces of a vector space:

LEMMA 1.1. If L and M are two linear subspaces of P n, then

dim〈L ∪M〉 = dimL+ dimM − dim(L ∪M).

Two disjoint linear subspaces L ⊂ P n and M ⊂ P n, dimL + dimM = n,are called complementary linear subspaces.

Let L ⊂ P n be a k-plane and M ⊂ P n be an (n − k − 1)-plane which aredisjoint subspaces. Note that L and M together span all of P n.


Suppose p is a point not on L. Then the span of L∪ p is a linear subspace L1

which has dimension one more than that of L, i.e., L1 is a (k+ 1)-plane. Henceby the dimension formula, we see that

dim(L ∩M) = dimL1 + dimM − dim〈L1 ∪M〉= (k + 1) + (n− k + 1)− n= 0,

so that L1 ∩M is a single point, in M of course.

DEFINITION 1.5. The projection from L to M is the mapping

π : P n − L −→M

defined by sending a point p ∈ P n − L to the intersection point of 〈L ∪ p〉 withM :

π(p) = 〈L ∪ p〉 ∩M.

The subspace L is called the center of projection.

It is easy to see that if L is defined by ζk+1 = ζk+2 = · · · = ζn = 0 and M isdescribed by ζ0 = ζ1 = · · · = ζk = 0, then

π(ζ0, ζ1, . . . , ζn) = (0, 0, . . . , 0, ζk+1, ζk+2, . . . , ζn).

One often suppresses the choice of the target subspace M in the language,and refers to π simply as “the projection from L”. The reason for this is thatif M1 and M2 are two complementary subspaces to L, with projections π1 fromL to M1 and π2 from L to M2, then the restriction of π2 to M1 is a projectivetransformation ϕ : M1 −→M2, and

ϕ ◦ π1 = π2.

So for most purposes it does not matter which subspace one is projecting to.

DEFINITION 1.6. Let V be a complex vector space. The dual projective spaceP (V )∗ is the set of codimension one subspaces of V .

Note that any codimension one subspaceW of V induces a hyperplane P (W ) ⊂P (V ); indeed, the dual space P (V )∗ may be identified with the set of hyper-planes in P (V ).

THEOREM 1.2. There is a natural bijection between P (V ∗) and P (V )∗ givenby associating to the span of nonzero functional f : V −→ C the codimensionone subspace which is the kernel of f .

We consider some connections with complex algebraic geometry.A meromorphic function f on P n is called rational if f = 0, or if there are

homogeneous polynomials F and G of the same degree such that F 6= 0 and

f(ζ0, . . . , ζn) =F (ζ0, . . . , ζn)

G(ζ0, . . . , ζn).


THEOREM 1.3. Every meromorphic function on P n is rational.

1.3. Modifications.

DEFINITION 1.7. Let f : X −→ Y be a proper surjective holomorphic mappingbetween two n-dimensional connected complex manifolds. The mapping f iscalled a (proper) modification of Y if there are nowhere dense analytic subsetsE ⊂ X and S ⊂ Y such that the following hold:

(i) f(E) ⊂ S.(ii) f maps X − E biholomorphically onto Y − S.

(iii) Every fiber f−1(y), y ∈ S, consists of more than one point.The set S is called the center of the modification andE = f−1(S) the exceptionalset.

Let U be a small neighborhood around the origin in Cn+1. We want toreplace the origin in U by an n-dimensional complex projective space P n ={(ζ1, . . . , ζn+1) ∈ P n}. If π : Cn+1 − 0 −→ P n is the canonical projection, thenevery line Cv through the origin determines an element ζ = π(v) in P n, and ζdetermines the line l(ζ) = π−1(ζ)∪0 such that Cv = l(π(v)). Now we insert P n

in such a way that we reach the point ζ by approaching the origin along l(ζ).We define

X = {(z, ζ) ∈ U × P n | z ∈ l(ζ)}and this can be rewritten in the form

X = {(z, ζ) ∈ U × P n | ζjzk − ζkzj = 0, j, k = 1, . . . , n+ 1}.

This is so-called incidence set. We first show that it is an (n + 1)-dimensionalcomplex manifold. Set Uj = {ζ ∈ P n | ζj 6= 0}, j = 1, . . . , n + 1. Then X is ananalytic subset of U × P n, with

X ∩ (U ×U1) ∼= {(z, w) ∈ U ×Cn | zk = wkz1, k = 2, 3, . . . , n+ 1}, wk = ζk/ζ1.

In U × Uj , j = 2, . . . , n + 1, there is a similar representations. It follows thatX is a submanifold of codimension n in the (2n+ 1)-dimensional manifold U ×P n. We have a holomorphic mapping q : (z, ζ) −→ z of X onto U which mapsX − q−1(0) biholomorphically onto U − 0 and q−1 : z −→ (z, π(z)). Obviously, qis a proper mapping.

The preimage q−1(0) is the exceptional set {(0, ζ) ∈ X | 0 ∈ l(ζ)} = 0 × P n.So q : X −→ U is a proper modification. It is called Hopf’s σ-process or theblowing up of U at the origin.

If z 6= 0 is a point of U and {λν} a sequence of nonzero complex numbersconverging to 0, then q−1(λνz) = (λνz, π(z)) converges to (0, π(z)). This is thedesired property.

We consider the case n = 1. Let M be a 2-dimensional connected complexmanifold and p ∈M a point. Let U ⊂M be a small neighborhood with complexcoordinates z, w such that (z(p), w(p)) = (0, 0). Let X ⊂ U × P 1 be the blowingup of U at the origin. Then

Qp(M) = (M − U) ∪X = (M − p) ∪ P 1


is again a 2-dimensional complex manifold. We call Qp(M) the qudratic trans-formation of M at p.

Let F : M1 −→ M2 be a holomorphic mapping between 2-dimensional com-plex manifolds and F (p1) = p2. Let q1 : Qp1(M1) −→ M1 and q2 : Qp2(M2) −→M2 be the quadratic transformations. Then there exists a biholomorphic map-ping F : Qp1(M1) −→ Qp2(M2) such that q2 ◦ F = F ◦ q1. This follows directlyfrom the construction, and it shows that the quadratic transformation is acanonical process.

2. COVERING SPACES AND COMPACT RIEMANN SURFACES

Riemann surfaces, i.e., connected 1-dimensional complex manifolds, orig-inated in function theory as a means of dealing with the problem of multi-valued functions. Such multi-valued functions occur because the analytic con-tinuation of a given holomorphic function element along different paths leadsin general to different branches of that function. It was the idea of Riemann toreplace the domain of the function with a many sheeted covering of the com-plex plane. If the covering is constructed so that it has as many points lyingover any given point in the plane as there are function elements at that point,then on this “covering surface” the analytic function becomes single-valued.

2.1. Elementary Properties of Holomorphic Mappings. We note some ofthe elementary topological properties of holomorphic mappings between Rie-mann surfaces.

THEOREM 2.1. Let X and Y be Riemann surfaces and f : X −→ Y be a non-constant holomorphic mapping and a ∈ X and b = f(a). Then there exists apositive integer k and coordinate systems ϕ : U −→ V on X and ψ : U ′ −→ V ′

on Y with the following properties:(i) a ∈ U,ϕ(a) = 0, b ∈ U ′, ψ(b) = 0.

