PLANE CURVES MEETING AT A POINT WITH HIGH INTERSECTION MULTIPLICITY

Abstract

As a generalization of an inflection point, we consider a point P on a smooth plane curve C of degree m at which another curve C' of degree n meets C with high intersection multiplicity. Especially, we deal with the existence of two curves of degree m and n meeting at the unique point.

1. INTRODUCTION AND PRELIMINARIES

Let Cm and Cn be smooth complex projective plane curve of degree m and n, respectively, with m, n ∈ ℕ. Let P be an intersection point of Cm and Cn. We denote I(Cm ⋂ Cn; P) the intersection multiplicity at P of two curves Cm and Cn.

For a point P of C = Cd (d ≥ 3) and for a general line L passing through P, the intersection multiplicity I(C ⋂ L; P) is one. If L = TP (C) is the tangent line of Cd at P then we have I(C ⋂ TP (C); P) ≥ 2 and equality holds for general point P. If I(C ⋂ TP (C); P) = e > 2, we call P an inflection point of Cd with intersection multiplicity e. In particular, if I(C ⋂ TP (C); P) = d, we call P a total inflection point of Cd. In this case the tangent line and the curve meet at only one point P by Bezout’s theorem.

Existence of an inflection point of high intersection multiplicity helps us to find Weierstrass points on C([1]). The canonical series of a smooth curve Cd is cut out by the system of degree d − 3 curves, hence e(d − 3)P is a special divisor. Thus, if d ≥ 4 and , then an inflection point with multiplicity e is a Weierstrass point. More generally, if there exists a curve of Cd−3 with I(Cd⋂Cd−3;P) ≥ g where , then the point P is a Weierstrass point of the curve Cd.

With this motivation we generalize the notion of an inflection point. We want to find two curves Cm and Cn with an intersection point P with high intersection multiplicity I(Cm ⋂ Cn; P).

To construct smooth plane curves satisfying our condition, we use the following theorems frequently.

Theorem 1.1 ([5, Bertini’s Theorem]). The genereic element of a linear system is smooth away from the base locus of the system.

Theorem 1.2 ([4, Namba’s Lemma]). Let C, C1 and C2 be plane curves. If P is a nonsingular point of C, then we have

Theorem 1.3 ([2, Bezout’s Theorem]). Let Cm and Cn be smooth plane curves of degree m and n. Then we have

 

2. PLANE CURVES MEETING WITH MAXIMAL INTERSECTION MULTIPLICITY

At first we give easy examples of smooth plane curves Cm and Cn with Cm◾Cn = mnP. Throughout this paper, the point P is the origin (0, 0) in the affine plane, i.e., the point (0, 0, 1) in homogeneous coordinate of the projective plane.

Example 2.1. (1) The case m = 1 :

Let C1 and Cn be the cur(ves defined by non-homogeneous equations as follows;

Then for general a, the curve Cn is smooth by 1.1 and we have

(2) The case m = 2 :

Let C2 and Cn(n ≥ 2) be the curves defined by non-homogeneous equations as follows;

Then for general a, the curve Cn is smooth by Bertini’s theorem and we have

(3) A generalization of (1) and (2) :

Let Cm and Cn with m ≤ n be the curves defined by non-homogeneous equations as follows;

Then for general k(x, y) and h(x, y), the curves Cm and Cn are smooth and we have

Remark 2.2. In (3) of above example, the point P is a total inflection point of Cm.

Now we are interested in the point P which is a total inflection point of neither Cm nor Cn. To consider such a problem, we prove some theorems concerning to the existence of such curves.

Theorem 2.3. Let m, n be positive integers. Suppose that there exist smooth curves Cm and Cn such that I(Cm ⋂ Cn; P) = mn. If k is a positive integer such that kn ≥ m, then there exists a smooth curve Ckn such that I(Cm ⋂ Ckn; P) = kmn.

