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Bull. Korean Math. Soc. 46 (2009), No. 1, pp. 171182

EMBEDDINGS OF LINE IN THE PLANE ANDABHYANKAR-MOH EPIMORPHISM THEOREM

Dosang Joe and Hyungju Park

Abstract. In this paper, we consider the parameter space of the rationalplane curves with uni-branched singularity. We show that such a param-eter space is decomposable into irreducible components which are ratio-nal varieties. Rational parametrizations of the irreducible componentsare given in a constructive way, by a repeated use of Abhyankar-MohEpimorphism Theorem. We compute an enumerative invariant of thisparameter space, and include explicit computational examples to recoversome classically-known invariants.

1. Introduction

Let C be a curve in the complex affine plane A2C

. We call C an un-parametrized affine embedding of line of degree d if it is isomorphic to A1

Cand

its projective closure C is a curve of degree d in P2C

. We call such a curve Ca smooth rational affine plane curve of degree d. Denote by A(d) the space ofall unparametrized affine embeddings of line of degree d. Note that the notionof the degree of C is well-defined with a fixed affine structure on A2

Csince any

affine automorphism can be extended to an automorphism ofP2C

.Many results are available on the local properties of the parameter space

A(d). For example, one knows that it is a union of smooth irreducible varieties[9]. Our main interest in this paper is in studying some of its global proper-ties. As a result, we show that A(d) is a disjoint union of smooth irreduciblerational varieties. In fact, we give explicit rational parametrizations of all ofits irreducible components, and compute the dimensions of the components. Inparticular, we show that the (maximal) dimension of A(d) is d + 2 for d 2,

and that there are exactly ( d(d+1)2 1) distinct unparametrized embeddings of

line (i.e., embeddings of line up to reparametrization) passing through generald + 2 points in A2

Cfor d 2.

Received May 20, 2008.2000 Mathematics Subject Classification. 14N10, 14Q05, 14R10.Key words and phrases. embedding of affine line, Abhyankar-Moh epimorphism theorem,

enumerative problems, Puiseux pairs.The first author was supported by the Faculty Research Fund of Konkuk University in

2004.

c2009 The Korean Mathematical Society

171

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172 DOSANG JOE AND HYUNGJU PARK

The main ingredient of the proof is the Abhyankar-Moh Epimorphism The-orem [1]. To explain this connection more explicitly, let us start by recalling a

basic definition.

Definition 1. A polynomial map = (f, g) : C1 C2 is called a parametrizedaffine embedding of line, or simply an embedding of line, ifC[t] = C[f(t), g(t)].

Note that the above condition amounts to the condition that the imagecurve in C2 is isomorphic to C1 under the polynomial map. The Abyhankar-Moh Epimorphism Theorem provides a necessary condition on the degrees off and g for (f, g) to be an embedding of line.

Theorem 1 ([1]). Suppose f, g are inC[t] with m := deg g n := deg f. IfC[f(t), g(t)] = C[t], then m divides n.

Among the curves parametrically defined by = (f, g) : C1 C2 with

m dividing n, there are smooth curves and non-smooth curves. An obviousquestion arises now:

How many such curves are smooth?

Since A(d) can be viewed as the space of all embeddings of line of degreed up to reparametrization, this question naturally prompts one to study theparameter space A(d). In fact, this was the initial motivation for the authorsto study the parameter space A(d).

We also study the parameter space M(d) of unicuspidal rational curveswith maximal tangent [4]. A(d) can be considered as a sub-variety of M(d),consisting of the curves whose singular points lie on the line at infinity whichis the maximal tangent line. Here, the maximal tangent line means the linewhich intersect the curve only at the singular point.

The variety A(d) has the main component Ad which has exactly the largestdimension among the irreducible components. The explicit parametrization ofa Zariski open subset Ad of the main component Ad is given by monomial termswith one exception in the homogeneous coordinates ofPH0(P2, O(d)). So, in asense the main component is almost a toric variety. Using this parametrization,we compute the degree of the variety A(d). Two proofs are given: one by usinga projection and Kouchinirenko theorem on the number of solutions of a systemof equations [16, 7], the other one by a direct method.