(ii) f(U) ⊂ U ′.(iii) The mapping F = ψ ◦ f ◦ ϕ−1 : V −→ V ′ is given by

F : z −→ F (z) = zk, z ∈ V.

The number k in Theorem 2.1 can be characterized in the following way. Forevery neighborhood U0 of a there exist neighborhoods U ⊂ U0 of a and W ofb = f(a) such that the set f−1(y) ∩ U contains exactly k elements for everypoint y ∈ W, y 6= b. We call k the multiplicity with which the mapping f takesthe value b at the point a or we just say that f has multiplicity k at the point a.

2.2. Branched and Unbranched Coverings. Nonconstant holomorphic map-pings between Riemann surfaces are “covering mappings”, possibly having“branch points”. For this reason we now gather together the most importantideas and results from the theory of covering spaces.

DEFINITION 2.1. Let X and Y be Riemann surfaces and p : Y −→ X be anon-constant holomorphic mapping. A point y ∈ Y is called a branch point of pif there is no neighborhood V of y such that p|V is injective. The mapping p iscalled an unbranched holomorphic mapping if it has no branch points.


THEOREM 2.2. Let X and Y be Riemann surfaces. A non-constant holomor-phic mapping p : Y −→ X has no branch points if and only if p is a localhomeomorphism.

Examples. (1) Let k be a natural number = 2 and pk : C −→ C be themapping defined by pk : z −→ pk(z) = zk. Then 0 ∈ C is a branch point of pkand the mapping pk|(C − 0) : C − 0 −→ C is unbranched.(2) Let p : Y −→ X be a nonconstant holomorphic mapping and y ∈ Y , x = p(y).Then y is a branch point if and only if the mapping p takes the value x at thepoint y with multiplicity = 2. By Theorem 2.1 the local behavior of p near y isjust the same as the local behavior of the mapping pk in Example (1) near theorigin.

DEFINITION 2.2. Let X and Y be topological spaces. A mapping p : Y −→ Xis called a covering map if the following hold:

Every point x ∈ X has an open neighborhood U such that its preimagep−1(U) can be represented as

p−1(U) =⋃j

Vj ,

where the Vj are disjoint open subsets of Y , and all the mappings p|Vj : Vj −→U are homeomorphisms. In particular, p is a local homeomorphism.

Examples. (1) Let D = {z ∈ C | |z| < 1} be the unit disk in the complexplane and let p : D −→ C be the canonical injection. Then p is a local home-omorphism, but not a covering map. For, no point a ∈ C with |a| = 1 has aneighborhood U with the property required in the definition.

(2) Let k be a natural number = 2 and let

pk : z −→ pk(z) = zk, z ∈ C − 0.

Then pk is a covering map. Proof : Suppose a ∈ C − 0 is arbitrary and chooseb ∈ C − 0 with pk(b) = a. Since pk is a local homeomorphism, there are openneighborhoods V0 of b and U of a such that pk|V0 : V0 −→ U is a homeomor-phism. Then

p−1k (U) = V0 ∪ ωV0 ∪ · · · ∪ ωk−1V0,where ω is a kth primitive root of unity, say ω = exp(2πi/k). It is clear that thesets Vj = ωjV0, j = 0, . . . , k−1, are pairwise disjoint and each pk|Vj : Vj −→ Uis a homeomorphism.

(3) The mapping exp : C −→ C − 0 is a covering map. Proof : Supposea ∈ C − 0 and b ∈ C with exp(b) = a. Since exp is a local homeomorphism,there exist open neighborhoods V0 of b and U of a such that exp |V0 : V0 −→ Uis a homeomorphism. Then

exp−1(U) =⋃n∈Z

Vn,

where Vn = V0 + 2πin. Clearly the Vn are pairwise disjoint and each mappingexp |Vn : Vn −→ U is a homeomorphism.


THEOREM 2.3. Let X and Y be Hausdorff spaces with X pathwise connectedand p : Y −→ X be a covering map. Then for any two points x0, x1 ∈ X the setsp−1(x0), p−1(x1) have the same cardinality.

The cardinality of p−1(x) for x ∈ X is called the number of sheets of thecovering and may be either finite or infinite.

Let X and Y be Riemann surfaces and f : X −→ Y be a proper nonconstantholomorphic mapping. It follows from Theorem 2.1 that the set A of branchpoints of f is closed and discrete. Since f is proper, B = f(A) is also closed anddiscrete. We call B the set of critical values of f .

Let Y ′ = Y − B and X ′ = X − f−1(B) ⊂ X − A. Then f |X ′ : X ′ −→ Y ′

is a proper unbranched holomorphic covering and it has a well-defined finitenumber of sheets n. This means that every value c ∈ Y ′ is taken exactly ntimes. In order to be able to extend this statement to the critical values b ∈ Bas well, we have to consider the multiplicities.

For x ∈ X denote by v(f, x) the multiplicity with which f takes the valuesf(x) at the point x. Then we shall say that f takes the value c ∈ Y , countingmultiplicities, m times on X, if

m =∑

x∈f−1(c)

v(f, x).

THEOREM 2.4. Let X and Y be Riemann surfaces and f : X −→ Y be a propernonconstant holomorphic mapping. Then there exists a natural number n suchthat f takes every value c ∈ Y , counting multiplicities, n times.

A proper nonconstant holomorphic map will be called an n-sheeted holomor-phic covering map, where n is the integer found in the previous theorem. Notethat holomorphic covering maps are allowed to have branch points. If we wishto emphasize that there are none, then we shall specifically say that the map isunbranched. If we speak of a topological covering map or if there is no complexstructure, then we mean a covering map in the sense of Definition 2.2.

DEFINITION 2.3. Let X and Y be topological spaces and p : Y −→ X be acovering map. By a covering transformation or deck transformation of thiscovering we shall mean a fiber-preserving homeomorphism f : Y −→ Y . Withoperation the composition of mappings, the set of all covering transformationof p : Y −→ X forms a group which we denote by Deck(Y/X). If there is anychance of confusion, then we will write Deck(Y

p−→ X) instead of Deck(Y/X).

DEFINITION 2.4. Let X and Y be connected Hausdorff spaces and p : Y −→ Xbe a covering map. The covering map is said to be Galois (the terms normaland regular are also in common usage) if for every pair of points y0, y1 ∈ Ywith p(y0) = p(y1) there exists a covering transformation f : Y −→ Y such thatf(y0) = y1.


Example. The mapping pk : z −→ pk(z) = zk, z ∈ C − 0, is a coveringmap. It is Galois since for any z1, z2 ∈ C − 0 with pk(z1) = pk(z2), we havez2 = ωz1, where ω is a kth root of unity and the mapping z −→ ωz is a coveringtransformation.

Let X be a Riemann surface and U be an open subset of X. We denote byM (U) the set of all meromorphic functions on U .

If π : Y −→ X is a nonconstant holomorphic mapping between Riemannsurfaces X and Y , then for any meromorphic function f on X the functionπ∗f = f ◦ π is a meromorphic function on Y . Thus there is a map

π∗ : M (X) −→M (Y )

which is a monomorphism of fields.The following theorem shows that the continuation of the covering is uniquely

determined up to isomorphism.

