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Polar Curves
JOURNAL OF ALGEBRA ARTICLE NO.
181, 449]477 Ž1996.
0129
Polar Curves Abramo HefezU Departamento de Matematica ´ Aplicada, Uni¨ ersidade Federal Fluminense, R. Sao ˜ Paulo srn-Campus do Valonguinho, 24020-005 Niteroi, RJ-Brazil
and Neuza Kakuta Departamento de Matematica-IBILCE, UNESP, R. Cristo¨ ao ´ ˜ Colombo 2265 J. Nazareth, 15054-000 S. Jose do Rio Preto, SP-Brazil Communicated by Walter Feit Received January 31, 1995
1. INTRODUCTION Polar curves of plane algebraic curves in characteristic zero have been studied extensively by several people and appear in the classical literature related to Plucker’s formulas Žsee for example w2, 8, or 10x.. ¨ This study however has not been carried out in depth for curves defined over fields of positive characteristic, and the aim of this work is to contribute to filling this gap. It turns out, as we will see, that the theory in positive characteristic is quite different from the characteristic zero case and establishes interesting interplays between geometry and number theory. Our motivation comes from w5x where, making a new use of polar curves in positive characteristic, we improved on the known upper bounds for the number of points of Fermat curves over finite fields. After the study we make in the present paper, it will be possible to apply the methods of w5x to other families of curves as well. The paper is organized as follows. Let Z be a projective plane curve of degree d and let l be an integer between 1 and d. We denote by DlP Z the * Partially supported by CNPq-Brazil. 449 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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HEFEZ AND KAKUTA
polar l-ic of Z at P. In Section 2 we show that the intersection multiplicity of Z and DlP Z at P is an upper semi-continuous function in P on Z. The minimum value of this function, achieved on an open dense Zariski subset of Z, will be denoted by hZ, l . It is shown in Theorem 2.5 that there exists an arithmetical function h Ž d, l. such that for any curve Z of degree d we have hZ, l G h Ž d, l., with equality holding for the general curve of degree d. In Section 3 we characterize the non-general curves Z, that is, the curves for which hZ, l ) h Ž d, l., in terms of the vanishing of certain derivatives of the defining polynomial of Z, assuming some mild behaviour on the singularities of the curve. In Section 4 we show how to get in some cases the shape of the equation of a non-general curve. In Section 5 we totally describe the arithmetical function h Ž d, l.. In Section 6 we give an enumerative formula for the stationary points on a suitably general curve with respect to its associated family of polar l-ics. More precisely, we count the number of points P on Z at which Z and DlP Z have intersection multiplicity greater than hZ, l . This result contains in particular the usual Plucker’s formula for the number of flexes of a non-singular plane ¨ projective curve. In Section 7 we introduce the polar morphisms of plane curves, which generalize the Gauss map, and prove a formula that generalizes Plucker’s formula relating the degree of a curve, the degree of its ¨ Gauss map, and the degree of its dual curve.
2. GENERAL THEORY Let K be an algebraically closed field, of characteristic p G 0, fixed once and for all. Let F g K w X 0 , X 1 , X 2 x be a homogeneous polynomial of degree d and let Q be an element of K 3. We consider, for l s 1, . . . , d, the homogeneous polynomials
Ž DlQ F . Ž X0 , X1 , X2 . s
Ý
i 0qi 1qi 2s l
Ž Di , i , i F . Ž Q . X 0i 0
1
2
0
X 1i1 X 2i 2 ,
where Di 0 , i1 , i 2 denotes the mixed partial Hasse differential operator, of order in with respect to the indeterminate Xn , for n s 0, 1, 2. We will also denote Di 0 , 0, 0 , D 0, i1 , 0 , and D 0, 0, i 2 respectively by D Xi 00 , D Xi11, and D Xi 22 . For the properties of these operators needed here we refer the reader to w3x. From the definition of the Hasse operators, we have, for P, Q g K 3 and indeterminates t 0 and t 1 , the identity F Ž t 0 P q t 1 Q . s t 0d F Ž P . q t 0dy 1 t 1 Ž D1P F . Ž Q . q ??? 1 d q t 0 t 1dy 1 Ž Ddy P F . Ž Q . q t1 F Ž Q .
Ž 1.
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POLAR CURVES
The above identity gives many relations involving the polynomials DlQ F, of which we use only a few. LEMMA 2.1. We ha¨ e the relations l .Ž . Ži. Ž DlP F .Ž Q . s Ž Ddy Q F P .
Žii. Ž DlP F .Ž P . s Ž ld . F Ž P .. Žiii. Let T be a linear automorphism of K 3. For any polynomial G in K w X 0 , X 1 , X 2 x and any Q in K 3, we denote GŽT Ž Q .. by G T Ž Q .. Then we ha¨ e
Ž DlP F .
T
s Ž DlT y1 Ž P . F T . .
Proof. Ži. is obtained by comparing the corresponding identity for Ž F t 1 Q q t 0 P . with Ž1.. Žii. follows easily from identity Ž1. when we put P s Q. Žiii. follows replacing Q by T Ž Q . in both members of Ž1. and then developing the first member using again formula Ž1.. Note that Lemma 2.1 Žii. is a generalization of Euler’s relation, which applied to the monomial X 0n 0 X 1n1 X 2n 2 , with n 0 q n1 q n 2 s d, gives in particular the well known relation, still called Euler’s relation,
Ý
i 0qi 1qi 2s l
n0 i0
n1 i1
n2 d ' . i2 l
ž /ž /ž /
ž /
Note also that what we established for polynomials in three indeterminates is valid in any number of indeterminates; in particular, also in the above Euler’s relation. DEFINITION. If Z is a projective plane curve defined by F s 0, and if some of the derivatives of F of order l evaluated at P is nonzero, then the curve DlP Z defined by DlP F s 0 is called the polar l-ic cur¨ e of Z at P. In particular, we have for l s 1 the polar line at P Žwhich is, for P g Z, the tangent line at P ., for l s 2 we have the polar conic at P, and for l s 3 the polar cubic at P. It is clear from Lemma 2.1 Ži. that our indices for polars are reversed with respect to the classical use, that is, our definition of DlP is the classical Ddy l Žas defined for example in w2x or in w8x.. We also have from this l formula that P g DlQ Z if and only if Q g Ddy P Z. It follows from Lemma l 2.1Žii. that if P g Z, then P g D P Z for all l. Lemma 2.1Žiii. says that the polar l-ic is a well defined projective object.
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HEFEZ AND KAKUTA
In order to study the behavior of the intersection multiplicity function of Z with its polar curves, we establish the following result. PROPOSITION 2.2. Let Z : F s 0 be an integral projecti¨ e plane cur¨ e and p let Z˜ ª Z be the normalization map of Z. Let l be an integer such that 1 F l - d, and some mixed deri¨ ati¨ e of F of order l is not identically zero. Then the integer-¨ alued function on Z˜ defined by the formula P˜ ¬ ord P˜ Ž Dlp Ž P˜. F . is upper semi-continuous. Proof. Let Q g Z˜ and suppose without loss of generality that p Ž Q . s Ž1; a; b .. Choose a rational function s g K Ž Z˜. such that its image in the local ring OZ,˜ Q is a uniformizing parameter. If U˜ is the open set of Z˜ of all points P˜ for which t s s y sŽ P˜. is an uniformizing parameter of OZ,˜ P˜ and p Ž P˜. f X 0 s 04 , then we have that
Ž Dlp Ž P˜. F . Ž 1, x, y . s
Ý
i 0qi 1qi 2s l
Ž Di , i , i F . Ž p Ž P˜. . x i 0
1
2
1
y i2 .
Consider the powers series Ý iG 0 Dsi Ž x (p . t i and Ý iG 0 Dsi Ž y (p . t i in K wU˜ xww t xx. These powers series with their coefficients evaluated at P˜ g U˜ represent the expansions of the functions x (p and y (p in the completion of the local ring OZ,˜ P˜. Consider now the expansion
Ž Dlp Ž P˜. F .
ž
1, Ý Dsi Ž x (p . Ž P˜. t i , Ý Dsi Ž y (p . Ž P˜. t i s iG0
iG0
/
Ý h i Ž P˜. t i , iG0
˜ Let h be the first index i for where each h i is a regular function on U. Ž which h i k 0 such an index exists because some derivative of F of order l is nonzero., then ord P˜Ž Dlp Ž P˜. F . is equal to h for P˜ in the open dense set ˜ so the P˜ g U˜ < h i Ž P˜. / 04 and is greater than h at the other points of U, ˜ hence on Z. ˜ function under consideration is upper semi-continuous on U, We will denote by I Ž P, F.G . the intersection multiplicity of the curves F s 0 and G s 0 at the point P. COROLLARY 2.3. If F is an irreducible polynomial of degree d in K w X 0 , X 1 , X 2 x, then for e¨ ery l s 1, . . . , d y 1 such that some deri¨ ati¨ e of order l of F is nonzero, the function P ¬ I Ž P, F.DlP F . is upper semi-continuous on Z : F s 0.
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POLAR CURVES
The upper semi-continuity of the above intersection multiplicity implies that its minimum value is achieved in a non-empty Zariski open set of Z. DEFINITION. We define hZ, l as being the minimum value of the above intersection multiplicity function. The number hZ, 1 is well known and it is the last order « 2 of the order sequence of the curve in the projective plane, that is, it is the intersection multiplicity at a general point P of Z with its tangent line at P Žsee for example w6x or w9x.. In order to compute the value of hZ, l , we will give the expansion of the polynomial DlQ F at a point P. Let f Ž x, y . be a polynomial in K w x, y x. If P X s Ž a, b . is a point in the affine plane, and r is an integer such that 0 F r F degŽ f ., we define f r , P X Ž x, y . s
Di , j f Ž P X . Ž x y a . Ž y y b . , i
Ý
j
iqjsr
where Di, j means the Hasse differential operator of order i with respect to x and of order j with respect to y. It is clear that we have degŽ f .
f Ž x, y . s
Ý
f r , P X Ž x, y . .
rs0
In the following we will denote by P the point Ž a0 , a1 , a2 ., with a0 / 0, in K 3 and by P X the point Ž a, b . s Ž a1ra0 , a2ra0 . in K 2 . LEMMA 2.4. Let F Ž X 0 , X 1 , X 2 . be a homogeneous polynomial of degree d in K w X 0 , X 1 , X 2 x, and let P s Ž a0 , a1 , a2 . g K 3, with a0 / 0, and l an integer such that 1 F l F d. Ži.
If Q g K 3, then l
Ž DlQ F . Ž 1, x, y . s Ý
l a0iqjy l Ž Ddy P D 0, i , j F . Ž Q . Ž x y a . Ž y y b . . i
j
iqjs0
Žii.
If f Ž x, y . denotes the polynomial F Ž1, x, y ., then
Ž DlP F . Ž 1, x, y . s
l
Ý a0ry l rs0
s a0dy l
dyr yr
l
rs0
i
D 0, i , j F Ž P . Ž x y a . Ž y y b .
žl / Ý Ý žl / Ž iqjsr
dyr f X x, y . . y r r, P
j
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HEFEZ AND KAKUTA
Proof. Ži. If I s Ž i 0 , i1 , i 2 ., and < I < s i 0 q i1 q i 2 , we have
Ž DlQ F . Ž 1, x, y . s Ý
i
< I
s
DI F Ž Q . Ž x y a q a . 1 Ž y y b q b . l
Ý Ý
< I
= DI F Ž Q . s
l
Ý Ý
< I
i1 i
i2 j
ž /ž / ž / ž / a1
i 1 yi
a0
a2 a0
i 2 yj i
Ž x y a. Ž y y b .
i
l
Ý
j
Ž 2.
a0iqjy l Ž Di 0 , i1yi , i 2yj ( D 0, i , j F . Ž Q .
=a0i 0 a1i1yia2i 2yj Ž x y a . Ž y y b . s
i2
j
Ž 3.
a0iqjy l Ž DlQyŽ iqj. D 0, i , j F . Ž P . Ž x y a . Ž y y b . i
j
iqjs0
s
l
Ý
l a0iqjy l Ž Ddy P D 0, i , j F . Ž Q . Ž x y a . Ž y y b . , Ž 4 . i
j
iqjs0
where equality Ž2. is obtained from the binomial expansion, equality Ž3. is a consequence of the composition law for the Hasse differential operators, and equality Ž4. follows from Lemma 2.1Ži.. Žii. This formula follows from Ži. and from Lemma 2.1Žii.. It follows from part Žii. of the above lemma that I Ž P , Z.DlP Z . G 2, DEFINITION. define
;P g Z.