The same parametrization can be used for degree counting for the parameterspace of rational projective plane curves with unique irreducible singular points.As an illustration of our technique, we attempt to compute the degree of theparameter space M(3) of cuspidal cubic curves in P2

C. An explicit computation

with the computer algebra package Singular [10] shows that this degree is 24,coinciding with a classical result [18, 15, 2].After writing up this paper, we learned that the degree of A(d) had been

independently obtained in a recent paper [13], as an example. The approaches,however, appear to be substantially different. It is to be noted that our method

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EMBEDDINGS OF LINE IN THE PLANE 173

produces not only the degree of the parameter space A(d) but also an explicitparametrization of A(d) itself.

2. Decomposition of embeddings of line

Let us start by recalling basic terminologies.

Definition 2 (c.f. [17]). A polynomial map : C2 C2 is called an elemen-tary polynomial map ofC2 if (x, y) is (x, y + f(x)) or (x + g(y), y) for somepolynomial f(x) C[x] or g(y) C[y].

The Abyhyankar-Moh can be used to reduce the degrees of by applyingan elementary polynomial map.

Lemma 1. Let = (f, g) : C1 C2 be an embedding of line inC2 where

f(t) = a0tn

+ a1tn1

+ + ang(t) = b0t

m + b1tm1 + + bm.

Suppose deg f = n = k deg g = km for a positive integer k. Then there is aunique elementary polynomial map (x, y) = (x c(y), y) such that (t) =(h(t), g(t)), c(0) = 0, and deg h is strictly less than m. Furthermore, if f and gare viewed as polynomials inZ[a0, . . . , an, b0, . . . , bm][t], then c is a polynomialinZ[a0, . . . , an, b0, . . . , bn][

1b0

][t].

Proof. There are unique c0, c1, . . . , ck1 C such that the degree of

h(t) := f(t) (c0g(t)k + c1g(t)

k1 + + ck1g(t))

is less than the degree of g. This is possible due to a successive application of

Theorem 1. Let c0 = a0/bk0 , and suppose c0, c1, . . . , cl1 are defined then cl isdetermined as

cl = (alm c0bk,nlm c1bk1,nlm cl1bkl+1,nlm)/bkl0 ,

where g(t)p = (m

i=0 bitmi)p =

nj=0 bp,jt

j .

Definition 3. (1) Denote by R Mor(C1,C2) the locus of all embeddings ofparametrized affine line and define

R(n, ) = {(f, g) R | deg f = n, deg g < n}.

Similarly, define

R(n, m) = {(f, g) R | deg f = n, deg g = m}

and

R(n) = {(f, g) R | deg f = n, deg g n}.

(2) Let Ck = C Ck1 for k > 1 and Ck1,...,kq =

qi=1 Cki for ki > 1.

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Note that R(n, ) =

m|n,m 1,

k,m : Ck R(m, ) R(mk, )

(c,g,h) (h +k1i=0

cigki, g).

Given ordered positive integers k1, . . . , kq withq

i=1 ki = n and ki > 1, byiteratively composing s we obtain a map

(k1,...,kq) : Ck1 (Ck2 (Ckq R(1, )) ) R(n, ),

where (k1,...,kq) := k1,k2kq kq1,kq kq,1.

Definition 4. Denote the image of this composition map by Rk

where k =(k1, . . . , kq).

Lemma 2. Rk

is a locally closed subset of Cn+1 Cm+1 under k, wheren =

qi=1 ki, m =

qi=2 ki. Furthermore it is isomorphic to Ck R(1, ) as a

variety.

Proof. Consider

k : Ck R(1, ) Cn+1 Cm+1

and its converse

k : U Ck R(1, )

in some open neighborhood U of Rk

in Cn+1 Cm+1. Then k k is theidentity. However k k() = if and only if R

k

. This shows that Rk

isa closed subset of U.