THEOREM 2.5. Let X,Y and Z be Riemann surfaces and π : Y −→ X, τ :Z −→ X be proper holomorphic covering maps. Let A ⊂ X be a closed discretesubset and let X ′ = X − A, Y ′ = π−1(X ′) and Z ′ = τ−1(X ′). Then every fiber-preserving biholomorphic mapping σ′ : Y ′ −→ Z ′ can be extended to a fiber-preserving biholomorphic mapping σ : Y −→ Z. In particular every coveringtransformation σ′ ∈ Deck(Y ′/X ′) can be extended to a covering transformationσ ∈ Deck(Y/X).

Theorem 2.5 makes the following definition meaningful.

DEFINITION 2.5. Let X and Y be Riemann surfaces and π : Y −→ X be abranched holomorphic covering. Let A ⊂ X be the set of critical values of πand let X ′ = X −A and Y ′ = π−1(X ′). Then the covering Y −→ X is said to beGalois if the covering Y ′ −→ X ′ is Galois.

THEOREM 2.6. Let X be a Riemann surface and

P (T ) = Tn + c1Tn−1 + · · ·+ cn ∈M (X)[T ]

be an irreducible polynomial of degree n. Then there exist a Riemann surfaceY , a branched holomorphic n-sheeted covering π : Y −→ X and a meromorphicfunction F ∈ M (Y ) such that (π∗P )(F ) = 0. The triple (Y, π, F ) is uniquelydetermined in the following sense. If (Z, τ,G) has the corresponding properties,then there exists exactly one fiber-preserving biholomorphic mapping σ : Z −→Y such that G = σ∗F .

To simplify the terminology (Y, π, F ) is called the algebraic function definedby the polynomial P (T ).

IfX and Y are Riemann surfaces and π : Y −→ X is a branched holomorphiccovering map, then Deck(Y/X) has a representation into the automorphismgroup of the field M (Y ) defined in the following way. For σ ∈ Deck(Y/X) let


σf = f ◦ σ−1. Clearly the correspondence f −→ σf is an automorphism ofM (Y ). The mapping

Deck(Y/X) −→ Aut(M (Y ))

is a group homomorphism. For suppose σ, τ ∈ Deck(Y/X). Then for everyf ∈M (Y )

(στ)f = f ◦ (στ)−1 = f ◦ τ−1 ◦ σ−1 = σ(f ◦ τ−1) = σ(τf).

Obviously every such automorphism f −→ σf leaves invariant the functionsof the subfield π∗M (X) ⊂ Y and thus is an element of the Galois groupAut(M (Y )/π∗M (X)).

THEOREM 2.7. Let X be a Riemann surface, K = M (X) be the field of mero-morphic functions onX and P (T ) ∈ K[T ] be an irreducible monic polynomial ofdegree n. Let (Y, π, F ) be the algebraic function defined by P (T ) and L = M (Y ).By means of the monomorphism π∗ : K −→ L consider K as a subfield ofL. Then L/K is a field extension of degree n and L ∼= K[T ]/P (T ). Everycovering transformation σ : Y −→ Y of Y over X induces an automorphismf −→ σf = f ◦ σ−1 of L leaving K fixed and the mapping

Deck(Y/X) −→ Aut(L/K)

which is so defined, is a group isomorphism. The covering map Y −→ X isGalois if and only if the field extension L/K is Galois.

2.3. Compact Riemann Surfaces. Among all Riemann surfaces the com-pact ones are especially important. They arise, for example, as those coveringsurfaces of the Riemann sphere defined by algebraic functions.

THEOREM 2.8. If X is a compact Riemann surface, then

dimH1(X,O) < +∞.

DEFINITION 2.6. Let X be a compact Riemann surface. Then

g = dimH1(X,O)

is called the genus of X.

LetX and Y be compact Riemann surfaces and f : X −→ Y be a nonconstantholomorphic mapping. For x ∈ X let v(f, x) be the multiplicity with which ftakes the value f(x) at the point x. The number

b(f, x) = v(f, x)− 1

is called the branching order of f at the point x. Note that b(f, x) = 0 if andonly if f is unbranched at x. Since X is compact, there are only finitely manypoints x ∈ X such that b(f, x) 6= 0. Thus

b(f) =∑x∈X

b(f, x),

the total branching order of f , is well-defined.


THEOREM 2.9 (Riemann-Hurwitz formula). Let f : X −→ Y be an n-sheetedholomorphic covering mapping between compact Riemann surfaces X and Ywith total branching order b = b(f). Let g be the genus of X and g′ be the genusof Y . Then we have

2g − 2 = n(2g′ − 2) + b.

3. ALGEBRAIC CURVES

3.1. Affine Algebraic Curves and Projective Algebraic Curves.

DEFINITION 3.1. A subset C ⊂ C2 is called an affine algebraic curve if thereexists a polynomial f ∈ C[X1, X2] such that deg f = 1 and

C = V (f) = {(x1, x2) ∈ C2 | f(x1, x2) = 0}.

Every polynomial f ∈ C[X1, X2],deg f = 1, has an associated curve V (f) ⊂C2. If f is a divisor of g ∈ C[X1, X2], i.e., if g = fh for some h ∈ C[X1, X2], thenV (f) ⊂ V (g). Since the ring of polynomials is a unique factorization domain,we have a good general idea of the divisibility properties of polynomials. Wewould like to use these to draw conclusions about the possibles subcurves of agiven curve. The following will help us find our way back from the loci of thecurves to the polynomials.

THEOREM 3.1 (Study’s lemma). Let f, g ∈ C[X1, X2]. If f is irreducible ofdegree = 1 and V (f) ⊂ V (g), then f is a divisor of g.

Of the numerous consequences of Study’s lemma, the first one we discussis the decomposition of an algebraic curve into “components”. Since rings ofpolynomials over fields are unique factorization domains, every nonconstantpolynomial f ∈ C[X1, X2] admits a factorization

f = fk11 · · · fkρρ · · · fkrr ,

where the fρ are irreducible and no two of them are associates. This factor-ization is unique up to units and the order in which the fρ occur. Hence wehave

V (f) = V (f1) ∪ · · · ∪ V (fρ) ∪ · · · ∪ V (fr).

In other words, the curve defined by f can be decomposed into componentsV (fρ).

If C = V (f) = V (g) for some other polynomial g, then we obtain

g = af l11 · · · f lρρ · · · f lrr ,

where a ∈ C − 0 and lρ ∈N .We shall say

f = f1 · · · fra minimal polynomial for the curve C.


DEFINITION 3.2. If C = V (f) ⊂ C2 is an affine algebraic curve and f a mini-mal polynomial, then

degC = deg f

is called the degree of the curve C.

DEFINITION 3.3. A subset C ⊂ P 2 is called a projective algebraic curve ifthere exists a homogeneous polynomial F ∈ C[X0, X1, X2] such that degF = 1and C = V (F ) = {(x0, x1, x2) ∈ P 2 | F (x0, x1, x2) = 0}.

LEMMA 3.1. Let f ∈ C[X1, X2] be a nonconstant polynomial, and let F ∈C[X0, X1, X2] be its homogenization. Then f is irreducible if and only if Fis irreducible.

For nonconstant homogeneous polynomial F ∈ C[X0, X1, X2] let

F = F k11 · · ·F kρρ · · ·F krrbe a prime factorization, where F kρρ is homogeneous polynomial. If C = V (F ) =V (G) for some other homogeneous polynomial G, then we obtain

G = aF l11 · · ·F lrr ,where a ∈ C − 0, lρ ∈N .