Proof. Consider a linear system . If Q(≠ P) is contained in the base locus of the linear system < Cm(1 + xkn−m + ykn−m), > then Q lies on the curve 1 + xkn−m + ykn−m and does not lie on Cm, since Cm and Cn meet only at P. Since Q is a smooth point of the curve 1 + xkn−m + ykn−m, it is a smooth point of a general member in the system. On the other hand P is a smooth point of Cm and not on the curve 1+xkn−m+ykn−m, P is a smooth point of a general member of the system. Let Ckn be a general member in the linear system. Then, by Bertini’s theorem, Ckn is a smooth curve. □

Remark 2.4. In fact, we may obtain Example 2.1 (3) from Example 2.1 (1) and Theorem 2.3.

Corollary 2.5. Let m be any positive integer and n be a positive even integer with n ≥ m. Then there exist smooth curves Cm and Cn such that I(Cm ⋂ Cn; P) = mn.

Proof. If m = 1, then it follows from Example 2.1 (1). So we assume that m ≥ 2. Let Cm and C2 be curves in (2) of Example 2 which satisfy I(Cm ⋂ C2; P) = 2m. Then for any even n ≥ m, there exists a smooth Cn such that I(Cm ⋂ Cn; P) = mn, by above theorem. □

Theorem 2.6. Let m > n ≥ k be natural integers. If Cm, Cn and Ck are smooth curves such that I(Cm ⋂ Cn; P) = mn and I(Cm ⋂ Ck; P) = mk, then n = k and Cn and Ck are the same curves.

Proof. By Namba’s lemma, I(Cn⋂Ck; P) ≥ mk which is bigger than nk, the product of the degrees of Cn and Ck. By Bezout’s theorem, Ck and Cn has a common component. However it is impossible since Ck and Cn are smooth and so irreducible unless Cn = Ck. □

Corollary 2.7. Let Cm be a smooth curve. Then there exists at most one smooth curve Ck with 1 ≤ k ≤ m − 1 such that I(Cm ⋂ Ck; P) = mk.

Proof. Obvious. □

Theorem 2.8. Let C3 and be distinct smooth cubics such that . Then I(C3 ⋂ C2; P) ≤ 5 for any irreducible conic C2.

Proof. Suppose I(C3 ⋂ C2; P) = 6 for some irreducible conic C2. Then, by Namba’s theorem, the point P can not be an inflection point of C3, and hence C3 · TPC3 = 2P + Q with P ≠ Q, where TPC3 is the tangent line to C3 at P. Then

Thus we have P ~ Q, which is a contradiction, since the genus of C3 is one. □

Theorem 2.9. Let Cm(m ≥ 3) and C2 satisfies I(Cm ⋂ C2; P) = 2m. Then there exists no smooth curve Cn of odd degree n such that I(Cm ⋂ Cn; P) = mn.

Proof. Note that TPCm = TPC2 and I(C2 ⋂ TPC2) = 2.

If n = 1 then for any C1, I(Cm⋂C1; P) ≤ I(Cm⋂TPCm; P) = I(C2⋂TPC2; P) = 2, by Namba’s lemma.

Suppose that for n = 2k + 1(k ≤ 1) there exists a smooth curve C2k+1 such that

On the other hand, since TPCm = TPC2 and I(C2 ⋂ TPC2; P) = 2, we have

where degD = m − 2. Then this implies the existence of the linear series on Cm, which is a contradiction. □

To state a generalization of the above theorem we need notations. Let is := is(Cm) = max{I(Cm ⋂ F; P) | degF = s}, and let Is(Cm) be a curve of degree s such that is(Cm) = I(Cm ⋂ Is(Cm); P). Note that I1(Cm) = TPCm and i1 = I(Cm ⋂ TPCm; P).

Theorem 2.10. Let m ≥ 3 and let Cm be smooth. Suppose that there exists a smooth curve Cr(r ≥ 2) such that I(Cm⋂Cr; P) = mr. If P is not a total inflection point of Cm, then, for k ≥ 1, there is no smooth curve Ckr±1 such that I(Cm ⋂ Ckr±1; P) = m(kr ± 1).