Lemma 3. Let k > 1. Endow R(l, ) with the induced Zariski topology as asubspace ofCl+1 Cl. Then the polynomial map : Ck R(m, ) R(mk, )

sending (c,g,h) (h +k1

i=0 cigki, g) is a closed map.

Proof. Let n = mk. Consider the sequences (c(j), g(j), h(j)) such that thelimit (f(), g()) of (c(j), g(j), h(j)) exists in R(m, ). Then the sequenceg(j) has the limit g() = b0t

m + + bm in Cm+1. Since the first component

of (c

(j)

, h

(j)

, g

(j)

)) has degree mk > 1, g

()

must not be zero.Case I: Suppose that b0 = 0. We will show that this cannot happen. Since

the coefficient oftmk in c(j)

0 (g(j))k converges to a nonzero number, |c

(j)0 | goes to

. By a change of coordinate t, we may assume that g()(t) = bmltl+O(tl+1)where n > l 1 and bml = 0. Now consider the coefficients oft

mk, tmk1, . . .,

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tm(k1)+l in h(j) +

c

(j)i (g

(j))ki, which are

c

(j)

0 (b

(j)

0 )

k

,c

(j)0 (b

(j)0 )

k1b(j)1 ,

c(j)0 (k(b

(j)0 )

k1b(j)2 +

k(k 1)

2(b

(j)0 )

k2(b(j)1 )

2),

c(j)0 (k(b

(j)0 )

k1b(j)3 +k(k 1)(b

(j)0 )

k2b(j)1 b

(j)2 +

k(k 1)(k 2)

3!(b

(j)0 )

k3(b(j)1 )

3),

...

c(j)

0 (k(b(j)0 )

k1b(j)ml+(b

(j)0 , . . . , b

(j)ml+1)),

where is a homogeneous polynomial of degree k. It shows that |b(j)0 | decreases

as fast as 1/|c(j)0 |

1k , |b

(j)1 | decreases as fast as (or faster than) 1/|c

(j)0 |

1k , . . .,

|b(j)ml| decreases as fast as (or faster than) 1/|c(j)0 |

1k . This is a contradiction to

b(j)ml bml = 0.

Case II: Suppose that b0 = 0. Let f() = a0t

n + + an with n = mk.

Then c(j)0 goes to a0/b

k0 = 0, and cl goes to

(alm c0bk,nlm c1bk1,nlm cl1bkl+1,nlm)/bkl0 ,

where (m

i=0 bitmi)p =

nj=0 bp,jt

j . Thus, c(j)i has the (bounded) limit for all

i and also the sequence h(j) has the limit in Cm.Since the Zariski closure and the closure by strong topology of a constructible

subset in a variety coincide, the above arguments prove the lemma.

Remark 1. (1) R(n, n) = C R(n, ) by sending (f, g) to ( b0a0 , (f, g b0a0

f)),

where f =

ni=0 ait

ni and g =

ni=0 bit

ni. Here the equivalence symbol =means that there are regular maps between open subsets containing Rn,n andC R(n, ) in Cn+1 Cn+1 such that those maps are inverses to each other

when restricted to R(n, n) and C R(n, ). Later we will show that R(n, n)and R(n, ) are subvarieties (but not closed ones).

(2) Similarly R(n) = C R(n, ) by sending (f, g) to ( b0a0 , (f, g b0a0

f))

where g =n

i=0 bitni. Note that b0 here could possibly be zero.

In what follows, we will call the maximal dimension (or simply, dimension)of a variety X to be the maximum among the dimensions of the irreduciblecomponents of X.

Theorem 2. (1) The parameter spaceR(k1,...,kq) inCn+1Cm+1 is isomorphic

to

R(k1,...,kq)=

qi=1

Cki C3

of dimension (

ki) + 3.

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(2) R(n, ) =

k,Q

ki=n

Rk.

(3) R(n) = C

k,Q

ki=n,

Rk

of the maximal dimension n + 4 if n 2or 4 if n = 1.