We shall sayF = F1 · · ·Fr

a minimal polynomial for the curve C.

DEFINITION 3.4. If C = V (f) ⊂ P 2 is a projective algebraic curve and F aminimal polynomial, then

degC = degF

is called the degree of the curve C. If F is not necessarily a minimal polynomial,one speaks of the degree of the divisor.

To get a first measure of the nastiness of a singularity, we consider higherderivative of the defining polynomial. Let f ∈ C[X1, X2] and p = (c1, c2) ∈ C2

be a fixed point. The substitution

Xj = (Xj − cj) + cj

gives the power series expansion of f about p:

f(X1, X2) =∑ν

fν(X1, X2),

where fν(X1, X2) =∑

m1+m2=ν

am1m2(X1−c1)m1(X2−c2)m2 , am1m2 =1

m1!m2!

∂m1+m2f(c1, c2)

∂Xm11 ∂Xm2

2

.

Thus the order of f at p can be defined as

ordp(f) = min{ν | fν 6= 0}.If f is a minimal polynomial of a curve C ⊂ C2, then

kp = ordp(C) = ordp(f)


is called the order of C at p. It is clear that(1) 0 5 ordp(C) 5 degC,(2) p ∈ C if and only if ordp(C) = 1,(3) C is singular at p if and only if ordp(C) > 1.

The extreme case ordp(C) = degC occurs if and only if f = fn, where n =degC.

THEOREM 3.2. If C ⊂ C2 is an affine algebraic curve and L is a line througha point p ∈ C, then

ordp(C) 5 (C.L)p,

and the inequality is strict for at most ordp(C) lines through p, where (C.L)pdenotes the intersection multiplicity of C and L at p.

To avoid splitting the definition into cases, we set (C.L)p = +∞ if p ∈ L ⊂ C.This allows us to define tangents of an algebraic curve at singular points aswell.

DEFINITION 3.5. Let C ⊂ C2 be an algebraic curve and L be a line through apoint p ∈ C. The line L is called a tangent line of C at p if

ordp(C) < (C.L)p.

DEFINITION 3.6. Let T be the tangent line of C at a smooth point p. If(C.T )p = 2 then T is called a simple tangent. If (C.T )p = 3 then T is calledan inflectional tangent and p is called an inflection point.

DEFINITION 3.7. Let F ∈ C[X0, X1, X2] be a homogeneous polynomial of de-gree = 2. Then the symmetric 3× 3 matrix

HF =

(∂2F

∂Xj∂Xk

)j,k=0,1,2

is called the Hessian matrix of F . If F is a minimal polynomial of a curveC = V (F ) ⊂ P 2 and deg(detHF ) = 1, then H(C) = V (detHF ) is called theHessian curve of C.

THEOREM 3.3. (1) The Hessian curve is independent of the coordinates.(2) deg(detHF ) = 3(n− 2) if detHF 6= 0.(3) Sing(C) ⊂ H(C), where Sing(C) denotes the set of singular points of C.

The following theorem shows that the significance of the Hessian curve.

THEOREM 3.4. Let C = V (F ) ⊂ P 2 be a curve that contains no lines. Then(1) detHF 6= ∅;(2) a smooth point p ∈ C is an inflection point if and only if p ∈ H(C);(3) C and H(C) have no common component;


(4) if p ∈ C is a simple inflection point, then

(C.H(C))p = 1,

where (C.H(C))p denotes the intersection multiplicity of C and H(C) atp.

Let C ⊂ P 2 be an algebraic curve, and S a Riemann surface. Then a map-ping

ϕ : S −→ C

is said to be holomorphic if it is holomorphic as a mapping to P 2. With thesepreliminaries out of the way, we can state the theorem.

THEOREM 3.5. For every irreducible algebraic curve C ⊂ P 2, there exists acompact Riemann surface S and a holomorphic mapping ϕ : S −→ C with thefollowing properties:

(i) Let C ′ = C − Sing(C) be the smooth part of C, and let S′ = ϕ−1(C ′) ⊂ S.Then

ϕ|S′ : S′ −→ C ′

is biholomorphic.(ii) For every point p ∈ C there is a bijection mapping

ϕ−1(p) −→ {branches of C at p}.

In particular, ϕ−1(p) is finite for every p ∈ C.

For any irreducible algebraic curve C ⊂ P 2, we define the genus of C as thegenus of S.

3.2. Dual Curves.

DEFINITION 3.8. Let C ⊂ P 2 be an algebraic curve. Then

C∗ = {L ∈ (P 2)∗ | L is a tangent line of C at some p ∈ C}

is called the dual curve of C.

By the definition, the condition on L means that ordp(C) < (C.L)p. For eachpoint p ∈ C there are only finitely many such lines. If C itself is a line, then C∗consists of a single point.

THEOREM 3.6. Let C ⊂ P 2 be an algebraic curve that has no lines as compo-nents. Then

(1) C∗ ⊂ (P 2)∗ is an algebraic curve;(2) if C is irreducible then C∗ is irreducible and degC∗ = 2;(3) C∗∗ = C.


Example. Let C ⊂ P 2 be a smooth quadric and p = (c0, c1, c2) be a point ofC. C has a corresponding symmetric matrix A = (ajk) ∈ GL(3,C). Denotingby (X0, X1, X2) the homogeneous coordinates on P 2,

F (X) =

2∑j,k=0

ajkXjXk

is a minimal polynomial of C. Since

∂F (X)

∂Xj= 2

2∑k=0

ajkXk,

the coordinates of Tp(C) in (P 2)∗ are given by 2∑j=0

aj0cj ,

2∑j=0

aj1cj ,

2∑j=0

aj2cj

.

Consider the map σ of P 2 into (P 2)∗ defined by

σ : x = (x0, x1, x2) −→ y = (y0, y1, y2) = σ(x) =

(2∑k=0

a0kxk,

2∑k=0

a1kxk,

2∑k=0

a2kxk

).

Then we obtain σ(C) = C∗. Hence y ∈ C∗ if and only if x ∈ C, provided that

yj =

2∑k=0

ajkxk, j = 0, 1, 2. We set A−1 =

(ajk)

and if yj =

2∑k=0

ajkxk, j = 0, 1, 2,

then the computation2∑

j,k=0

ajkyjyk =

2∑j,k=0

ajk

(2∑

α=0

ajαxα

) 2∑β=0

akβxβ

=

2∑k,α,β=0

δ kαakβxαxβ

=

2∑α,β=0

aαβxαxβ

shows that the condition x ∈ C can be rewritten in the form2∑

j,k=0

ajkyjyk = 0.

Therefore, C∗ is the quadric described by A−1.


4. MAIN RESULTS

Galois group at each point for some self-dual curvesHIROYUKI HAYASHI AND HISAO YOSHIHARA

ABSTRACT. We study the Galois group defined by a point projection for planecurve. First we present a sufficient condition that the group is primitive andthen determine the structure at each point for some self-dual curves.

4.1. Introduction. This is a continuation of [M], [Y2, Y3, Y4] and etc. Ingeneral it is not easy to determine the Galois group GP at every point P forplane curve, in particular for curve with singular point. When we determinethe structure of GP , it is important to know whether it is primitive or not.However, there are not so many results which are useful for our purpose (cf.[S]). In this article we give a geometrical criterion and then determine thegroup at each point for some self-dual curves.