Let d and l be two integers such that 1 F l - d. We
½ž
h Ž d, l . s min s
dy1 dys k mod p . ly1 lys
/ ž
/
5
It is clear that 2 F h Ž d, l . F l q 1 F d. In particular, for any d ) 1 we have h Ž d, 1. s 2. We also have that if p s 0, then h Ž d, l. is always equal to 2.
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DEFINITION. If d s d 0 q d1 p q ??? qd n p n is the p-adic expansion of d, then l s d 0 q ??? qd m p m , for m - n, is called the truncation of d at order m, or simply a truncation of d. 1. Remark 1. When p ) 0 and l / 0 is a truncation of d, then Ž ld y ' y1 1 mod p, and
h Ž d, l . s l q 1. The above equality follows from the following well known congruence, which we will refer to as the fundamental congruence,
ž
a0 q a1 p q a2 p 2 q ??? b 0 q b1 p q b 2 p q ??? 2
a0 b0
a1 b1
a2 ??? b2
/ ž /ž /ž / '
mod p.
The following result will illustrate the importance of this arthimetical function. THEOREM 2.5. Let Z : F s 0 be an irreducible projecti¨ e plane cur¨ e of degree d G 2, and let l be an integer such that 1 F l - d. Then we ha¨ e Ži. hZ, l G h Ž d, l., Žii. hZ, l ) h Ž d, l., if and only if, h Ž d , l.
Ý
žD
h Ž d , l .yr r F X i ( DX j
/ ŽyF
Xi
.
h Ž d , l .yr
Ž FX . j
r
'0
mod F ,
Ž 5.
rs0
for some i, j s 0, 1, 2, with i / j. Žiii. If Z is a general cur¨ e of degree d, then hZ, l s h Ž d, l.. Proof. Ži. Let P s Ž1; a; b . be a general point of Z, and let P X s Ž a, b .. If f Ž x, y . s F Ž 1, x, y . s Ý r G 0 f r , P X Ž x, y . , and Ž D lP X f .Ž x, y . s r. Ý r G 0 Ž ld y f r, P X Ž x, y ., then we have from Lemma 2.4Žii. that yr I Ž P , F.DlP F . s I Ž P X , f.DlP X f . . Now, since I Ž P X , f.DlP X f . s I P X , f. DlP X f y
ž ž
dy1 f y2
žl / //
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HEFEZ AND KAKUTA
and DlP X f y
dy1 fs y1
žl /
Ý rG h Ž d , l .
dyr dy1 y yr y1
žl / žl /
f r , P X Ž x, y . ,
Ž 6.
we have that I Ž P , F.DlP F . G h Ž d, l . . Žii. Remark that Ž5. is always true if i s j. We keep the notation of Ži. and assume that f y Ž x, y . k 0 Žbecause if f y Ž x, y . ' 0, then we have f x Ž x, y . k 0, and the argument is similar.. We may then choose a parametrization of Z at P as follows: P Ž t . s Ž 1; a q t ; b q b1 t q b 2 t 2 q ??? . .
Ž 7.
This implies, for h s h Ž d, l., that DlP X f y s
dy1 f y1
žl /
ž
Ž PŽ t..
dyh dy1 y lyh ly1
/ ž /
h
Ý bh1 yr Ž Dr , hyr f . Ž PX . t h rs0
q higher powers of t.
Ž 8.
dy h 1. Since Ž l y h . k Ž ld y mod p, we have that I Ž P, F.DlP F . ) h if and only y1
if h
Ý bh1 yr Ž Dr , hyr f . Ž PX . s 0 rs0
Now, since b1 s yf x Ž a, b .rf y Ž a, b . s yFX 1Ž1, a, b .rFX 2Ž1, a, b ., it follows that I Ž P, F.DlP F . ) h if and only if h
r
Ý Ž D 0, r , hyr F . Ž P . Ž FX Ž P . . Ž yFX Ž P . . 2
1
h yr
s 0,
Ž 9.
rs0
and since P is general, this is equivalent to Ž5. for i s 1 and j s 2. The other cases are proved taking P s Ž a; 1; b . or P s Ž a; b; 1.. Žiii. Part Žii. shows that hZ, l s h Ž d, l. is an open condition in the open subset of irreducible curves in the space of all curves of degree d, so the only thing we have to show is that this condition is not empty, which amounts to produce any polynomial F such that Ž5. is not satisfied.
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POLAR CURVES
Given d and l as above, let F Ž X 0 , X 1 , X 2 . s X 0h X 2dy h q X 1 X 2dy 1 , where h s h Ž d, l.. In this case the left-hand side of the congruence in Ž5., for i s 1, j s 2, is X 2dq h Ž dy2., which is clearly not divisible by F. DEFINITION. An irreducible curve Z of degree d will be called Ž d, l.general if hZ, l s h Ž d, l.. In particular, a Ž d, 1.-general curve is a curve with hZ, 1 s h Ž d, 1. s 2. Thus if char K / 2, then Ž d, 1.-generality is equivalent to reflexivity. Remark 2. The proof of Theorem 2.5Žii. shows that if Ž5. is true for some i, j s 0, 1, 2, with i / j, then it is true for all i, j s 0, 1, 2. COROLLARY 2.6. Let h s h Ž d, l.. If for some i and j, with i / j, we ha¨ e s 0, for all r s 0, . . . , h , then the cur¨ e F s 0 is not Ž d, l.general.
D Xr i ( DhXyr F j
EXAMPLE 1. We give here an example of a family of non-general curves. Let d and l be such that d ' h Ž d, l. y 1 mod p s with h Ž d, l. - p s, for some positive integer s. If F is of the form s
Fs
Ý i 0qi 1qi 2s h Ž d , l .y1
s
s
Pi 0 , i1 , i 2 Ž X 0p , X 1p , X 2p . X 0i 0 X 1i1 X 2i 2 ,
for some homogeneous polynomials Pi 0 , i1 , i 2 , then the curve Z : F s 0 is not Ž d, l.-general. Indeed, for all non-negative integers j0 , j1 , and j2 such that j0 q j1 q j2 s h Ž d, l., we have that Dj 0 , j1 , j 2 F s 0, and the result follows from the above corollary. COROLLARY 2.7. Let Z be a projecti¨ e irreducible cur¨ e of degree d G 2 defined o¨ er a field K of characteristic p / 2. Let l be an integer such that 1 F l - d. Then hZ, l ) 2 if and only if either Z is non-reflexi¨ e or dy1 dy2 ' y1 y2
žl / žl /
mod p.
Proof. Since hZ, l G h Ž d, l. G 2, it follows from Theorem 2.5 that hZ, l s 2 if and only if h Ž d, l. s 2 and for some i, j s 0, 1, 2, i / j, 2
Ý ž DXr ( DX2yr F / Ž yFX . i
j
i
2yr
Ž FX . j
r
k0
mod F ,
rs0
which, according to w3, Proposition 4.12x, is equivalent to reflexivity.
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HEFEZ AND KAKUTA
Remark 3. The above collary tells us that in characteristic zero we always have hZ, l s 2, and since h Ž d, l. s 2, we always have Ž d, l.-generality in this case. Remark 4. In general, the polar l-ic curve does not coincide with the osculating curve of degree l, since for a general curve Z these curves have different intersection multiplicities with Z at a general point.
3. NON-GENERAL CURVES In order to characterize the non-general curves in terms of their defining equations we have to make some restrictions on the singularities of the curve. DEFINITION. We will say that a curve Z of degree d has mild singularities with respect to the pair Ž d, l. if the following inequality holds;
Ý
e P - d Ž h Ž d, l . y 1 . ,
PgZ
where e P is the multiplicity of the Jacobian ideal of Z at P. Note that all non-singular curves satisfy the above condition. PROPOSITION 3.1. Let Z : F s 0 be an irreducible cur¨ e of degree d with mild singularities with respect to Ž d, l.. Suppose that for some r G h Ž d, l. and for all i, j s 0, 1, 2, we ha¨ e r
Ý Ž DXr ( DXryr F . Ž yFX . i
j
i
r yr
Ž FX . j
r
'0
mod F ,
Ž 10 .
rs0
then for all i, j s 0, 1, 2, and all r s 0, 1, . . . , r , we ha¨ e D Xr i ( D Xryr F s 0. j Proof. We may choose coordinates in PK2 in such a way that for every singular point Q of Z we have for all i s 0, 1, 2 that eQ s I Ž Q, F.FX i . , and such that no partial derivative of F of the first order vanishes at the smooth points of Z located on any of the coordinates axes. In particular, FX i / 0, for all i s 0, 1, 2.
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Now, from Ž10. we get, for all i, j s 0, 1, 2, that
Ž DXr F . Ž FX . j
r
i
' FX j .G
mod F
Ž 11 .
for some polynomial G. Suppose by reductio ad absurdum that D Xrj F / 0 for some j. From Ž11. it follows, for all i s 0, 1, 2 and for all P g Z, that I Ž P , F. D Xrj F . q r I Ž P , F.FX i . G I Ž P , F.FX j . .
Ž 12 .
If P is a regular point of Z, then for some k s 0, 1, 2 we have I Ž P , F.FX k . s e P s 0, therefore from Ž12. taking i s k, we get that I Ž P , F. D Xrj F . G I Ž P , F.FX j . , hence I Ž P , F. D Xrj F . q e P G I Ž P , F.FX j . .
Ž 13 .
If P is any singular point of Z, we have clearly that I Ž P , F. D Xrj F . q e P G e P s I Ž P , F.FX j . .
Ž 14 .
Summing Ž13. over all regular points of Z, with Ž14. over all singular points of Z, we have
Ý I Ž P , F. DXr F . q Ý
eP G
j
PgZ
PgZ
Ý I Ž P , F.FX . , j
PgZ
which together with Bezout’s Theorem yields Žrecall that D Xrj F, FX j / 0. ´ dŽ d y r . q
Ý
eP G d Ž d y 1. .
PgZ
It then follows that
Ý
eP G d Ž r y 1. G d Ž h y 1. ,
PgZ
contradicting the assumption that Z has mild singularities with respect to the pair Ž d, l., hence D Xrj F s 0,
; j s 0, 1, 2.
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HEFEZ AND KAKUTA
F. If i s j, we have We will analyze now the vanishing of D 1X i ( D Xry1 j D 1X i ( D Xry1 F s Ž r y 1 . D Xri F s 0. j So we may assume i / j and, again by reductio ad absurdum, assume that D 1X i ( D Xry1 F / 0. Since D Xr j F s 0, from Ž10. we have that there is a j polynomial GX such that
žD
1 r y1 X i ( DX j F
/ ŽyF
Xi
.
ry1
' FX j .GX
mod F ,
and consequently we have for any P g Z that I P , F. D 1X i ( D Xry1 F q Ž r y 1 . I Ž P , F.FX i . G I Ž P , F.FX j . . j
ž
/
Ž 15 .
Let now P be a regular point of Z. If FX iŽ P . / 0, then we have from Ž15. that F G I Ž P , F.FX j . . I P , F. D 1X i ( D Xry1 j
ž
/
Ž 16 .