Proof. The proof of statement (1) is proven in Lemma 2. The second statementfollows from Lemmas 2 and 3. For the proof of statement (3). Consider

: (C Cn) Cn+1 C (C Cn) Cn

sending (a, b) to (b0/a0, a, b b0a0

(a1, . . . , an)). It has the inverse (c, a, d) =

(a, d + ca).

Note that the maximal dimension of R(n, m) is (n/m) + m + 3, and themaximal dimension of R(n, n) is n + 4, and the maximal dimension of R(n, )is n + 3.

Example 1.

R(8, 8) = C R(8, )

= C (C4 C2

C2 C2 C2

C2 C4

C8) R(1, ),

R(8, 4) = (C2 C2 C2

C2 C4) R(1, ),

R(8) = C (C4 C2

C2 C2 C2

C2 C4

C8) R(1, ).

Note that R Mor(C1,C2) has an action by Aff(C2) = GL2(C)C2 induced

from its action of affine automorphisms of the target C2. Since Aff(C2) is aconnected algebraic group and R is a disjoint union of components of types,we obtain the following definition.

Definition 5. An element of Rk

is called a parametrized affine embedding ofline of type k. In general we define the type of a closed parametrized affineembedding of the line A1

Cto the affine plane A2

Cafter choosing appropriate

affine coordinates ofA1C

and A2C

.

3. Applications

3.1. Parameter spaces of (unparametrized affine) embeddings of line

Given (t) :C

1

C

2

, consider its unique projectivization :(t) : C1 C2

: P1 P2.

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In terms of homogeneous coordinates ofP1, the projectivization can be writ-ten as

([t : s])

= [a0tn+a1t

n1s + + ansn : b0t

msnm + b1tm1snm+1 + + bms

n : sn],

where (t) = (a0tn + a1t

n1 + + an, b0tm + b1t

m1 + + bm).This rational projective curve of degree n meets the line at infinity exactly

at one point, which is [a0 : b0 : 0] if n = m, [1 : 0 : 0] if n > m. It is obviousthat this point is the unique singular point of the given curve, if there is any.Moreover the line l at infinity is a unique tangent line at this singular pointof the curve. The intersection multiplicity C l is n. Let us denote thatcuspidal singularity of plane curves defined to be irreducible singular point.Also unicuspidal curve meant to be a curve having only one singular point.

Definition 6. Denote by M(n) P(H0(P2, O(n)) the space of unicuspidalrational curves of degree n with maximal tangent line.

Example 2 ([8]). There is a rational plane curve with one cusp and the uniquetangent line at the cuspidal point.

In P2, consider the plane curve C defined by the equation

(x2 yz)2 zy3 = 0.

By a parametrization, we see that it is a rational curve having only one singularpoint (0, 0, 1) with a unique tangent line:

x = t(t2 1), y = (t2 1)2, z = 1.

However the curve C \ (0, 0, 1) is not isomorphic to C: Otherwise, there is aregular map : P1 \ {1} C P1. Now considering the degree of the

extension : P1 P1, we obtain a contradiction.

Example 3. There is a rational unicuspidal quintic not having maximal tan-gent line [6, 20]. The equation is the following:

(y x2)(y x2 2xy) + y5 = 0.

The affine plane curve given by the parametrization (t) : C1 C2 has thedefining equation

h(x, y) := Rest(f(t) x, g(t) y) PH0(P2, OP2(n)),

where Rest(f(t) x, g(t) y) is the resultant of f(t) x and g(t) y withrespect to t (c.f. Theorem 0.4 of [19]). Define A(n, m) by

A(n, m) = {h(x, y) = Rest(f(t) x, g(t) y)|(f, g) R(n, m)},

which is the locus of the curves corresponding to R(n, m). We also define Ak

to be the locus corresponding to Rk

. The locus A(n, m) is viewed as a subsetofPH0(P2, OP2(n)). Let Mk be the subset of M(n) consisting of those thatcan be transformed to elements of A

kby an automorphism ofP2.

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Theorem 3. Each component Mk is a rational variety of dimension 4 +

ki

for n 3.