Let k be an algebraically closed field of characteristic zero. We fix it as theground field of our discussions. Let C be an irreducible plane curve of degreed (≥ 2) and K = k(C) the rational function field of C. Let (X : Y : Z) bea set of homogeneous coordinates on P2 and put P1 = (0 : 0 : 1), P2 = (0 :1 : 0), P3 = (1 : 0 : 0). Let F (X,Y, Z) be the defining equation of C and putf(x, y) = F (X,Y, Z)/Zd where x = X/Z, y = Y/Z.

4.1.1. Galois group. Let r : C −→ C be the resolution of singularities of C. Fora point P ∈ P2, let P be the dual line in the dual space P2 of P2 correspondingto P . We define the morphism πP by

πP : C 3 Q 7→ PR ∈ P ∼= P1,

where PR is the point in P2 corresponding to the line `PR, which passes throughP and R = r(Q) if P 6= R. In case P = R, the line `PR is the tangent line to thebranch of C at R. Clearly we have deg πP = d −mP (C) and a field extensionπP∗ : k(P1) ↪→ K = k(C), where mP (C) denotes the multiplicity of C at P . In

case P /∈ C we understand mP (C) = 0. We put n(P ) = d−mP (C), if there is nofear of confusion we simply denote it by n. Since the extension depends only onP , we denote k(P1) by KP , i.e., we have πP ∗ : KP ↪→ K. Let LP be the Galoisclosure of K/KP and GP the Galois group Gal(LP /KP ).

Definition 1. We call GP the Galois group at P for C. In case K/KP is aGalois extension, the point P is said to be a Galois point.

In case k is the field of complex numbers, GP is isomorphic to the mon-odromy group of the covering πP : C −→ P1 [C, H].

4.1.2. self-dual curve.

Definition 2. A point Q ∈ C is said to be a cusp of C if it is a singular pointand r−1(Q) consists of a single point. Furthermore, if µ : BQ(P2) −→ P2 is ablow-up and µ−1(Q) is a nonsingular point of the proper transform of µ−1(C),the point Q is said to be a simple cusp.

Denote by C the dual curve of C.


Definition 3. If C is projectively equivalent to C, then C is said to be a self-dual curve.

Suppose C is smooth. Then, C is self-dual if and only if d = 2. However, ifC has a singular point, the condition that C is self-dual becomes complicated.The following proposition has been known (cf. [Y1]).

Proposition 4. Suppose C is one of the following curves:(1) C has just one singular point.(2) C is rational and has only simple cusps as singular points.

Then, C is a self-dual curve if and only if C is projectively equivalent to thecurve defined by y = xd.

Example 5. It seems that only a few self-dual curves have been known. Herewe present some of them.

(I) C(e,d) : the curve defined by Y eZd−e = Xd, gcd(e, d) = 1, 1 ≤ e ≤ d− 1

(II) C(4) : the curve defined by (Y Z −X2)2 = X3Y (cf. [I-U-N])(III) C54 : the curve defined by (XY −XZ + Y Z)3 + 54X2Y 2Z2 = 0 (cf. [O])

For the curve C(e,d), if 1 < e < d− 1, then P1 = (0 : 0 : 1) and P2 = (0 : 1 : 0) arenot simple cusps and C(e,d) has no flex. The curve C(4) has two cusps P1 andP2, where P1 is not a simple cusp. The curve C54 has three cusps P1, P2 and P3

and the normalization is an elliptic curve. It is easy to find the dual curve ofC(e,d), however, in the other curves we need some consideration, for the details,see [I-U-N, O].

Remark 1. Let ΦC be the rational map P2 99K P2 giving the dual of C, i.e.,

ΦC(X : Y : Z) = (∂XF : ∂Y F : ∂ZF ),

where F is the defining equation of C. In the case where C = C(e,d), the mapΦC turns out to be a quadratic transformation of P2:

ΦC(X : Y : Z) = (−dY Z : eZX : (d− e)XY ).

We use the following notation:• Zm : the cyclic group of order m• Sd : the symmetric group of degree d• i(X1, X2 ;Q) : the intersection number of two curves X1 and X2 at Q• `PQ : the line passing through P and Q, P 6= Q• `P : a line passing through P• TQ = TQ(C) : the tangent line to C at Q

4.2. Statement of results. We need some preparations before stating the re-sults. A curve means a nonsingular projective algebraic curve. Let X1 and X2

be curves and f : X1 −→ X2 a surjective morphism, which we call a coveringfor short. We denote by e(R, f) be the ramification index of f at R ∈ X1. Ifthere is no fear of confusion, we simply denote it by e(R).

Definition 6. Let f : X1 −→ X2 be the covering above. If there exists a curveX3 and coverings α : X1 −→ X3 and β : X3 −→ X2 such that f = βα, degα ≥2 and deg β ≥ 2, then f is said to be decomposable and X3 an intermediatecovering. If such a curve X3 does not exist, then f is said to be indecomposable(cf. [P-S]).


Definition 7. Let f : X1 −→ X2 be the covering above and R1, . . . , Rr all theramification points for f . Put e(Ri) = ei (1 ≤ i ≤ r). The covering f is saidto be an s-covering over f(Ri) if there exists no ramification point in f−1f(Ri)except Ri. The f is said to be an s-covering if it is an s-covering over eachf(Ri) (1 ≤ i ≤ r).

Definition 8. With the same notation as in Definition 7 we call{(R1, . . . , Rr), (e1, . . . , er)} (or, simply (e1, . . . , er)) the ramification data for f .

We give several sufficient conditions that f is indecomposable. Some of themwill not be used later in this article.

Proposition 9. Let f : X1 −→ X2 be the covering above and n = deg f . If oneof the following conditions are satisfied, then f is indecomposable.

(1) For some i (1 ≤ i ≤ r), ei is prime and n < 2ei.(2) e1 = n− 1.(3) X2 is a rational curve, f is an s-covering except over f(R1) and ei is

prime for each i ≥ s+ 1, where f−1f(R1) = {R1, . . . , Rs}.

Proposition 10. With the same notation as in Proposition 9, if f is an s-covering and satisfies one of the following conditions, then f is indecomposable.

(1) X1 is a rational curve, e1 ≥ e2, n− 1 ≥ e2 and ei is prime for each i ≥ 3.(2) X1 is a rational curve and ei is prime for each i ≥ 2.(3) X2 is a rational curve and ei is prime for each i.

Hereafter we follow the notation in Section 1. By taking a suitable projec-tive change of coordinates, we can assume the projection center is P1 with-out changing the structure of GP . Putting y = tx, we have KP = k(t) andK = k(x, y) = k(t, x). Put g(x) = f(x, tx)/xm ∈ k(t)[x], where m = mP (C) andlet {x1, . . . , xn} (n = n(P )) be the roots of g(x) = 0. Then we can consider GPas a permutation subgroup of Sn. Note that GP is a transitive subgroup ofSn. Hence GP is a primitive group if and only if the isotropy subgroup of anelement of {x1, . . . , xn} is a maximal subgroup of Sn.

Theorem 11. The group GP is primitive if and only if πP is indecomposable.In particular, if n(P ) is a prime number, then GP is primitive for P ∈ P2.

Definition 12. Assume Q ∈ C is a smooth point or a cusp. A line ` = `PQ issaid to be a simple e-tangent line to C if the following conditions are satisfied:

(1) If Q 6= P (resp. Q = P ), then i(C, ` ;Q) = e (resp. e + m), where e ≥ 2and m = mP (C).