If FX iŽ P . s 0, then from Euler’s relation we get, for k / i and k / j, that Pj FX jŽ P . q Pk FX kŽ P . s 0, and since our coordinates have been chosen in order that the point P is not on any of the coordinate axes, we must have FX jŽ P . / 0. This implies that Ž16. is also true in this case. Therefore we have for all regular points P of Z that I P , F. D 1X i ( D Xry1 F q e P G I Ž P , F.FX j . . j
ž
/
If P is any singular point, then we have an equation analogous to Ž14. with D 1X i ( D Xry1 F in place of D Xrj F, and the proof proceeds exactly as j above to show that D 1X i ( D Xry1 F s 0. j The same argument can now be used to show that D X2 i ( D Xry2 F s 0, j and so on. Let d and l be positive integers with l - d, and p a prime number. Consider the following open statements on integers n : RŽ n . :
ž l nn / ž l / dy y
k
dy1 y1
and S Ž n . : ' i , j s 0, 1, 2, ' r s 0, . . . , n ; D Xr i ( D Xnyr F / 0. j
POLAR CURVES
461
THEOREM 3.2. Let Z : F s 0 be an irreducible projecti¨ e plane cur¨ e of degree d with mild singularities with respect to a pair Ž d, l.. We ha¨ e that hZ, l is the smallest integer in the set
h Ž d, l . F n F l < R Ž n . and S Ž n . 4 j n ) l < S Ž n . 4 . Proof. Let r be the smallest integer in the above set. Let P s Ž1; a; b . be a general point of Z, then we have a formula like Ž6. with r instead of h Ž d, l.. This expression evaluated at a parametrization of Z at P like Ž7. yields a formula like Ž8. with r instead of h , hence hZ, l s I Ž P, F.DlP F . G r . Also, hZ, l ) r implies a formula like Ž9. with r instead of h. The same argument will show that if hZ, l ) r , then Ž10. holds for all i, j s 0, 1, 2 with i / j, therefore from Proposition 3.1 it follows that for all i, j s 0, 1, 2 and all r s 0, . . . , r , D Xr i ( D Xryr F s 0, j which contradicts the definition of r . So hZ, l s r . Remark 5. If l / 0 is a truncation of d and Z has mild singularities, then we have that
hZ , l s min n ) l < S Ž n . 4 The following result is a generalization for all l of w7, Theorem 2.1; 3, Theorem 5.1; and 1, Theorem 3x, all established for l s 1. COROLLARY 3.3. Let Z : F s 0 be an irreducible cur¨ e of degree d with mild singularities with respect to Ž d, l.. The cur¨ e Z is not Ž d, l.-general if and only if for all i, j s 0, 1, 2, and all integers m and n such that m q n s h Ž d, l., we ha¨ e D Xmi ( D Xn j F s 0. EXAMPLE 2. Let p s 5 and d s 38. If F s X 017 X 121 q X 19 X 229 q X 038 q X 238 , then the curve Z : F s 0 is smooth of degree 38 over the algebraic closure of F5 . So we may apply to it Theorem 3.2 and the above corollary. For l s 3, the curve Z is Ž d, l.-general because, as it is easy to verify, we have h Ž d, l. s 4, and D X4 1 F s X 15 X 229 / 0. For l s 13, the curve Z is not Ž d, l.-general, because we have in this case h Ž d, l. s 14, and for all i, j s 0, 1, 2, and all m, n with n q m s 14, that D Xn i D Xmj F s 0. 2 21 Now since we have D 15 X 0 F s 2 X 0 X 1 / 0, it follows that hZ, l s 15.
462
HEFEZ AND KAKUTA
PROPOSITION 3.4. Let Z : F s 0 be an irreducible projecti¨ e plane cur¨ e of degree d with mild singularities with respect to a pair Ž d, l.. Ži. We ha¨ e hZ, l - d. If moreo¨ er l is a truncation of d, then Žii. ;lX , h Ž d, lX . G h Ž d, l. « hZ, lX G hZ, l . l Žiii. If P, Q g PK2 and P g Z l Ddy Q Z, then l I Ž P , Z l Ddy Q Z . G hZ , l y l .
Proof. Ži. and Žii. follow immediately from Theorem 3.2, while Žiii. is proved also using Lemma 2.4 Ži., replacing l by d y l. COROLLARY 3.5. With the same hypotheses as in Proposition 3.4 Žiii., and l assuming that Ddy Q F / 0, we ha¨ e l a Ž Z l Ddy Q Z. F
d Ž d y l.
hZ , l y l
.
4. THE EQUATION OF A NON-GENERAL CURVE In order to give explicitly the shape of the equation of a curve which is not Ž d, l.-general, we will need the following two lemmas: LEMMA 4.1. Let non-negati¨ e integers n 0 , n1 , n 2 , h , and p be gi¨ en such that p is prime, 2 F h - p, and h y 1 ' n 0 q n1 q n 2 mod p. If for all non-negati¨ e integers a and b with a q b s h , and for all i, j s 0, 1, 2, we ha¨ e ni a
ž /ž
nj
b
/
'0
mod p,
then for all non-negati¨ e integers i 0 , i1 , and i 2 with i 0 q i1 q i 2 s h , we ha¨ e n0 i0
n1 i1
n2 '0 i2
ž /ž /ž /
mod p.
Proof. Write the p-adic expansion of n i as n i s n i , 0 q n i , 1 p q ??? . Since a q b s h - p, we have from the hypotheses and from the fundamental congruence that 0'
ni a
ž /ž
nj
b
ni , 0 a
n j, 0
/ ž /ž / '
b
.
463
POLAR CURVES
It then follows that n i , 0 q n j, 0 F h y 1,
; i , j s 0, 1, 2,
hence 2 Ž n 0, 0 q n1, 0 q n 2, 0 . F 3 Ž h y 1 . .
Ž 17 .
Since n 0 q n1 q n 2 ' h y 1 mod p, we have, for some r g 0, 1, 24 , that n 0, 0 q n1, 0 q n 2, 0 s h y 1 q rp.
Ž 18 .
From Ž17. and Ž18. it follows that 2 rp F h y 1 - p. So r s 0 and n 0, 0 q n1, 0 q n 2, 0 s h y 1. So for all Ž i 0 , i1 , i 2 . with i 0 q i1 q i 2 s h , we have n0 i0
n1 i1
n2 n 0, 0 ' i2 i0
n1, 0 i1
n 2, 0 '0 i2
ž /ž /ž / ž /ž /ž /
mod p.
PROPOSITION 4.2. Let d, h , and p be integers such that p is prime, 2 F h - p, and d ' h y 1 mod p. If F g K w X 0 , X 1 , X 2 x is homogeneous of degree d, where char K s p, and D Xa i D Xbj F s 0
; i , j s 0, 1, 2, ;a , b ; a q b s h ,
then ; i 0 , i1 , i 2 ; i 0 q i1 q i 2 s h .
Di 0 , i 1 , i 2 F s 0
Proof. It is sufficient to prove the assertion for monomials X 0n 0 X 1n1 X 2n 2 ;
n 0 q n1 q n 2 s d,
and in this case the result follows from Lemma 4.1. The assertion in Proposition 4.2 may fail if the hypotheses are not satisfied, as one can see in the following example. EXAMPLE 3. Put p s 5 and let F s X 02 X 12 X 23 . It is easy to verify for all i, j s 0, 1, 2 that D Xa i D Xbj F s 0
;a , b ; a q b s 6,
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HEFEZ AND KAKUTA
but D X2 0 D X2 1 D X2 2 F / 0. This does not contradict Proposition 4.2, since h s 6 ) p s 5. Given a multi-index I s Ž i 0 , i1 , . . . , i n ., we will use the notation < I < s i 0 q i 2 q ??? qi n . We will also use the notation a Fp b to mean that a is less than or equal to b with respect to the p-adic ordering, that is, a i F bi , for all i, where a i and bi are respectively the coefficients of the p-adic expansions of a and b. It follows immediately from the fundamental congruence that a k 0 mod p m b Fp a. b
ž/
LEMMA 4.3. Gi¨ en non-negati¨ e integers m and r such that r Fp m and gi¨ en J s Ž j0 , . . . , jn . such that < J < s m , then there exists I s Ž i 0 , i1 , . . . , i n . with < I < s r such that i 0 Fp j0 , . . . , i n Fp jn . Proof. Suppose by reductio ad absurdum that for all I such that < I < s r , there exists an integer sŽ I . such that 0 F sŽ I . F n and i sŽ I . gp j sŽ I . . It then follows that for all such I there exists sŽ I . such that j sŽ I .
ž / i sŽ I .
'0
mod p.
From Euler’s relation we get that
m r s
ž /
Ý
< I
j sŽ I . j0 jn ??? ??? '0 i0 i sŽ I . in
ž/ ž / ž/
mod p,
whence r gp m , a contradiction. PROPOSITION 4.4. Suppose that Di 0 , . . . , i n are Hasse differential operators acting on a function space. Assume that for some function f all deri¨ ati¨ es of some order r are zero, that is, Di 0 , . . . , i n f s 0,
;I s Ž i 0 , . . . , i n . ; < I < s r ,
then all deri¨ ati¨ es of order m of f are zero if m Gp r .
POLAR CURVES
465
Proof. Suppose that J s Ž j0 , . . . , jn . is such that < J < s m. From Lemma 4.3 there exists I s Ž i 0 , . . . , i n . with < I < s r such that i 0 Fp j0 , . . . , i n Fp jn . Since 0 s Dj 0yi 0 , . . . , j nyi n( Di 0 , . . . , i n f s
j0 jn ??? Dj 0 , . . . , j n f , i0 in
ž/ ž/
j j and Ž i 0 . ??? Ž i n . k 0, it follows that Dj 0 , . . . , j n f s 0. 0
n
THEOREM 4.5. Let Z : F s 0 be a projecti¨ e plane cur¨ e of degree d with mild singularities with respect to the pair Ž d, l.. If 1 F l - p y 1 and d ' l mod p, then the following assertions are equi¨ alent: Ži. Žii. Žiii. such that
Z is Ž d, l.-non-general hZ, l s p a for some integer a G 1. There exist homogeneous polynomials Q i 0 , i1 , i 2 g K w X 0 , X 1 , X 2 x F is of the form Fs
Ý
i 0qi 1qi 2s l
X 0i 0 X 1i1 X 2i 2 Q ip0 , i1 , i 2 .
Proof. Since 1 F l - p y 1 and d ' l mod p, it follows that l is a truncation of d, hence from Remark 1 we have that h Ž d, l. s l q 1 - p. Ži. « Žii.: Since Z is Ž d, l.-non-general with mild singularities then from Corollary 3.3 we have, for all i, j s 0, 1, 2 and all integers m and n such that m q n s l q 1, that D Xmi ( D Xn j F s 0. It follows from Proposition 4.4 that for all i, j s 0, 1, 2, D Xmi ( D Xn j F s 0,
; m, n; m q n Gp l q 1.
Now, since l q 1 - p, we have that the least possible value for m q n with m q n ) l q 1 such that D Xmi ( D Xn j F / 0 is p a for some a G 1. Hence from Theorem 3.2 we have hZ, l s p a. Žii. « Žiii.: If hZ, l s p a for some a G 1, then from Theorem 3.2 we have for all i, j s 0, 1, 2, D Xmi ( D Xn j F s 0,
; m, n; m q n s l q 1.
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HEFEZ AND KAKUTA
So from Proposition 4.2 we have Di 0 , i1 , i 2 F s 0,
; i 0 , i1 , i 2 ; i 0 q i1 q i 2 s l q 1.
Hence for all i 0 , i1 , i 2 such that i 0 q i1 q i 2 s l, and for all i s 0, 1, 2, we have D 1X iŽ Di 0 , i1 , i 2 F . s 0. So Di 0 , i1 , i 2 F, with i 0 q i1 q i 2 s l, is of the form Q ip0 , i1 , i 2 . Now from the generalized Euler’s relation we get that Fs
Ý
i 0qi 1qi 2s l
Žiii. « Ži.: p. Then
Ž Di , i , i F . X 0i 0
1
2
0
X 1i1 X 2i 2 s
Ý
i 0qi 1qi 2s l
Q ip0 , i1 , i 2 X 0i 0 X 1i1 X 2i 2 .
From Theorem 3.2 we have that Žiii. implies that hZ, l G
hZ , l G p ) l q 1 s h Ž d, l . , so Z is Ž d, l.-non-general. Remark 6. Example 2 shows us that Ži. may not imply Žii. if h Ž d, l. G p. 5. THE FUNCTION h Ž d, l. DEFINITION.
We define the integral function m Ž d, l. as
½ ž
m Ž d, l . s max m
dy1 dyr ' mod p, ; r , 1 F r F m . ly1 lyr
/ ž
5
/
It is clear that we have
h Ž d, l . s m Ž d, l . q 1. In this section we will study the function m Ž d, l., and consequently the function h Ž d, l.. We introduce the notation aŽ l . to denote the truncation of the p-adic expansion of a s a0 q a1 p q ??? up to order l, that is, aŽ l . s a0 q a1 p q ??? qal p l . LEMMA 5.1. Ži.