Proof. Note that there is a fibration : Mk F(3) where F(3) is a flagmanifold and assigns to a curve the flag of its singular point and the tangentline at the singular point. The fibration has a rational section by the followingconstruction. Near the flag

((0, 1, 0), (0, 1, 0), (1, 0, 0))

of a point in F(3), there is an affine chart of (,,) C3 parametrizing flags(, 1, , (, 1, ), (1, 0, )). Pick any f(x,y,z) Ek, then f(x,y,z x +( )y) defines a rational section. Thus Mk is a rational variety.

Corollary 1. A(n) =k

Ak is a quasi projective subvariety, and each com-ponent Ak is a rational variety of dimension 2 + ki for n 2.Example 4. Note that A(1) is P2 minus one point. In general An is an affinebundle over P1 with fiber C Cn for n 2.

3.2. The degree ofA(n)

Lemma 4. Let n be an nonnegative integer. The lattice volume (c.f. [7]) ofthe convex hull of the exponents of

1, yk0 , x1, x1yk1 , x2, x2y

k2 , . . . , xn, xnykn

inZn+1 isn

i=0 ki if ki are nonnegative integers not vanishing simultaneously.

Proof. When n = 0, it is clear. Assume that n > 0. By the Kouchnirenkotheorem [16], it is equal to the number N of common zeros of generic n + 2equations ax = 0, where runs over all exponents in the set of monomialsabove.

Note that the equations can be rewritten as Ax = 0, where

x = (1, x1, . . . , xn)transpose

and the entries aij of the matrix A satisfies aij C + Cykj if n j 0. By

taking the determinant of A, we conclude that the number N is less than orequal to the degree of the determinant of A in y. Thus N

ni=1 ki. However

we can achieve the upper bound by simply taking A as the maximal cyclic form

yk0 0 0 0 11 yk1 0 0 00 1 yk2 0...

...0 1 ykn

Theorem 4. The degree of (the maximal dimension component of) A(n) isn(n+1)

2 1.

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Proof. Since we have

C An,1 = n

i=0

ai(y x)ni x P(H0(P2, O(n)) | a Cn, C

,

it is enough to find the degree of the closure of the image

: (C)n+2 P(n+22 )1

defined by sending (a0, . . . , an, ) to (a0, a0 , . . ., a0n, a1, a1 , . . ., a1

n1, . . .,an2, an2, an2

2, an1, an1 1, an).Let be the composition A of the affine transformation

A(z0, . . . , zN, zN+1) (z0, . . . , zN zN+1, zN+1)

ofPN+1 and the projection ofPN+1 to PN centered at (0, . . . , 0, 1) along thesubspace P(CN {0}). Consider

: (C)n+2 PN

sending (a0, . . . , an, ) to (a0, a0 , . . ., a0n, a1, a1 , . . ., a1

n1, . . ., an2,an2, an2

2, an1, an1, an, 1).Note that is defined by monomials and = . Now applying Lemma 4,

one notes that the degree of the closure of Im is n(n+1)2

. Since this variety isdefined by the obvious quadratic equations, it is easy to check that the center(0, . . . , 0, 1, 0, 1) is a nonsingular point, which, together with the injectivity of, implies that the degree of Im is one less than the degree of Im.

Another Proof of Theorem 4. Denote the general hyperplane P(H0(P2, OP2(n))by Hi. To compute the intersection number

H1 Hn+2 A(n),

we may assume that all the intersection points occur in

C An,1 =

ni=0

ai(y x)ni x P(H0(P2, O(n))|a Cn, C

.

The equations can be expressed by a matrix equation Ab = 0 where A hasentries aij that are polynomials in with degree kj and j = 0, 1, . . . , n + 1.Note that the first column entries ai0 are constants, and b is the transpose of

(1, b1, . . . , bn+1) := (1, an, an1, . . . , a0).

There is one more constraint that the constant term ai0 equals the negative ofthe leading coefficient of ai1. By considering the determinant of A, it is clearthat there are at most (

n+1i=0 ki) 1 solutions. However one can achieve the

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upper bound by taking the specialization of A to

1 k1 0 0 1

0 k11 0 0 10 1 k2 0...