(2) the curves C and ` have normal crossings except at Q.Sometimes we call ` a simple e-tangent for short.

Note that a simple e-tangent `PQ yields an s-covering over πP (Q).

Lemma 13. We have the following assertions for GP .(1) If each line `P has normal crossings with C or is a simple e-tangent line

to C such that e is a prime number, then GP is primitive (cf. [S, Lemma4.4.4]).

(2) If there exists a simple 2-tangent line `P , then GP contains a transposi-tion.


The following lemma is well-known.

Lemma 14. If a permutation group G ⊂ Sn is primitive and contains a trans-position, then it is a full symmetric group.

Combining the results above, we get the following corollary.

Corollary 15. If the covering πP : C −→ P1 is one of the coverings in Propo-sitions 9 or 10 and πP is an s-covering over πP (Ri) with ei = 2 for somei (1 ≤ i ≤ r), then GP is a full symmetric group. In particular, if each line`P has normal crossings with C or is a simple 2-tangent, then GP is a full sym-metric group.

Corollary 15 implies [Y2, Theorem 1 and 1′]. Now we can state the structureof GP as follows:

Theorem 16. For the curves C in Example 5 the Galois groups GP are asfollows, where Z1 indicates the trivial group.

(I) the case C = C(e,d)

P P1 P2 P3 P ∈ C \ {P1, P2} P ∈ P2 \ C ∪ {P3}GP Zd−e Ze Zd Sd−1 Sd

(II) the case C = C(4)

P P1, P2 P ∈ C \ {P1, P2} P ∈ P2 \ CGP Z2 S3 S4

(III) the case C = C54

P P1, P2, P3 P ∈ C \ {P1, P2, P3} P ∈ P2 \ CGP S3 S5 S6

Remark 2. For the curves in Theorem 16, P is a Galois point if and only if GPis a cyclic group. However, the same assertion does not hold true in general,see for example [Y3].

4.3. Proofs. First we prove Propositions 9 and 10.

Claim 1. Suppose f and a ramification point R ∈ X1 satisfy the followingconditions:

(1) f is an s-covering over f(R).(2) e(R) is prime.

If there exists an intermediate covering β : X3 −→ X2, then β is unramified atR′ = α(R).

Proof. Suppose β is ramified atR′. Then, since e(R) is prime, we have e(R′, β) =e(R, f), hence R′ is not a branch point for α. Then, there will appear anotherramification point for f in f−1(f(R)). This is a contradiction. �

The proof of Proposition 9 is as follows. Suppose f is decomposable andthere exists a covering β : X3 −→ X2 as in Definition 6. First we prove theassertion (1). Since ei is prime, β is unramified at R′i by Claim 1. Hence wehave e(Ri, α) = e(Ri, f). Since there exists at least two points in β−1(f(Ri)), wehave n = deg f ≥ 2e(Ri, f), which contradicts the assumption. Next we prove(2). Clearly α and β are ramified at R1 and R′1, respectively. Put B1 = f(R1).


Then, since e1 = n − 1, β−1(B1) consists of one or two points. In the formercase α−1(β−1(B1)) consists of two points, on the other hand in the latter caseα−1(B1i) (i = 1, 2) consists of one point, where β−1(B1) = {B11, B12}. In eachcase we infer the inequality n = deg f ≥ (n − 1) + 2, which is a contradiction.We go to the proof of (3). Then, by Claim 1, Bi (i ≥ s+ 1) is not a branch pointfor β. Thus B1 is the only branch point for β. Then, by Hurwitz’s Formula,we have 2g(X3) − 2 = −2b + c, where g(X3) is the genus of X3, b is the degreeof β and c ≤ b − 1. Since g(X3) ≥ 0, this inequality implies b ≤ 1, which is acontradiction.

Next we prove Proposition 10. In each case we use the reduction to absur-dity, i.e., suppose f is decomposable. So we use the notation R′i = α(Ri) (1 ≤i ≤ r). In the case (1), by Claim 1, β is unramified at R′i (i ≥ 3). Since X2

and X3 are rational, from Hurwitz’s Formula, we infer that β is ramified withthe index e(R′1, β) = e(R′2, β) = deg β. Then, since there exists no ramificationpoints in f−1(f(Ri)) except Ri (i = 1, 2), α must branch at R′1 and R′2. However,there exists an unramified point in f−1(f(R2)), this is a contradiction. There-fore f is indecomposable. In the case (2), by Claim 1, β is unramified at R′i fori ≥ 2. Since X3 is rational, by Hurwitz’s Formula, we have a contradiction.In the case (3) similarly, by Claim 1, β is unramified at every point R′i, how-ever, since X2 is rational, β must be an identity, which is a contradiction. Thiscompletes the proof of Proposition 10.

The proof of Theorem 11 is as follows: supposeGP is not primitive and letGxbe the isotropy group of x = x1 in GP . Then, there exists a subgroup H of GPsuch that Gx ( H ( GP . Let CH be the nonsingular model of the intermediatefield which corresponds to H by the Galois correspondence. Then there existthe coverings α : C −→ CH and β : CH −→ P1 such that πP = βα. Thus πP isdecomposable. The converse assertion is clear from the Galois corresponding.

The proof of Lemma 13 is simple. In view of Definition 12 we see that theassertion (1) is another expression of (3) in Proposition 10. The assertion (2)may be well-known (cf. [H]).

Now we proceed to the proof of Theorem 16. The structure of GP depends onthe covering πP and πP depends on the position of P . We prove by examiningthe cases where P lies on the tangent line to C at the cusp or at the flex.Hereafter we assume C is the curve in Theorem 16. Since C is a self-dualcurve and has only cusps as the singularity, the following remark is clear.

Remark 3. Suppose a line ` satisfies the following conditions:(1) ` does not pass through any cusp.(2) ` is not the tangent line to C at the flex.

Then, ` is a simple 2-tangent line to C or ` and C have normal crossings.

Proof of the case (I)Assume C = C(e,d). It has the following property.

Claim 2. The tangent line TP1 (resp. TP2) is Y = 0 (resp. Z = 0) and TP1 ∩TP2 ={P3}, which does not lie on C. In case e = 1 (resp. d − 1) C has one flex atP1 (resp. P2). On the other hand, in case 1 < e < d− 1, C has no flex.

Proof. Calculating the Hessian of Xd − Y eZd−e (cf. [F]), we infer readily theassertions. �


If P = P1, P2 or P3, then GP can be determined directly. In fact, if P = P1,then consider the affine part Z 6= 0 of C, i.e., the affine defining equation isye − xd = 0. Then, putting y = tx, we get te − xd−e = 0, hence GP ∼= Zd−e.The other case P = P2 is similarly determined. If P = P3, then consider theaffine part X 6= 0, we get yezd−e = 1. Putting z = ty, we get td−eyd = 1, henceGP ∼= Zd. As we have seen above, these points are Galois ones.