If a F p g and b F p g , then
ž Žii.
Let a, b, a , b , and g be positi¨ e integers.
a pg y a by a a y 1 ' Ž y1 . by1 b pg y b
/
ž
m Ž a p g , b p g . s p gm Ž a , b ..
/ž
by1 ay1
/
mod p
467
POLAR CURVES
Proof. Ži. First note that it is easy to prove by induction on a, observing that the case a s 1 is Wilson’s Theorem, that
Ž p y 1 . ! Ž a y 1 . !' Ž y1.
a
mod p.
Using this and proceeding by induction on g , the case a s b s 1 is proved. The general statement follows from observing that from the fundamental congruence we have
ž
a pg y a Ž a y 1. pg q pg y a ay1 ' ' by1 b pg y b Ž b y 1. pg q pg y b
/ ž
/ ž /ž
pg y a . pg y b
/
Žii. This follows from Ži. and from the fundamental congruence. From now on, d and l will be integers such that 1 F l - d, and p s Žrespectively p t . will be the highest power of p that divides l Žrespectively d .. Remark 7. It is easy to verify that the condition dy1 k0 y1
žl /
mod p
Ž 19 .
is equivalent to Ža. Žb.
s - t, l t - d t , and l i F d i , ; i G t q 1, or s s t and l i F d i , ; i G t.
Note that when d 0 / 0, then t s 0 and necessarily we are in case Žb., hence in such case we have that condition Ž19. is equivalent to l Fp d. PROPOSITION 5.2. Let d and l be integers such that 1 F l - d, and 1. suppose that Ž ld y k 0 mod p. Then we ha¨ e y1 Ži. If m Ž d, l. ) 1 then d ' l mod p. Žii. Suppose that d ' l mod p. If l s max i < d Ž i. s lŽ i.4 , then we ha¨ e
m Ž d, l . s
½
dŽl.
if d Ž l . / 0
p lq1
if d Ž l . s 0.
1. Proof. Ži. Remark that m Ž d, l. ) 1 is equivalent to Ž ld y ' y1 2. dy 2. Ž ld y Ž mod p, which in turn is equi ¨ alent to ' 0 mod p. y2 ly1
468
HEFEZ AND KAKUTA
Case d 0 s 0. If l0 G 1, then we have
ž
py2 dy2 ' l0 y 1 ly1
/
ž
py1 py1 ??? l1 l ty1
/ž / ž /ž
dt y 1 lt
d tq1 ??? l tq1
/ž /
which is not congruent to zero mod p in view of Remark 7. So l0 s d 0 . Case d 0 / 0. From Remark 7 we have that l0 / 0 and l Fp d, hence 0 - l0 F d 0 . So if d 0 s 1, we have that l 0 s 1. If d 0 G 2, then d 0 y 2 l0 y 1, hence d 0 - l0 q 1, and therefore d 0 s l0 . Žii. Suppose that d ' l mod p, and write d s d Ž l . q a p lq1 and l s l q b p lq1. Žl.
Case d Ž l . s lŽ l . / 0. From Lemma 5.1Ži. and from the fundamental congruence we have, for all r with 1 F r F d Ž l ., that
ž
a dyr ' b lyr
/ ž /ž
a lŽ l . y r s . Žl. b d yr
/ ž/
Now, since from Remark 7 we have that l i F d i , for all i G l q 1, then from the definition of l it follows that l lq1 - d lq1 and hence
ž
Ž d lq1 y 1 . q pd lq2 q ??? ay1 s b l lq1 q pl lq2 q ???
/ ž
/
k0
mod p.
It then follows from Stifel’s relation and from Lemma 5.1Ži. that
ž
ay1 a d y dŽ l. y 1 d y dŽ l. ' k ' Žl. b y 1 b lyd y1 l y dŽ l.
/ ž
/ ž / ž
/
mod p,
hence m Ž d, l . s d Ž l .. Case d Ž l . s lŽ l . s 0. From Lemma 5.1Žii. we have that
m Ž d, l . s m Ž a p lq1 , b p lq1 . s p lq1m Ž a , b . . 1 1. . k 0, and from Since Ž ld y k 0, it follows from Lemma 5.1Ži. that Ž ab y y1 y1 the definition of l we have that a k b mod p, so from Ži. we have that m Ž a , b . s 1, from which the result follows.
PROPOSITION 5.3. Let d and l be integers such that 1 F l - d, and Ž . ' 0 mod p. Then we ha¨ e dy 1 ly 1
m Ž d, l . s
½
if l gp d
d Ž m. , d
Ž sy1.
,
if l Fp d
with m s max i < d Ž i. - lŽ i.4 , and p s the highest power of p that di¨ ides l.
469
POLAR CURVES
Proof. Case l gp d. Writing d s d Ž m. q a p mq 1 and l s lŽ m. q b p mq 1, it follows from the definition of m that d Ž m. - lŽ m., d mq 1 ) l mq1 , and a Gp b . First of all let us observe that d Ž m. / 0. This is so because otherwise we would have as a consequence of d mq 1 ) l mq1 that dy1 ' ly1
ž
/
ž
p mq 1 y 1
l
Ž m.
y1
ay1 / 0. b
/ž /
Now, for all r such that 1 F r F d Ž m., since d Ž m. - lŽ m., it follows that
ž
d Ž m. y r dyr ' Ž m. lyr l yr
/ ž
a ' 0. b
/ž /
On the other hand,
ž
d y Ž d Ž m. q 1 .
ly Žd
Ž m.
q 1.
/ ž '
p mq 1 y 1
l
Ž m.
yd
Ž m.
y1
ay1 / 0, b
/ž /
hence in this case we have m Ž d, l. s d Ž m.. Case l Fp d. In this case the hypotheses imply that s ) t G 0, hence d Ž sy1. / 0. Write d s d Ž sy1. q a p s and l s b p s. Now, since d Ž dy1. - p s, we have for all r, such that 1 F r F d Ž sy1., that
ž
d Ž sy1. y r dyr ' ps y r lyr
/ ž
/ž
a ' 0. by1
/
On the other hand,
ž
d y Ž d Ž sy1. q 1 .
ly Žd
Ž sy1.
q 1.
/ ž '
ps y 1 p yd s
Ž sy1.
y1
/ž
ay1 k 0, by1
/
hence in this case we have m Ž d, l. s d Ž sy1.. COROLLARY 5.4. we ha¨ e that
For any pair Ž d, l. of positi¨ e integers with 1 F l - d,
m Ž d, m Ž d, l . . s m Ž d, l . . Proof. This follows from Propositions 5.2 and 5.3 by a direct verification. PROPOSITION 5.5. we ha¨ e that
For any pair Ž d, l. of positi¨ e integers with 1 F l - d,
m Ž d, d y l . s m Ž d, l . .
470
HEFEZ AND KAKUTA
Proof. The proof of this result is elementary and may be done using the definition of m Ž d, l. and twice Stifel’s relation. Alternatively, this can be proved using Propositions 5.2 and 5.3. COROLLARY 5.6. Let Z be a plane projecti¨ e cur¨ e of degree d and let l be an integer such that 1 F l - d. Then Ži. Žii.
Z is Ž d, l.-general if and only if Z is Ž d, d y l.-general. If Z has mild singularities, and l is a truncation of d, then
hZ , dy l G hZ , l . Proof. Ži. is a consequence of Theorem 2.5 and Proposition 5.5, while Žii. is a consequence of Corollary 3.4Žii. and of Proposition 5.5.
6. STATIONARY POINTS Let Z : F s 0 be a projective irreducible plane curve of degree d. Let l be an integer such that 1 F l - d and some mixed derivative of F of order l is nonzero. DEFINITION.
A point P g Z will be called l-stationary if I Ž P , Z.DlP Z . ) hZ , l .
By the very definition of hZ, l we have that there are finitely many such points, and the aim of this section is to determine their number. We will compute this number when either Z is Ž d, l.-general, or when l is a truncation of d, and Z has mild singularities with respect to the pair Ž d, l.. We will make this assumption from now on in this section. For any integer r and distinct integers, i, j, k s 0, 1, 2, we define for i - j, SkŽ r . s
r
r
Ý Ž DXr ( DXryr F . Ž FX . Ž yFX . i
j
j
i
r yr
.
rs0
Note that from the arguments we used in the proof of Theorem 2.5, and from Remark 5 and Proposition 3.1, we have that P g Z is a l-stationary point if and only if SkŽh Z, l. Ž P . s 0, for all k s 0, 1, 2. Remark that from the above definition we have that all singular points of Z are stationary points. To establish the main result of this section, namely Theorem 6.7, we will need several auxiliary results.
471
POLAR CURVES
LEMMA 6.1. Let a be an integer. If b is minimal among the integers greater than a but not p-adically greater, then there exists some r G 1 such that b s a y aŽ ry1. q p r . Proof. Let a s a s p s q a sq1 p sq1 q ??? qa n p n, with a s / 0, be the padic expansion of a. It is clear that the first integer greater than a which is not p-adically greater than a is b 0 s Ž a sq1 q 1 . p sq 1 q ??? qan p n s a y aŽ s. q p sq1 . Now we can restrict our search to the integers which are greater than b 0 but not p-adically greater, and the result follows clearly. PROPOSITION 6.2. Let Z be a projecti¨ e irreducible plane cur¨ e of degree d, and let l s d Ž l . for some l. Suppose that Z is not Ž d, l.-general and with mild singularities, then we ha¨ e for some r,
hZ , l s d Ž l . y d Ž ry1. q p r .
Ž 20 .
Proof. Since l is a truncation of d, we have from Remark 5 that
hZ , l s min n G l q 1 < S Ž n . 4 . Now, in view of Proposition 4.4, hZ, l must be an integer greater than h Ž d, l. s l q 1, but not p-adically greater than l q 1, and minimal with respect to the p-adic ordering. So from Lemma 6.1 we have that
hZ , l s Ž l q 1 . y Ž l q 1 .
Ž ry1 .
q pr
for some r, and the proposition follows. COROLLARY 6.3. Let Z: F s 0 be a plane projecti¨ e irreducible cur¨ e of degree d and l s d Ž l . for some l. If Z is not Ž d, l.-general, with mild singularities, then hZ, l ' 0 mod p. LEMMA 6.4. ha¨ e
ž
If Ž d, l. is a pair of integers such that 1 F l - d, then we
dys ' 0 mod p h Ž d, l . y s
/
;s; s s 2, . . . , h Ž d, l . y 1.
Proof. From Stifel’s relation we have that
ž
d y Ž s y 1. dys dys q s . h Ž d, l . y s m Ž d, l . y s m Ž d, l . y Ž s y 1 .
/ ž
/ ž
/
Ž 21 .
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HEFEZ AND KAKUTA
Now, from Corollary 5.4 it follows that
ž
d y m Ž d, l . dy1 dy2 ' ' ??? ' . m Ž d, l . y 1 m Ž d, l . y 2 m Ž d, l . y m Ž d, l .
/ ž
ž
/
/
From this and from Ž21., we get that
ž
dys ' 0 mod p, h Ž d, l . y s
s s 2, . . . , h Ž d, l . y 1.
/
LEMMA 6.5. Let d and l be integers such that 1 F l - d. If l is a truncation of d, and hZ, l is as in Ž20., then
ž
dys hZ , l y s ' 0 mod p,
s s 2, . . . , l .
/
Proof. Suppose that l s d Ž l . for some l G 0, and let hZ, l s d Ž l . y d q p r , for some r with 1 F r F l q 1. We will initially suppose that r - l q 1. From the fundamental congruence we get, for all s s 0, . . . , d Ž l ., that Ž ry1.