...0 1 kn+1

.

The above theorem combined with Kleimans transversality theorem [14, 5]applied to (P2)n+2 with the group action (GL2(C))

n+2 implies the following.

Corollary 2. Let n 2. There are exactly (n(n+1)2 1) distinct plane curvesin A(n), passing through general dimA(n) points inP2.

3.3. The degree ofM(3)

Corollary 3 ([15, 2]). Let M(3) be the space of cuspidal cubic curves. Thenthe deg M(3) = 24.

Let (f(t), g(t)) be a smooth polynomial parametrization of a cubic curve.Then its general form must be (a0t

3 + a1t2 + a2t + a3, t). And their defining

equations are parametrized as follows:

a0y3 + a1y

2 + a2y + a3 = x a0y3 + a1y

2z + a2yz2 + a3z

3 = xz2.

The defining equations near singular point [1 : 0 : 0] are

a0y3 + a1y

2z + a2yz2 + a3z

3 z2 = f(y, z).

Now we consider PGL(3,C) action on them. The local action changes thesingular point and the unique singular tangent:

f(y, z) f(y + b1, y + z + b2).Hence we have local parametrization of cuspidal cubics as follows:

a0(y + b1)3 + a1(y + b1)

2(y + z + b2) + a2(y + b1)(y + z + b2)2

+a3(y + z + b2)3 (y + z + b2)

2 = 0.

We can use deg lex order y > z with descending order.

(a0 + a1 + a22 + a3

3)y3

+(a1 + 2a2 + 3a32)y2z

+(a2 + 3a3)yz2

+a3z3

+(3a0b1 + a1b2 + 2a1b1 + 2a2b2 + a2b12 + 3a3b2

2 2)y2

+(2a1b1 + 2a2b2 + 2a2b1 + 6a3b2 2)yz+(a2b1 + 3a3b2 1)z

2

+(3a0b21 + a1b

21 + 2a1b1b2 + a2b

22 + 2a2b1b2 + 3a3b

22 2b2)y

+(a1b21 + 2a2b1b2 + 3a3b

22 2b2)z

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+(a0b31 + a1b

21b2 + a2b1b

22 + a3b

32 b

22).

Define polynomials Pi,j(a0, a1, a2, a3, b1, b2, ) by

f(y + b1, y + z + b2) =

i+j3

Pi,j(a0, a1, a2, a3, b1, b2, )yizj .

Hence the projective closure, Im(), of the image of the map

: C C6 C9 CP9

is the parameter space of cuspidal cubic curves in CP2. It is an explicit rationalparametrization of the parameter space of cuspidal cubic curves.

An explicit computation with the computer algebra package Singular [10]

shows that the degree of Im() is 24. Moreover another numerical invariantwhich counts the number of cuspidal cubics with a fixed singular point can becomputed by computing the degree of the subvariety Im((b1 = b2 = 0)). A

computation shows that this number is 2. We also have checked the degree ofthe subvariety Im((b1 = 0). This number counts the cuspidal cubics whosesingular points lie on a fixed line. Our computation shows it is 12.

Remark 2. (1) These numerical invariants have been known for years [18,15, 2, 11, 12], going all the way back to a very old Enriques formula.

(2) After writing up this paper, we learned that the degree of A(d) hadbeen independently obtained in a recent paper [13] as an example. Theapproaches, however, appear to be substantially different.

(3) Note that our method produces not only the degree of the parameterspace A(d) but also an explicit parametrization of A(d) itself.

Acknowledgements. The authors are grateful to Ziv Ran for helpful com-

ments. Dosang Joe thanks the staff at Korea Institute for Advanced Study,for providing him with pleasant research atmosphere during his visits as anassociate member. The second author was supported by KRF Grant #2005-070-C00004.

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no. 4, 152155.

Dosang Joe

Department of Mathematics Education

Konkuk University

Seoul 143-701, Korea

E-mail address: [email protected]

Hyungju Park

School of Computational Sciences

Korea Institute for Advanced Study

Seoul 130-722, Korea

E-mail address: [email protected]

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