Next we treat the case P ∈ C \ {P1, P2}. First we prove the sub-case 1 <e < d − 1. Since C is a self-dual curve and has no flex, we see that, if a line`P passes through neither P1 nor P2, then it has normal crossings with C orit is a simple 2-tangent line to C. Furthermore, by Hurwitz’s Formula, we seethere exists a simple 2-tangent. Then, by (1) in Proposition 10 and Lemma 14,we have GP ∼= Sd−1. Next we prove the sub-case e = 1. Then, P1 (resp. P2) is aflex (resp. cusp) and the tangent line at P1 (resp. P2) does not meet C except atP1 (resp. P2). If a line `P does not pass through P2, then it has normal crossingswith C or it is a simple 2-tangent line to C. By (2) in Proposition 10 and Lemma14, we have GP ∼= Sd−1. The proof of the case e = d− 1 is the same.

Now we prove the case where P ∈ P2 \ C and P 6= P3. If P ∈ `P1P2and 1 <

e < d − 1, then πP has two ramification points R1 and R2 such that e(R1) = e,e(R2) = d− e and πP (R1) = πP (R2). Thus πP is not an s-covering. If `P passesthrough neither P1 nor P2, then `P is a simple 2-tangent to C or has normalcrossings with C. By (3) in Proposition 9, πP is indecomposable. Since thereexists a simple 2-tangent `P , we conclude GP ∼= Sd. In case P ∈ `P1P2

ande = 1 or d − 1, πP is an s-covering and e1 = d − 1 and e2 = 2, hence by (2) inProposition 9, GP is primitive and there exists a simple 2-tangent line `P , thuswe conclude GP ∼= Sd. In view of Remark 3 we conclude easily from the similarargument that GP ∼= Sd when P ∈ P2 \ (C ∪ `P1P2

).

Proof of the case (II)Assume C = C(4). It has the following property.

Claim 3. The TP1(resp. TP2

) is Y = 0 (resp. Z = 0) and TP1∩TP2

= {P3}, whichdoes not lie on C. Furthermore TP1

∩ C = {P1} and TP2∩ C = {P2, (1 : 1 : 0)}.

The C has one flex F of order 1, i.e., i(C, TF ;F ) = 3 and TF does not pass thoughP3.

Proof. The last assertion is checked by Hurwitz’s Formula and the others aresimple. �

Remark 4. The coordinates of the flex F is computed as (−576 : −4096 : 135).

Clearly, if P = P1 or P2, then GP ∼= Z2. If P ∈ C \ {P1, P2}, then n = 3, henceGP is primitive. We divide the proof into three cases

(1) P = F(2) P = (1 : 1 : 0)(3) P is the other point.

In any case, by Hurwitz’s Formula, we infer that there exists at least onesimple 2-tangent line passing through P , hence GP ∼= S3. Then consider thecase P ∈ P2 \ C. If P ∈ `P1P2

, then πP has ramification points R1 and R2

such that e(R1) = e(R2) = 2 and πP (R1) = πP (R2). Thus πP is not an s-covering. Consider πP for the most special case `P1P2

∩ TF = {P}. We inferfrom Hurwitz’s Formula that the ramification data is (3, 24) := (3, 2, 2, 2, 2). By


(3) in Proposition 9 we have GP ∼= S4. There are several cases of position ofP which yield different ramification data, however it is easy to see that thereexists i such that ei = 2. Then from Proposition 9 or 10 we conclude GP ∼= S4.

Proof of the case (III)Assume C = C54. It has the following property. There exists a projective

transformation σ such that σ(C) = C and σ(X,Y, Z) = (Y,X,−Z), (−X,Z, Y )or (Z, Y,X). So that σ interchanges Pi (i = 1, 2, 3).

Claim 4. The flexes of C are F1 = (4 : −1 : 4), F2 = (1 : −4 : 4) and F3 =(4 : −4 : 1), hence the tangent line to C at them are L1 : X + 8Y + Z = 0, L2 :8X + Y −Z = 0 and L3 : −X + Y + 8Z = 0, respectively. On the other hand, thetangent lines to C at P1, P2 and P3 are L4 : X = Y, L5 : X = −Z and L6 : Y = Z,respectively. There exist just three points Qi (i = 1, 2, 3) satisfying the followingconditions:

(1) Qi /∈ C.(2) If ` = `Qi does not pass through any cusp, then ` and C have normal

crossings or there exist two points Q′ ∈ C satisfying i(C, ` ;Q′) ≥ 3.Such Qi is an intersection Lj ∩ Lk, where {i, j, k} = {1, 2, 3}, indeed Q1 = (1 :−7 : 1), Q2 = (7 : −1 : 1) and Q3 = (1 : −1 : 7). Therefore, if P ∈ P2 \(C ∪ {Q1, Q2, Q3}), then there exists a line ` passing through P such that ` is asimple 2-tangent line to C.

L1

L2

L6L4 L5

L3

(1 : 1 : 1)••

•

•

••• P1P2P3

Q3

Q2

Q1

HHHHH

HHHHHH

HHHHHH

HHHHHH

��

��

��

��

��

��

��

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LLLLLLLLLLLLLLLLLLLLLL

Proof. Making use of the results in [O] and observing the self-duality of C, wecan check the assertions by direct computations. �


Now let us begin the proof. If P = P1, then n = 3, hence GP is primitive. Thelines `P1P2

and `P1P3yield the ramification points of order three of πP , hence

we infer from Hurwitz’s Formula that there exists i such that ei = 2. Thus weget GP1

∼= S3. For P = P2 or P3, using the projective transformation σ above,we see GPi ∼= S3 (i = 2, 3).

Next consider the case P ∈ C \{P1, P2, P3}. Then we have n = 5, hence GP isprimitive. Using Hurwitz’s Formula or the self-duality of C, we see that thereexists a simple 2-tangent line to C, thus we have GP ∼= S5.

Finally, we consider the remaining case P ∈ P2 \ C.

Claim 5. Let ni be the number of ramification points with index i. Then wehave n2 + 2n3 + 3n4 = 12, where n4 ≤ 3. In particular, if n4 = 3 (resp. 2), thenP = (1 : 1 : 1) (resp. Qi), furthermore n3 = 0 (resp. 3) and n2 = 3 (resp. 0).

Proof. The former assertion is clear from Claim 4 and Hurwitz’s Formula. Theproof of the latter assertion is as follows: observing Claim 4, we infer that, ifn4 = 3, then P is unique (1 : 1 : 1), which is the intersections of the threelines L4, L5 and L6. Similarly observing Claim 4, we infer that, if n4 = 2, thenP = Q1, Q2 or Q3. In this case we have i(C, `PPi ;Pi) = 3, hence n3 = 3. �

Claim 6. If πP is an s-covering, then πP is indecomposable.