ž
d Ž l . q a p lq1 y s dŽ l. y s dys s ' ' 0. Ž l . hZ , l y s d y d Ž ry1. q p r y s d Ž l . y d Ž ry1. q p r y s
/
ž
/ ž
/
Suppose now that r s l q 1, then hZ, l s p r , and from the fundamental congruence we have for s G 1 that
ž
d Ž ry1. q b p r y s dys d Ž ry1. y s s ' s 0. r r p ys pr y s p ys
/ ž
/ ž
/
PROPOSITION 6.6. Let Z : F s 0 be a projecti¨ e irreducible plane cur¨ e of degree d and let l be an integer such that 1 F l - d. Suppose that either Z is Ž d, l.-general, or Z has mild singularities and l is a truncation of d, then we ha¨ e for h s hZ, l , and for all integers i, j s 0, 1, 2, that X jh Si Žh . ' X ih Sj Žh .
mod F.
Proof. If i s j, there is nothing to prove. We will only prove the case j s 0 and i s 2, since the other cases are similar.
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POLAR CURVES
Let P s Ž a0 , a1 , a2 . g K 3. From Lemma 2.4Žii. we have X 0h Ž DhP F . 1,
ž
h s ay 0
X1 X0
,
X2 X0
/
h
Ý X 0hyr rs0
dyr yr
žh /
a Ý Ž D 0, a , b F . Ž P . Ž a0 X1 y a1 X 0 . Ž a0 X 2 y a2 X 0 . b ,
=
aq b sr
and X 2h Ž DhP F . s
ž
h ay 2
X0 X2
,
X1 X2
,1
h
Ý X 2hys ss0
=
/ dys ys
žh /
g Ý Ž Dg , d , 0 F . Ž P . Ž a2 X 0 y a0 X 2 . Ž a2 X1 y a1 X 2 . d .
gq d ss
Now, since obviously X 0h Ž DhP F . 1,
ž
X1 X0
X2
,
X0
/
s X 2h Ž DhP F .
ž
X0 X2
,
X1 X2
/
,1 ,
we get by comparing the coefficients of X 1t X 2hyt of both expressions that h D 0, t , hyt F Ž P . s ay 2
h
Ý ss0
=
dys ys
žh /
Ý Ž Dg , d , 0 F . Ž P . Ž ya0 .
g
gq d ss
žd/ t
a2t Ž ya1 .
d yt
.
Now, from the above expression, Lemmas 6.4 and 6.5, and Theorem 3.2, we have that d X 2h D 0, t , hyt F s h
dy1 yt 0 X 2t Ž yX 0 . F q hy1 t
ž /ž / ž /Ž =
0 t
q
Ý
ž
yX 0 . Ž yX1 .
gq d s h
žd/Ž t
yt
X 2t FX 0 q g
/ 1 t
ž /Ž
yX1 .
Dg , d , 0 F . Ž yX 0 . X 2t Ž yX1 .
1yt
d yt
.
X 2t FX 1
474
HEFEZ AND KAKUTA
Using Euler’s identity and the above one, we get the following equalities and congruences mod F, X 0h S 2Žh . s '
( D Xd 1 F . X 0g Ž yX 0 FX 0 .
Ý Ž DXg
0
( D Xd 1 F . X 0g Ž X 1 FX 1 q X 2 FX 2 .
gq d s h
h
Ý Ý
gq d s h ts0
žd/Ž t
= X 2t Ž yFX 1 .
hyt
Ž FX . 1
g
Dg , d , 0 F . Ž yX 0 . Ž yX1 .
Ž FX .
d
hyt
1
Ž FX .
t
2
y
ts0
d yt
ž
dy1 hy1
h
= Ž yX 0 . FX 0Ž yFX 1 . q Ž yX1 . FX 1Ž yFX 1 . qX 2 FX 1Ž yFX 1 . h
Ý Ž DXt ( DhXyt F . Ž yFX . 1
g
1
t
Ý Ž X 2h D 0, t , hyt F . Ž yFX .
' X 2h
Ž FX .
2
h
'
g
0
gq d s h
s
d
Ý Ž DXg
2
hyt
1
hy1
Ž FX . 2
t
/
h
FX 2
s X 2h S 0Žh . .
ts0
Remark 8. The above proposition tells us that the data
½ž
Ua ,
SaŽh . Xah
/5
, a s0, 1, 2
where Ua s Ž X 0 ; X 1; X 2 .< Xa / 04 , define a section of the fiber bundle OZ Ž dŽh q 1. y 3h ., whose zeros are the stationary points of Z with respect to Ž d, l.. DEFINITION. We define the l-weight of P as being the order of vanishing w Ž P . at P of the above section. So P g Z is a l-stationary point if and only if w Ž P . ) 0. THEOREM 6.7. Let Z : F s 0 be a projecti¨ e irreducible plane cur¨ e of degree d and let l be an integer such that 1 F l - d. Suppose that either Z is Ž d, l.-general, or Z has mild singularities and l is a truncation of d; then we ha¨ e, for h s hZ, l , that
Ý w Ž P . s d d Ž h q 1. y 3h PgZ
.
POLAR CURVES
475
Proof. This follows immediately from the above remark, and from the definition of w Ž P .. Remark 9. The method of proof of the above theorem is inspired from w4x, where the result is proved for l s 1. Observe that if a curve Z of degree d is reflexive, then it is Ž d, 1.-general. And if Z is smooth and Ž d, 1.-non-general, then d ' 1 mod p Žsee for example w3x or w7x., then 1 is a truncation of d, so that Theorem 6.7 gives us the classical formula for the weighted number of flexes of a non-singular plane curve Z of degree d. Namely, f s d d Ž « 2 q 1. y 3« 2 , where « 2 is the intersection multiplicity of the curve with its tangent line at a general point, or alternatively, the inseparable degree of the dual map of the curve. Žsee for example w10x and w6x or w9x.. Remark also that in characteristic zero, a point is l-stationary if and only if it is a flex, so in this case we are just counting the flexes.
7. POLAR MORPHISMS Let Z : F s 0 be an irreducible smooth plane curve of degree d, and let l be an integer such that 1 F l - d, and suppose that some mixed derivative of F of order l is nonzero. Let Nl s lŽ l q 3.r2. DEFINITION.
The l th polar morphism is the morphism defined by pl : Z ª PKNl P ¬ Ž Di 0 , i 1 , i 2 F Ž P . ; i 0 q i 1 q i 2 s l .
We will denote the image of pl by Zl . So for l s 1, we have that pl is the Gauss map and Zl is the dual curve of Z. We will also denote by ¨l the Veronese l-uple embedding, U
¨l : PK2 ª Ž PKNl .
Q ¬ Ž Q0i 0 Q1i1 Q2i 2 ; i 0 q i1 q i 2 s l . . THEOREM 7.1. Let Z be a smooth cur¨ e of degree d, and let l be an integer such that 1 F l - d. Then we ha¨ e deg Ž pl . deg Ž Zl . s d Ž d y l . .
476
HEFEZ AND KAKUTA
Proof. We will denote by deg s Ž pl . Žrespectively, deg i Ž pl .. the separable Žrespectively, inseparable . degree of pl. Let P1 , . . . , Pr g Z be the points such that U s Zl R pl Ž P1 . , . . . , pl Ž Pr . 4 is the open set of points R for which aply1 Ž R . s deg s Ž pl . . Let Q be a point of PK2 , and consider the hyperplane HQ in PKNl whose coefficients are the coordinates of the point ¨lŽ Q . g ŽPKNl .U . So HQ :
Ý
i 0qi 1qi 2s l
Q0i 0 Q1i1 Q2i 2 Zi 0 , i1 , i 2 s 0,
where Zi 0 , i1 , i 2 are the coordinate functions of PKNl. For Q general in PK2 we have that HQ l Zl ; U, because the above condition means that the general point Q of PK2 is not a zero of any of the nonzero polynomials DlP i F, for i s 1, . . . , r. Now, from standard ramification theory and from Lemma 2.1Ži. we have that l deg i Ž pl . I Ž pl Ž P . , HQ l Zl . s I Ž P , Z l Dly F Ž Q . . s I Ž P , Z l Ddy Q Z.,
where Dly F Ž Q . . s
Ý
i 0qi 1qi 2s l
Q0i 0 Q1i1 Q2i 2 Di 0 , i1 , i 2 F Ž X 0 , X 1 , X 2 . .
Now, from Bezout’s Theorem we have ´ d Ž d y l. s
l Ý I Ž P , Z l Ddy Q Z . s deg i Ž pl . Ý I Ž pl Ž P . , HQ l Zl .
PgZ
PgZ
s deg i Ž pl . deg s Ž pl .
Ý I Ž R , HQ l Zl . s degŽ pl . degŽ Zl . . RgZl
The above theorem is a generalization of Plucker’s formula, which was ¨ known for l s 1, relating the degree of a curve with that of the Gauss map and of the dual curve.
REFERENCES 1. E. Ballico and A. Hefez, Non-reflexive projective curves of low degree, Manuscripta Math. 70 Ž1991., 385]396.
POLAR CURVES
477
2. J. L. Coolidge, ‘‘A Treatise on Algebraic Plane Curves,’’ 1931; Dover, New York, 1959. 3. A. Hefez, Non-reflexive curves, Compositio Math. 69 Ž1989., 3]35. 4. A. Hefez, ‘‘Introduc¸˜ ao ` a geometria projetiva,’’ Monografias de Matematica, Vol. 46, ´ IMPA, 1991. 5. A. Hefez and N. Kakuta, New bounds for Fermat curves over finite fields, in ‘‘Proc. Zeuthen Symp.,’’ Contemporary Math., Vol. 123, pp. 89]97, Amer. Math. Soc., Providence, RI, 1991. 6. D. Laksov, Wronskians and Plucker formulas for linear systems on curves, Ann. Sci. ¨ ´ Ecol. Norm. Sup. 17 Ž1984., 45]66. 7. R. Pardini, Some remarks on plane curves over fields of finite characteristic, Compositio Math. 60 Ž1986., 3]17. 8. G. Salmon, ‘‘A Treatise on the Higher Plane Curves,’’ 3rd ed., Chelsea, New York. 9. K. O. Stohr, ¨ and J. F. Voloch, Weierstrass points and curves over finite fields, Proc. London Math. Soc. Ž 3 . 52 Ž1986., 1]19. 10. R. Walker, ‘‘Algebraic Curves,’’ Princeton Univ. Press, Princeton, NJ, 1950.
181, 449]477 Ž1996.
0129
Polar Curves Abramo HefezU Departamento de Matematica ´ Aplicada, Uni¨ ersidade Federal Fluminense, R. Sao ˜ Paulo srn-Campus do Valonguinho, 24020-005 Niteroi, RJ-Brazil
and Neuza Kakuta Departamento de Matematica-IBILCE, UNESP, R. Cristo¨ ao ´ ˜ Colombo 2265 J. Nazareth, 15054-000 S. Jose do Rio Preto, SP-Brazil Communicated by Walter Feit Received January 31, 1995
1. INTRODUCTION Polar curves of plane algebraic curves in characteristic zero have been studied extensively by several people and appear in the classical literature related to Plucker’s formulas Žsee for example w2, 8, or 10x.. ¨ This study however has not been carried out in depth for curves defined over fields of positive characteristic, and the aim of this work is to contribute to filling this gap. It turns out, as we will see, that the theory in positive characteristic is quite different from the characteristic zero case and establishes interesting interplays between geometry and number theory. Our motivation comes from w5x where, making a new use of polar curves in positive characteristic, we improved on the known upper bounds for the number of points of Fermat curves over finite fields. After the study we make in the present paper, it will be possible to apply the methods of w5x to other families of curves as well. The paper is organized as follows. Let Z be a projective plane curve of degree d and let l be an integer between 1 and d. We denote by DlP Z the * Partially supported by CNPq-Brazil. 449 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
450
HEFEZ AND KAKUTA
polar l-ic of Z at P. In Section 2 we show that the intersection multiplicity of Z and DlP Z at P is an upper semi-continuous function in P on Z. The minimum value of this function, achieved on an open dense Zariski subset of Z, will be denoted by hZ, l . It is shown in Theorem 2.5 that there exists an arithmetical function h Ž d, l. such that for any curve Z of degree d we have hZ, l G h Ž d, l., with equality holding for the general curve of degree d. In Section 3 we characterize the non-general curves Z, that is, the curves for which hZ, l ) h Ž d, l., in terms of the vanishing of certain derivatives of the defining polynomial of Z, assuming some mild behaviour on the singularities of the curve. In Section 4 we show how to get in some cases the shape of the equation of a non-general curve. In Section 5 we totally describe the arithmetical function h Ž d, l.. In Section 6 we give an enumerative formula for the stationary points on a suitably general curve with respect to its associated family of polar l-ics. More precisely, we count the number of points P on Z at which Z and DlP Z have intersection multiplicity greater than hZ, l . This result contains in particular the usual Plucker’s formula for the number of flexes of a non-singular plane ¨ projective curve. In Section 7 we introduce the polar morphisms of plane curves, which generalize the Gauss map, and prove a formula that generalizes Plucker’s formula relating the degree of a curve, the degree of its ¨ Gauss map, and the degree of its dual curve.