Proof. Suppose that there exists a smooth curve Ckr+1[resp: Ckr−1]. Then

On the other hand,

where D is the divisor such that Cm ◾ TPCm = i1P + D, hence its degree is m − i1 which satisfies 1 ≤ m−i1 ≤ m−2. Comparing two divisors, we conclude that there exists a linear series on Cm, which is impossible on a smooth plane curve of degree m. □

Theorem 2.11. Let m ≥ 7 and let Cm be smooth. Suppose that there exists a smooth curve Cr with such that I(Cm ⋂ Cr; P) = mr. Then, for k ≥ 1, there is no smooth curve Ckr±2 such that I(Cm ⋂ Ckr±2; P) = m(kr ± 2).

Proof. Suppose that there exists such a curve Ckr+2[resp: Ckr−2]. Then

On the other hand,

where D is the divisor such that Cm◾ I2(Cm) = i2P +D, hence its degree is 2m−i2. Comparing two divisors, we conclude that there exists a linear series . By Namba’s lemma and since the dimension of conics is 5, we have 5 ≤ i2 ≤ 2r ≤ m. By Coppens’ results([3]), we can not have such a linear series on a smooth plane curve of degree m. □

Theorem 2.12. Let m ≥ s2 + 2 and let Cm be smooth. Suppose that there exists a smooth curve Cr with such that I(Cm⋂Cr; P) = mr and is(Cm) ≥ s2+1. Then, for k ≥ 1, there is no smooth curve Ckr±s such that I(Cm ⋂ Ckr±s; P) = m(kr ± s).

Proof. Suppose that there exists such a curve Ckr+s[resp: Ckr−s]. Then

On the other hand,

where D is the divisor such that Cm ◾ Is(Cm) = isP +D, hence its degree is sm−is. Comparing two divisors, we conclude that there exists a linear series . By Namba’s lemma and our assumption, we have s2 + 1 ≤ is ≤ sr < m. By Coppens’ results([3]), we can not have such a linear series on a smooth plane curve of degree m. □

 

3. SOME EXAMPLES

Now we give examples of Cm and Cn meeting at the unique point P which is not an inflection point of any curve and I(Cm ⋂ Cn; P) = mn.

If a smooth curve C passing through the origin P(0, 0) is given by the equation

then TP (C) is given by the equation y = 0 and I(C ⋂ TP (C); P) = 2 so P is not an inflection point of C.

We found examples using the mathematics package, Maple.

Example 3.1. The case m = 3 and n = 3, 4 or 6.

(1) m = n = 3 : Let C3 and be given by the equations

Then the equation for C3 and satisfies

so

Also we can represent C3 and using a parameter t as follows

C3 : (t, t2 − t5 + 2t8 − 5t11 + 14t14 − 42t17 + . . . )

and we get again.

In fact the resultant of C3 and is x9 so .

(2) m = 3 and n = 4 : Let C3 and C4 be given by the equations

Then

so

We can represent C3 and C4 with a parameter t as follows

C3 : (t, t2 − t6 + 3t10 − 12t14 + 55t18 + . . .)

C4 : (t, t2 − t6 + 3t10 − t12 − 12t14 + 9t16 + 53t18 + . . .)

so I(C3 ⋂ C4; P) = 12 again.

In fact the resultant of C3 and C4 is x12 so I(C3 ⋂ C4; P) = 12.

(3) m = 3 and n = 6 : Using C3 and in (1), we construct curves for m = 3 and n = 6.

With the similar method as above we have

Also the resultant of C3 and C6 is x18.

Example 3.2. The case m = 3 and n = 5.

Let C3 and C5 be given by the equations

We can see that C3 and C5 are smooth plane curves and the resultant of C3 and C5 is x15 by Maple so I(C3 ⋂ C5; P) = 15.

Example 3.3. The case m = 3 and n = 7.

Let C3 and C7 be given by the equations

We can see that C3 and C7 are smooth and that the resultant of C3 and C7 is 33554432x21 using Maple. So we get C3 and C7 with I(C3⋂C7; P) = 21 which is the maximal possible intersection multiplicity that two smooth plane curves of degree 3 and 7 can have.