Proof. By Claim 5 the ramification index is 2, 3 or 4. Suppose πP is decompos-able. Then, deg β = 2 or 3. By Claim 1 β is unramified at R′i = α(Ri) whereei = 2 or 3. By Claim 5 we have n4 ≤ 3. As we have seen in the proof ofProposition 9, β cannot be ramified at only one point. Thus we have n4 6= 1.If n4 = 0, then the proof is clear by (3) in Proposition 10. If n4 = 2, thenP = Qi (i = 1, 2, 3). In case deg β = 2, β is ramified at R′1 and R′2. Sincedegα = 3, this cannot occur. In case deg β = 3, β is ramified at R′1 and R′2with e(R′1, β) = e(R′2, β) = 2, however these do not satisfy Hurwitz’s Formula.If n4 = 3, then P = (1 : 1 : 1) and from Claim 4 and Hurwitz’s Formula weinfer that the ramification data is (43, 23) := (4, 4, 4, 2, 2, 2). If deg β = 2, thenβ is ramified at R′i, (i = 1, 2, 3). However, since degα = 3, this case cannotoccur. Then, we have deg β = 3. We see that easily that β is ramified at R′i withe(R′i, β) = 2 (i = 1, 2, 3). However, this does not satisfy Hurwitz’s Formula.Therefore πP is indecomposable. �

Now we resume the proof. We prove by examining the cases:(i) P = (1 : 1 : 1)

(ii) P = Qi (i = 1, 2, 3)(iii) P ∈ `PiPj (1 ≤ i, j ≤ 3), P 6= (1 : 1 : 1) and P 6= Qi (i = 1, 2, 3)(iv) P is the point not appearing in the above case.By Claims 5 and 6 the proof is complete for (i) and (iv). So let us treat

the case (ii). By Claim 6 GP is primitive. However, there exists no simple 2-tangent line. Take Q1 = (1 : −7 : 1) and consider the affine part Z 6= 0. Thedefining equation is (xy − x + y)3 + 54x2y2 = 0. Putting u = x − 1, v = y + 7and v = tu, we get h(t, u) := (tu2 − 8u + 2tu − 15)3 + 54(u + 1)(tu − 7)2 = 0.Here we consider the Galois group obtained by the special value t = 2. Bythe aid of a software, for example PARI, we see that the polynomial h(2, u) =(2u2−4u−15)3 +54(u+1)(2u−7)2 in Q[u] is irreducible and the Galois group ofthis polynomialis S6. Let u1(t), . . . , u6(t) be the roots of h(t, u) = 0 with respect


to u. Note that ui(t) (1 ≤ i ≤ 6) is regular near t = 2 and {u1(2), . . . , u6(2)} arethe roots of h(2, u) = 0. We can find ci ∈ Q (1 ≤ i ≤ 6) satisfying the conditions:u(t) = c1u1(t)+ · · ·+c6u6(t) (resp. u(2) = c1u1(2)+ · · ·+c6u6(2)) is a generator ofthe minimal splitting field of h(t, u) (resp. h(2, u)) over k(t) (resp. Q). Supposethe degree of u(t) is less than 6!. Then, so is u(2), which is a contradiction.Hence we have [k(t, u) : k(t)] = 6!, thus we conclude GP ∼= S6. The proof of theother two cases Q2 and Q3 are almost the same.

The proof of the case (iii) is as follows: Here we notice that if P ∈ `PiPj , i 6=j, (i, j = 1, 2, 3), then πP is not an s-covering. First we consider the special casewhere P is in some TFi , for example, `P1P2

∩ TF1= {P}. Then the ramification

data is {(F1, P1, P2, P3, R5, R6, R7), (4, 33, 23)} and πP (P1) = πP (P2). SupposeπP is decomposable. Then, by Claim 1, β : X3 −→ P1 is unramified at α(P3)and R′i, (i ≥ 5). Namely, β is ramified at just two points. Then the ramificationdata of β is {(α(F1), α(P1)), (2, 2)} or {(α(F1), α(P1)), (3, 3)}, where deg β = 2or 3, respectively. However it is easy to see that this is impossible consideringα and πP , so πP is indecomposable. Since there exist ei = 2 (i = 5, 6, 7) weconclude GP ∼= S6. On the other hand, if P is not in TFi for each i (i = 1, 2, 3),then, by (3) in Proposition 9, πP is indecomposable. Since there exists a simple2-tangent, we have GP ∼= S6.

Thus we complete all the proofs.

Remark 5. In the list of Theorem 16 only two kinds of group appear. Of course,other kinds will appear in other examples, for example, let us take the Fermatquartic X4 + Y 4 + Z4 = 0. Then there exist 12 points such that GP is thedihedral group of order 8 (cf. [M-Y]).

Problem. Concerning the Galois groups for C(e,d) (1 < e < d−1), full symmet-ric group Sd degenerates into the cyclic group. How does the symmetric groupdegenerate for various curves?

Acknowledgments. The authors would like to express their thanks to Okafor teaching the example of self-dual curve C54. They thank also the reviewersfor carefully reading the manuscript and giving the suitable suggestions forimprovements.

REFERENCES OF SECTION 4[ C ] F. Cukierman, Monodromy of projections, Mat. Contemp., 16 15th

School of Algebra (Portuguese) (1999), 9–30.[ F ] W. Fulton, Algebraic Curves, Math. Lecture Note Series, Benjamin,

New York (1969).[ H ] J. Harris, Galois groups of enumerative problems, Duke Math. J., 46

(1979), 685–724.[ I-U-N ] S. Iitaka, K. Ueno and Y. Namikawa, Descartes no Seishin to Daisûkika,

Nippon Hyoron Sha, 1980 (in Japanese).[ M ] K. Miura, Field theory for function fields of singular plane quartic

curves, Bull. Austral. Math. Soc., 62 (2000), 193–204.[ M-H ] K. Miura and H. Yoshihara, Field theory for function fields of plane

quartic curves, J. Algebra, 226 (2000), 283–294.[ O ] M. Oka, Elliptic curves from sextics, J. Math. Soc. Japan, 54 (2002),

349–371.


[ P-S ] G.-P. Pirola and E. Schlesinger, Monodromy of projective curves, J.Algebraic Geometry, 14 (2005), 623–642.

[ S ] J.-P. Serre, Topics in Galois Theory, Res. Notes in Math., Jones &Bartlett (1992).

[ Y1 ] H. Yoshihara, An application of Plüker’s relations (in Japanese), Sug-aku, 32 (1980), 367–369.

[ Y2 ] , Function field theory of plane curves by dual curves, J. Alge-bra, 239 (2001), 340–355.

[ Y3 ] , Galois points for plane rational curves, Far east J. Math., 25(2007), 273–284

[ Y4 ] , Rational curve with Galois point and extendable Galois auto-morphism, J. Algebra, 321 (2009), 1463–1472.


REFERENCES

[1] M. Field, Several Complex Variables and Complex Manifolds I. Cambridge University Press,1982.

[2] G. Fischer, Plane Algebraic Curves. American Mathematical Society, 2001.[3] O. Forster, Lectures on Riemann Surfaces. Springer-Verlag, 1981.[4] K. Fritzsche and H. Grauert, From Holomorphic Functions to Complex Manifolds. Springer-

Verlag, 2002.[5] P. Griffiths, Introduction to Algebraic Curves. Princeton University Press, 1989.[6] H. Hayashi and H. Yoshihara, Galois Group at Each Point for Some Self-Dual Curves. Hin-

dawai Publishig Corporation, Geometry, vol. 2013 (2013), pp. 1-6.[7] L. Kaup and B. Kaup, Holomorphic Functions of Several Variables. Walter de Gruyter, 1983.[8] F. Kirwan, Complex Algebraic Curves. Cambridge University Press, 1992.[9] R. Miranda, Algebraic Curves and Riemann Surfaces. American Mathematical Society, 1995.

[10] M. Namba, GEOMETRY OF PROJECTIVE ALGEBRAIC CURVES. MARCEL DEKKER,INC., 1984.

[11] R. Walker, Algebraic Curves. Springer-Verlag, 1978.[12] H. Whitney, COMPLEX ANALYTIC VARIETIES. Addison-Wesley Publishing Company,

1972.

galois group at each point for some self-dual

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