2. GENERAL THEORY Let K be an algebraically closed field, of characteristic p G 0, fixed once and for all. Let F g K w X 0 , X 1 , X 2 x be a homogeneous polynomial of degree d and let Q be an element of K 3. We consider, for l s 1, . . . , d, the homogeneous polynomials
Ž DlQ F . Ž X0 , X1 , X2 . s
Ý
i 0qi 1qi 2s l
Ž Di , i , i F . Ž Q . X 0i 0
1
2
0
X 1i1 X 2i 2 ,
where Di 0 , i1 , i 2 denotes the mixed partial Hasse differential operator, of order in with respect to the indeterminate Xn , for n s 0, 1, 2. We will also denote Di 0 , 0, 0 , D 0, i1 , 0 , and D 0, 0, i 2 respectively by D Xi 00 , D Xi11, and D Xi 22 . For the properties of these operators needed here we refer the reader to w3x. From the definition of the Hasse operators, we have, for P, Q g K 3 and indeterminates t 0 and t 1 , the identity F Ž t 0 P q t 1 Q . s t 0d F Ž P . q t 0dy 1 t 1 Ž D1P F . Ž Q . q ??? 1 d q t 0 t 1dy 1 Ž Ddy P F . Ž Q . q t1 F Ž Q .
Ž 1.
451
POLAR CURVES
The above identity gives many relations involving the polynomials DlQ F, of which we use only a few. LEMMA 2.1. We ha¨ e the relations l .Ž . Ži. Ž DlP F .Ž Q . s Ž Ddy Q F P .
Žii. Ž DlP F .Ž P . s Ž ld . F Ž P .. Žiii. Let T be a linear automorphism of K 3. For any polynomial G in K w X 0 , X 1 , X 2 x and any Q in K 3, we denote GŽT Ž Q .. by G T Ž Q .. Then we ha¨ e
Ž DlP F .
T
s Ž DlT y1 Ž P . F T . .
Proof. Ži. is obtained by comparing the corresponding identity for Ž F t 1 Q q t 0 P . with Ž1.. Žii. follows easily from identity Ž1. when we put P s Q. Žiii. follows replacing Q by T Ž Q . in both members of Ž1. and then developing the first member using again formula Ž1.. Note that Lemma 2.1 Žii. is a generalization of Euler’s relation, which applied to the monomial X 0n 0 X 1n1 X 2n 2 , with n 0 q n1 q n 2 s d, gives in particular the well known relation, still called Euler’s relation,
Ý
i 0qi 1qi 2s l
n0 i0
n1 i1
n2 d ' . i2 l
ž /ž /ž /
ž /
Note also that what we established for polynomials in three indeterminates is valid in any number of indeterminates; in particular, also in the above Euler’s relation. DEFINITION. If Z is a projective plane curve defined by F s 0, and if some of the derivatives of F of order l evaluated at P is nonzero, then the curve DlP Z defined by DlP F s 0 is called the polar l-ic cur¨ e of Z at P. In particular, we have for l s 1 the polar line at P Žwhich is, for P g Z, the tangent line at P ., for l s 2 we have the polar conic at P, and for l s 3 the polar cubic at P. It is clear from Lemma 2.1 Ži. that our indices for polars are reversed with respect to the classical use, that is, our definition of DlP is the classical Ddy l Žas defined for example in w2x or in w8x.. We also have from this l formula that P g DlQ Z if and only if Q g Ddy P Z. It follows from Lemma l 2.1Žii. that if P g Z, then P g D P Z for all l. Lemma 2.1Žiii. says that the polar l-ic is a well defined projective object.
452
HEFEZ AND KAKUTA
In order to study the behavior of the intersection multiplicity function of Z with its polar curves, we establish the following result. PROPOSITION 2.2. Let Z : F s 0 be an integral projecti¨ e plane cur¨ e and p let Z˜ ª Z be the normalization map of Z. Let l be an integer such that 1 F l - d, and some mixed deri¨ ati¨ e of F of order l is not identically zero. Then the integer-¨ alued function on Z˜ defined by the formula P˜ ¬ ord P˜ Ž Dlp Ž P˜. F . is upper semi-continuous. Proof. Let Q g Z˜ and suppose without loss of generality that p Ž Q . s Ž1; a; b .. Choose a rational function s g K Ž Z˜. such that its image in the local ring OZ,˜ Q is a uniformizing parameter. If U˜ is the open set of Z˜ of all points P˜ for which t s s y sŽ P˜. is an uniformizing parameter of OZ,˜ P˜ and p Ž P˜. f X 0 s 04 , then we have that
Ž Dlp Ž P˜. F . Ž 1, x, y . s
Ý
i 0qi 1qi 2s l
Ž Di , i , i F . Ž p Ž P˜. . x i 0
1
2
1
y i2 .
Consider the powers series Ý iG 0 Dsi Ž x (p . t i and Ý iG 0 Dsi Ž y (p . t i in K wU˜ xww t xx. These powers series with their coefficients evaluated at P˜ g U˜ represent the expansions of the functions x (p and y (p in the completion of the local ring OZ,˜ P˜. Consider now the expansion
Ž Dlp Ž P˜. F .
ž
1, Ý Dsi Ž x (p . Ž P˜. t i , Ý Dsi Ž y (p . Ž P˜. t i s iG0
iG0
/
Ý h i Ž P˜. t i , iG0
˜ Let h be the first index i for where each h i is a regular function on U. Ž which h i k 0 such an index exists because some derivative of F of order l is nonzero., then ord P˜Ž Dlp Ž P˜. F . is equal to h for P˜ in the open dense set ˜ so the P˜ g U˜ < h i Ž P˜. / 04 and is greater than h at the other points of U, ˜ hence on Z. ˜ function under consideration is upper semi-continuous on U, We will denote by I Ž P, F.G . the intersection multiplicity of the curves F s 0 and G s 0 at the point P. COROLLARY 2.3. If F is an irreducible polynomial of degree d in K w X 0 , X 1 , X 2 x, then for e¨ ery l s 1, . . . , d y 1 such that some deri¨ ati¨ e of order l of F is nonzero, the function P ¬ I Ž P, F.DlP F . is upper semi-continuous on Z : F s 0.
453
POLAR CURVES
The upper semi-continuity of the above intersection multiplicity implies that its minimum value is achieved in a non-empty Zariski open set of Z. DEFINITION. We define hZ, l as being the minimum value of the above intersection multiplicity function. The number hZ, 1 is well known and it is the last order « 2 of the order sequence of the curve in the projective plane, that is, it is the intersection multiplicity at a general point P of Z with its tangent line at P Žsee for example w6x or w9x.. In order to compute the value of hZ, l , we will give the expansion of the polynomial DlQ F at a point P. Let f Ž x, y . be a polynomial in K w x, y x. If P X s Ž a, b . is a point in the affine plane, and r is an integer such that 0 F r F degŽ f ., we define f r , P X Ž x, y . s
Di , j f Ž P X . Ž x y a . Ž y y b . , i
Ý
j
iqjsr
where Di, j means the Hasse differential operator of order i with respect to x and of order j with respect to y. It is clear that we have degŽ f .
f Ž x, y . s
Ý
f r , P X Ž x, y . .
rs0
In the following we will denote by P the point Ž a0 , a1 , a2 ., with a0 / 0, in K 3 and by P X the point Ž a, b . s Ž a1ra0 , a2ra0 . in K 2 . LEMMA 2.4. Let F Ž X 0 , X 1 , X 2 . be a homogeneous polynomial of degree d in K w X 0 , X 1 , X 2 x, and let P s Ž a0 , a1 , a2 . g K 3, with a0 / 0, and l an integer such that 1 F l F d. Ži.
If Q g K 3, then l
Ž DlQ F . Ž 1, x, y . s Ý
l a0iqjy l Ž Ddy P D 0, i , j F . Ž Q . Ž x y a . Ž y y b . . i
j
iqjs0
Žii.
If f Ž x, y . denotes the polynomial F Ž1, x, y ., then
Ž DlP F . Ž 1, x, y . s
l
Ý a0ry l rs0
s a0dy l
dyr yr
l
rs0
i
D 0, i , j F Ž P . Ž x y a . Ž y y b .
žl / Ý Ý žl / Ž iqjsr
dyr f X x, y . . y r r, P
j
454
HEFEZ AND KAKUTA
Proof. Ži. If I s Ž i 0 , i1 , i 2 ., and < I < s i 0 q i1 q i 2 , we have
Ž DlQ F . Ž 1, x, y . s Ý
i
< I
s
DI F Ž Q . Ž x y a q a . 1 Ž y y b q b . l
Ý Ý
< I
= DI F Ž Q . s
l
Ý Ý
< I
i1 i
i2 j
ž /ž / ž / ž / a1
i 1 yi
a0
a2 a0
i 2 yj i
Ž x y a. Ž y y b .
i
l
Ý
j
Ž 2.
a0iqjy l Ž Di 0 , i1yi , i 2yj ( D 0, i , j F . Ž Q .
=a0i 0 a1i1yia2i 2yj Ž x y a . Ž y y b . s
i2
j
Ž 3.
a0iqjy l Ž DlQyŽ iqj. D 0, i , j F . Ž P . Ž x y a . Ž y y b . i
j
iqjs0
s
l
Ý
l a0iqjy l Ž Ddy P D 0, i , j F . Ž Q . Ž x y a . Ž y y b . , Ž 4 . i
j
iqjs0
where equality Ž2. is obtained from the binomial expansion, equality Ž3. is a consequence of the composition law for the Hasse differential operators, and equality Ž4. follows from Lemma 2.1Ži.. Žii. This formula follows from Ži. and from Lemma 2.1Žii.. It follows from part Žii. of the above lemma that I Ž P , Z.DlP Z . G 2, DEFINITION. define
;P g Z.
Let d and l be two integers such that 1 F l - d. We
½ž
h Ž d, l . s min s
dy1 dys k mod p . ly1 lys
/ ž
/
5
It is clear that 2 F h Ž d, l . F l q 1 F d. In particular, for any d ) 1 we have h Ž d, 1. s 2. We also have that if p s 0, then h Ž d, l. is always equal to 2.
455
POLAR CURVES
DEFINITION. If d s d 0 q d1 p q ??? qd n p n is the p-adic expansion of d, then l s d 0 q ??? qd m p m , for m - n, is called the truncation of d at order m, or simply a truncation of d. 1. Remark 1. When p ) 0 and l / 0 is a truncation of d, then Ž ld y ' y1 1 mod p, and
h Ž d, l . s l q 1. The above equality follows from the following well known congruence, which we will refer to as the fundamental congruence,
ž
a0 q a1 p q a2 p 2 q ??? b 0 q b1 p q b 2 p q ??? 2
a0 b0
a1 b1
a2 ??? b2
/ ž /ž /ž / '
mod p.
The following result will illustrate the importance of this arthimetical function. THEOREM 2.5. Let Z : F s 0 be an irreducible projecti¨ e plane cur¨ e of degree d G 2, and let l be an integer such that 1 F l - d. Then we ha¨ e Ži. hZ, l G h Ž d, l., Žii. hZ, l ) h Ž d, l., if and only if, h Ž d , l.
Ý
žD
h Ž d , l .yr r F X i ( DX j
/ ŽyF
Xi
.
h Ž d , l .yr
Ž FX . j
r
'0
mod F ,
Ž 5.
rs0
for some i, j s 0, 1, 2, with i / j. Žiii. If Z is a general cur¨ e of degree d, then hZ, l s h Ž d, l.. Proof. Ži. Let P s Ž1; a; b . be a general point of Z, and let P X s Ž a, b .. If f Ž x, y . s F Ž 1, x, y . s Ý r G 0 f r , P X Ž x, y . , and Ž D lP X f .Ž x, y . s r. Ý r G 0 Ž ld y f r, P X Ž x, y ., then we have from Lemma 2.4Žii. that yr I Ž P , F.DlP F . s I Ž P X , f.DlP X f . . Now, since I Ž P X , f.DlP X f . s I P X , f. DlP X f y
ž ž
dy1 f y2
žl / //
456
HEFEZ AND KAKUTA
and DlP X f y
dy1 fs y1
žl /
Ý rG h Ž d , l .
dyr dy1 y yr y1
žl / žl /
f r , P X Ž x, y . ,
Ž 6.
we have that I Ž P , F.DlP F . G h Ž d, l . . Žii. Remark that Ž5. is always true if i s j. We keep the notation of Ži. and assume that f y Ž x, y . k 0 Žbecause if f y Ž x, y . ' 0, then we have f x Ž x, y . k 0, and the argument is similar.. We may then choose a parametrization of Z at P as follows: P Ž t . s Ž 1; a q t ; b q b1 t q b 2 t 2 q ??? . .
Ž 7.
This implies, for h s h Ž d, l., that DlP X f y s
dy1 f y1
žl /
ž
Ž PŽ t..
dyh dy1 y lyh ly1
/ ž /
h
Ý bh1 yr Ž Dr , hyr f . Ž PX . t h rs0
q higher powers of t.
Ž 8.
dy h 1. Since Ž l y h . k Ž ld y mod p, we have that I Ž P, F.DlP F . ) h if and only y1
if h
Ý bh1 yr Ž Dr , hyr f . Ž PX . s 0 rs0
Now, since b1 s yf x Ž a, b .rf y Ž a, b . s yFX 1Ž1, a, b .rFX 2Ž1, a, b ., it follows that I Ž P, F.DlP F . ) h if and only if h
r
Ý Ž D 0, r , hyr F . Ž P . Ž FX Ž P . . Ž yFX Ž P . . 2
1
h yr
s 0,
Ž 9.
rs0
and since P is general, this is equivalent to Ž5. for i s 1 and j s 2. The other cases are proved taking P s Ž a; 1; b . or P s Ž a; b; 1.. Žiii. Part Žii. shows that hZ, l s h Ž d, l. is an open condition in the open subset of irreducible curves in the space of all curves of degree d, so the only thing we have to show is that this condition is not empty, which amounts to produce any polynomial F such that Ž5. is not satisfied.
457
POLAR CURVES
Given d and l as above, let F Ž X 0 , X 1 , X 2 . s X 0h X 2dy h q X 1 X 2dy 1 , where h s h Ž d, l.. In this case the left-hand side of the congruence in Ž5., for i s 1, j s 2, is X 2dq h Ž dy2., which is clearly not divisible by F. DEFINITION. An irreducible curve Z of degree d will be called Ž d, l.general if hZ, l s h Ž d, l.. In particular, a Ž d, 1.-general curve is a curve with hZ, 1 s h Ž d, 1. s 2. Thus if char K / 2, then Ž d, 1.-generality is equivalent to reflexivity. Remark 2. The proof of Theorem 2.5Žii. shows that if Ž5. is true for some i, j s 0, 1, 2, with i / j, then it is true for all i, j s 0, 1, 2. COROLLARY 2.6. Let h s h Ž d, l.. If for some i and j, with i / j, we ha¨ e s 0, for all r s 0, . . . , h , then the cur¨ e F s 0 is not Ž d, l.general.
D Xr i ( DhXyr F j
EXAMPLE 1. We give here an example of a family of non-general curves. Let d and l be such that d ' h Ž d, l. y 1 mod p s with h Ž d, l. - p s, for some positive integer s. If F is of the form s
Fs
Ý i 0qi 1qi 2s h Ž d , l .y1
s
s
Pi 0 , i1 , i 2 Ž X 0p , X 1p , X 2p . X 0i 0 X 1i1 X 2i 2 ,
for some homogeneous polynomials Pi 0 , i1 , i 2 , then the curve Z : F s 0 is not Ž d, l.-general. Indeed, for all non-negative integers j0 , j1 , and j2 such that j0 q j1 q j2 s h Ž d, l., we have that Dj 0 , j1 , j 2 F s 0, and the result follows from the above corollary. COROLLARY 2.7. Let Z be a projecti¨ e irreducible cur¨ e of degree d G 2 defined o¨ er a field K of characteristic p / 2. Let l be an integer such that 1 F l - d. Then hZ, l ) 2 if and only if either Z is non-reflexi¨ e or dy1 dy2 ' y1 y2
žl / žl /
mod p.
Proof. Since hZ, l G h Ž d, l. G 2, it follows from Theorem 2.5 that hZ, l s 2 if and only if h Ž d, l. s 2 and for some i, j s 0, 1, 2, i / j, 2
Ý ž DXr ( DX2yr F / Ž yFX . i
j
i
2yr
Ž FX . j
r
k0
mod F ,
rs0
which, according to w3, Proposition 4.12x, is equivalent to reflexivity.
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HEFEZ AND KAKUTA
Remark 3. The above collary tells us that in characteristic zero we always have hZ, l s 2, and since h Ž d, l. s 2, we always have Ž d, l.-generality in this case. Remark 4. In general, the polar l-ic curve does not coincide with the osculating curve of degree l, since for a general curve Z these curves have different intersection multiplicities with Z at a general point.
3. NON-GENERAL CURVES In order to characterize the non-general curves in terms of their defining equations we have to make some restrictions on the singularities of the curve. DEFINITION. We will say that a curve Z of degree d has mild singularities with respect to the pair Ž d, l. if the following inequality holds;
Ý
e P - d Ž h Ž d, l . y 1 . ,
PgZ
where e P is the multiplicity of the Jacobian ideal of Z at P. Note that all non-singular curves satisfy the above condition. PROPOSITION 3.1. Let Z : F s 0 be an irreducible cur¨ e of degree d with mild singularities with respect to Ž d, l.. Suppose that for some r G h Ž d, l. and for all i, j s 0, 1, 2, we ha¨ e r
Ý Ž DXr ( DXryr F . Ž yFX . i
j
i
r yr
Ž FX . j
r
'0
mod F ,
Ž 10 .
rs0
then for all i, j s 0, 1, 2, and all r s 0, 1, . . . , r , we ha¨ e D Xr i ( D Xryr F s 0. j Proof. We may choose coordinates in PK2 in such a way that for every singular point Q of Z we have for all i s 0, 1, 2 that eQ s I Ž Q, F.FX i . , and such that no partial derivative of F of the first order vanishes at the smooth points of Z located on any of the coordinates axes. In particular, FX i / 0, for all i s 0, 1, 2.
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POLAR CURVES
Now, from Ž10. we get, for all i, j s 0, 1, 2, that
Ž DXr F . Ž FX . j
r
i
' FX j .G
mod F
Ž 11 .
for some polynomial G. Suppose by reductio ad absurdum that D Xrj F / 0 for some j. From Ž11. it follows, for all i s 0, 1, 2 and for all P g Z, that I Ž P , F. D Xrj F . q r I Ž P , F.FX i . G I Ž P , F.FX j . .
Ž 12 .
If P is a regular point of Z, then for some k s 0, 1, 2 we have I Ž P , F.FX k . s e P s 0, therefore from Ž12. taking i s k, we get that I Ž P , F. D Xrj F . G I Ž P , F.FX j . , hence I Ž P , F. D Xrj F . q e P G I Ž P , F.FX j . .
Ž 13 .
If P is any singular point of Z, we have clearly that I Ž P , F. D Xrj F . q e P G e P s I Ž P , F.FX j . .
Ž 14 .
Summing Ž13. over all regular points of Z, with Ž14. over all singular points of Z, we have
Ý I Ž P , F. DXr F . q Ý
eP G
j
PgZ
PgZ
Ý I Ž P , F.FX . , j
PgZ
which together with Bezout’s Theorem yields Žrecall that D Xrj F, FX j / 0. ´ dŽ d y r . q
Ý
eP G d Ž d y 1. .
PgZ
It then follows that
Ý
eP G d Ž r y 1. G d Ž h y 1. ,
PgZ
contradicting the assumption that Z has mild singularities with respect to the pair Ž d, l., hence D Xrj F s 0,
; j s 0, 1, 2.
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HEFEZ AND KAKUTA
F. If i s j, we have We will analyze now the vanishing of D 1X i ( D Xry1 j D 1X i ( D Xry1 F s Ž r y 1 . D Xri F s 0. j So we may assume i / j and, again by reductio ad absurdum, assume that D 1X i ( D Xry1 F / 0. Since D Xr j F s 0, from Ž10. we have that there is a j polynomial GX such that
žD
1 r y1 X i ( DX j F
/ ŽyF
Xi
.
ry1
' FX j .GX
mod F ,
and consequently we have for any P g Z that I P , F. D 1X i ( D Xry1 F q Ž r y 1 . I Ž P , F.FX i . G I Ž P , F.FX j . . j
ž
/
Ž 15 .
Let now P be a regular point of Z. If FX iŽ P . / 0, then we have from Ž15. that F G I Ž P , F.FX j . . I P , F. D 1X i ( D Xry1 j
ž
/
Ž 16 .
If FX iŽ P . s 0, then from Euler’s relation we get, for k / i and k / j, that Pj FX jŽ P . q Pk FX kŽ P . s 0, and since our coordinates have been chosen in order that the point P is not on any of the coordinate axes, we must have FX jŽ P . / 0. This implies that Ž16. is also true in this case. Therefore we have for all regular points P of Z that I P , F. D 1X i ( D Xry1 F q e P G I Ž P , F.FX j . . j
ž
/
If P is any singular point, then we have an equation analogous to Ž14. with D 1X i ( D Xry1 F in place of D Xrj F, and the proof proceeds exactly as j above to show that D 1X i ( D Xry1 F s 0. j The same argument can now be used to show that D X2 i ( D Xry2 F s 0, j and so on. Let d and l be positive integers with l - d, and p a prime number. Consider the following open statements on integers n : RŽ n . :
ž l nn / ž l / dy y
k
dy1 y1
and S Ž n . : ' i , j s 0, 1, 2, ' r s 0, . . . , n ; D Xr i ( D Xnyr F / 0. j
POLAR CURVES
461
THEOREM 3.2. Let Z : F s 0 be an irreducible projecti¨ e plane cur¨ e of degree d with mild singularities with respect to a pair Ž d, l.. We ha¨ e that hZ, l is the smallest integer in the set
h Ž d, l . F n F l < R Ž n . and S Ž n . 4 j n ) l < S Ž n . 4 . Proof. Let r be the smallest integer in the above set. Let P s Ž1; a; b . be a general point of Z, then we have a formula like Ž6. with r instead of h Ž d, l.. This expression evaluated at a parametrization of Z at P like Ž7. yields a formula like Ž8. with r instead of h , hence hZ, l s I Ž P, F.DlP F . G r . Also, hZ, l ) r implies a formula like Ž9. with r instead of h. The same argument will show that if hZ, l ) r , then Ž10. holds for all i, j s 0, 1, 2 with i / j, therefore from Proposition 3.1 it follows that for all i, j s 0, 1, 2 and all r s 0, . . . , r , D Xr i ( D Xryr F s 0, j which contradicts the definition of r . So hZ, l s r . Remark 5. If l / 0 is a truncation of d and Z has mild singularities, then we have that
hZ , l s min n ) l < S Ž n . 4 The following result is a generalization for all l of w7, Theorem 2.1; 3, Theorem 5.1; and 1, Theorem 3x, all established for l s 1. COROLLARY 3.3. Let Z : F s 0 be an irreducible cur¨ e of degree d with mild singularities with respect to Ž d, l.. The cur¨ e Z is not Ž d, l.-general if and only if for all i, j s 0, 1, 2, and all integers m and n such that m q n s h Ž d, l., we ha¨ e D Xmi ( D Xn j F s 0. EXAMPLE 2. Let p s 5 and d s 38. If F s X 017 X 121 q X 19 X 229 q X 038 q X 238 , then the curve Z : F s 0 is smooth of degree 38 over the algebraic closure of F5 . So we may apply to it Theorem 3.2 and the above corollary. For l s 3, the curve Z is Ž d, l.-general because, as it is easy to verify, we have h Ž d, l. s 4, and D X4 1 F s X 15 X 229 / 0. For l s 13, the curve Z is not Ž d, l.-general, because we have in this case h Ž d, l. s 14, and for all i, j s 0, 1, 2, and all m, n with n q m s 14, that D Xn i D Xmj F s 0. 2 21 Now since we have D 15 X 0 F s 2 X 0 X 1 / 0, it follows that hZ, l s 15.
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HEFEZ AND KAKUTA
PROPOSITION 3.4. Let Z : F s 0 be an irreducible projecti¨ e plane cur¨ e of degree d with mild singularities with respect to a pair Ž d, l.. Ži. We ha¨ e hZ, l - d. If moreo¨ er l is a truncation of d, then Žii. ;lX , h Ž d, lX . G h Ž d, l. « hZ, lX G hZ, l . l Žiii. If P, Q g PK2 and P g Z l Ddy Q Z, then l I Ž P , Z l Ddy Q Z . G hZ , l y l .
Proof. Ži. and Žii. follow immediately from Theorem 3.2, while Žiii. is proved also using Lemma 2.4 Ži., replacing l by d y l. COROLLARY 3.5. With the same hypotheses as in Proposition 3.4 Žiii., and l assuming that Ddy Q F / 0, we ha¨ e l a Ž Z l Ddy Q Z. F
d Ž d y l.
hZ , l y l
.
4. THE EQUATION OF A NON-GENERAL CURVE In order to give explicitly the shape of the equation of a curve which is not Ž d, l.-general, we will need the following two lemmas: LEMMA 4.1. Let non-negati¨ e integers n 0 , n1 , n 2 , h , and p be gi¨ en such that p is prime, 2 F h - p, and h y 1 ' n 0 q n1 q n 2 mod p. If for all non-negati¨ e integers a and b with a q b s h , and for all i, j s 0, 1, 2, we ha¨ e ni a
ž /ž
nj
b
/
'0
mod p,
then for all non-negati¨ e integers i 0 , i1 , and i 2 with i 0 q i1 q i 2 s h , we ha¨ e n0 i0
n1 i1
n2 '0 i2
ž /ž /ž /
mod p.
Proof. Write the p-adic expansion of n i as n i s n i , 0 q n i , 1 p q ??? . Since a q b s h - p, we have from the hypotheses and from the fundamental congruence that 0'
ni a
ž /ž
nj
b
ni , 0 a
n j, 0
/ ž /ž / '
b
.
463
POLAR CURVES
It then follows that n i , 0 q n j, 0 F h y 1,
; i , j s 0, 1, 2,
hence 2 Ž n 0, 0 q n1, 0 q n 2, 0 . F 3 Ž h y 1 . .
Ž 17 .
Since n 0 q n1 q n 2 ' h y 1 mod p, we have, for some r g 0, 1, 24 , that n 0, 0 q n1, 0 q n 2, 0 s h y 1 q rp.
Ž 18 .
From Ž17. and Ž18. it follows that 2 rp F h y 1 - p. So r s 0 and n 0, 0 q n1, 0 q n 2, 0 s h y 1. So for all Ž i 0 , i1 , i 2 . with i 0 q i1 q i 2 s h , we have n0 i0
n1 i1
n2 n 0, 0 ' i2 i0
n1, 0 i1
n 2, 0 '0 i2
ž /ž /ž / ž /ž /ž /
mod p.
PROPOSITION 4.2. Let d, h , and p be integers such that p is prime, 2 F h - p, and d ' h y 1 mod p. If F g K w X 0 , X 1 , X 2 x is homogeneous of degree d, where char K s p, and D Xa i D Xbj F s 0
; i , j s 0, 1, 2, ;a , b ; a q b s h ,
then ; i 0 , i1 , i 2 ; i 0 q i1 q i 2 s h .
Di 0 , i 1 , i 2 F s 0
Proof. It is sufficient to prove the assertion for monomials X 0n 0 X 1n1 X 2n 2 ;
n 0 q n1 q n 2 s d,
and in this case the result follows from Lemma 4.1. The assertion in Proposition 4.2 may fail if the hypotheses are not satisfied, as one can see in the following example. EXAMPLE 3. Put p s 5 and let F s X 02 X 12 X 23 . It is easy to verify for all i, j s 0, 1, 2 that D Xa i D Xbj F s 0
;a , b ; a q b s 6,
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HEFEZ AND KAKUTA
but D X2 0 D X2 1 D X2 2 F / 0. This does not contradict Proposition 4.2, since h s 6 ) p s 5. Given a multi-index I s Ž i 0 , i1 , . . . , i n ., we will use the notation < I < s i 0 q i 2 q ??? qi n . We will also use the notation a Fp b to mean that a is less than or equal to b with respect to the p-adic ordering, that is, a i F bi , for all i, where a i and bi are respectively the coefficients of the p-adic expansions of a and b. It follows immediately from the fundamental congruence that a k 0 mod p m b Fp a. b
ž/
LEMMA 4.3. Gi¨ en non-negati¨ e integers m and r such that r Fp m and gi¨ en J s Ž j0 , . . . , jn . such that < J < s m , then there exists I s Ž i 0 , i1 , . . . , i n . with < I < s r such that i 0 Fp j0 , . . . , i n Fp jn . Proof. Suppose by reductio ad absurdum that for all I such that < I < s r , there exists an integer sŽ I . such that 0 F sŽ I . F n and i sŽ I . gp j sŽ I . . It then follows that for all such I there exists sŽ I . such that j sŽ I .
ž / i sŽ I .
'0
mod p.
From Euler’s relation we get that
m r s
ž /
Ý
< I
j sŽ I . j0 jn ??? ??? '0 i0 i sŽ I . in
ž/ ž / ž/
mod p,
whence r gp m , a contradiction. PROPOSITION 4.4. Suppose that Di 0 , . . . , i n are Hasse differential operators acting on a function space. Assume that for some function f all deri¨ ati¨ es of some order r are zero, that is, Di 0 , . . . , i n f s 0,
;I s Ž i 0 , . . . , i n . ; < I < s r ,
then all deri¨ ati¨ es of order m of f are zero if m Gp r .
POLAR CURVES
465
Proof. Suppose that J s Ž j0 , . . . , jn . is such that < J < s m. From Lemma 4.3 there exists I s Ž i 0 , . . . , i n . with < I < s r such that i 0 Fp j0 , . . . , i n Fp jn . Since 0 s Dj 0yi 0 , . . . , j nyi n( Di 0 , . . . , i n f s
j0 jn ??? Dj 0 , . . . , j n f , i0 in
ž/ ž/
j j and Ž i 0 . ??? Ž i n . k 0, it follows that Dj 0 , . . . , j n f s 0. 0
n
THEOREM 4.5. Let Z : F s 0 be a projecti¨ e plane cur¨ e of degree d with mild singularities with respect to the pair Ž d, l.. If 1 F l - p y 1 and d ' l mod p, then the following assertions are equi¨ alent: Ži. Žii. Žiii. such that
Z is Ž d, l.-non-general hZ, l s p a for some integer a G 1. There exist homogeneous polynomials Q i 0 , i1 , i 2 g K w X 0 , X 1 , X 2 x F is of the form Fs
Ý
i 0qi 1qi 2s l
X 0i 0 X 1i1 X 2i 2 Q ip0 , i1 , i 2 .
Proof. Since 1 F l - p y 1 and d ' l mod p, it follows that l is a truncation of d, hence from Remark 1 we have that h Ž d, l. s l q 1 - p. Ži. « Žii.: Since Z is Ž d, l.-non-general with mild singularities then from Corollary 3.3 we have, for all i, j s 0, 1, 2 and all integers m and n such that m q n s l q 1, that D Xmi ( D Xn j F s 0. It follows from Proposition 4.4 that for all i, j s 0, 1, 2, D Xmi ( D Xn j F s 0,
; m, n; m q n Gp l q 1.
Now, since l q 1 - p, we have that the least possible value for m q n with m q n ) l q 1 such that D Xmi ( D Xn j F / 0 is p a for some a G 1. Hence from Theorem 3.2 we have hZ, l s p a. Žii. « Žiii.: If hZ, l s p a for some a G 1, then from Theorem 3.2 we have for all i, j s 0, 1, 2, D Xmi ( D Xn j F s 0,
; m, n; m q n s l q 1.
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HEFEZ AND KAKUTA
So from Proposition 4.2 we have Di 0 , i1 , i 2 F s 0,
; i 0 , i1 , i 2 ; i 0 q i1 q i 2 s l q 1.
Hence for all i 0 , i1 , i 2 such that i 0 q i1 q i 2 s l, and for all i s 0, 1, 2, we have D 1X iŽ Di 0 , i1 , i 2 F . s 0. So Di 0 , i1 , i 2 F, with i 0 q i1 q i 2 s l, is of the form Q ip0 , i1 , i 2 . Now from the generalized Euler’s relation we get that Fs
Ý
i 0qi 1qi 2s l
Žiii. « Ži.: p. Then
Ž Di , i , i F . X 0i 0
1
2
0
X 1i1 X 2i 2 s
Ý
i 0qi 1qi 2s l
Q ip0 , i1 , i 2 X 0i 0 X 1i1 X 2i 2 .
From Theorem 3.2 we have that Žiii. implies that hZ, l G
hZ , l G p ) l q 1 s h Ž d, l . , so Z is Ž d, l.-non-general. Remark 6. Example 2 shows us that Ži. may not imply Žii. if h Ž d, l. G p. 5. THE FUNCTION h Ž d, l. DEFINITION.
We define the integral function m Ž d, l. as
½ ž
m Ž d, l . s max m
dy1 dyr ' mod p, ; r , 1 F r F m . ly1 lyr
/ ž
5
/
It is clear that we have
h Ž d, l . s m Ž d, l . q 1. In this section we will study the function m Ž d, l., and consequently the function h Ž d, l.. We introduce the notation aŽ l . to denote the truncation of the p-adic expansion of a s a0 q a1 p q ??? up to order l, that is, aŽ l . s a0 q a1 p q ??? qal p l . LEMMA 5.1. Ži.
If a F p g and b F p g , then
ž Žii.
Let a, b, a , b , and g be positi¨ e integers.
a pg y a by a a y 1 ' Ž y1 . by1 b pg y b
/
ž
m Ž a p g , b p g . s p gm Ž a , b ..
/ž
by1 ay1
/
mod p
467
POLAR CURVES
Proof. Ži. First note that it is easy to prove by induction on a, observing that the case a s 1 is Wilson’s Theorem, that
Ž p y 1 . ! Ž a y 1 . !' Ž y1.
a
mod p.
Using this and proceeding by induction on g , the case a s b s 1 is proved. The general statement follows from observing that from the fundamental congruence we have
ž
a pg y a Ž a y 1. pg q pg y a ay1 ' ' by1 b pg y b Ž b y 1. pg q pg y b
/ ž
/ ž /ž
pg y a . pg y b
/
Žii. This follows from Ži. and from the fundamental congruence. From now on, d and l will be integers such that 1 F l - d, and p s Žrespectively p t . will be the highest power of p that divides l Žrespectively d .. Remark 7. It is easy to verify that the condition dy1 k0 y1
žl /
mod p
Ž 19 .
is equivalent to Ža. Žb.
s - t, l t - d t , and l i F d i , ; i G t q 1, or s s t and l i F d i , ; i G t.
Note that when d 0 / 0, then t s 0 and necessarily we are in case Žb., hence in such case we have that condition Ž19. is equivalent to l Fp d. PROPOSITION 5.2. Let d and l be integers such that 1 F l - d, and 1. suppose that Ž ld y k 0 mod p. Then we ha¨ e y1 Ži. If m Ž d, l. ) 1 then d ' l mod p. Žii. Suppose that d ' l mod p. If l s max i < d Ž i. s lŽ i.4 , then we ha¨ e
m Ž d, l . s
½
dŽl.
if d Ž l . / 0
p lq1
if d Ž l . s 0.
1. Proof. Ži. Remark that m Ž d, l. ) 1 is equivalent to Ž ld y ' y1 2. dy 2. Ž ld y Ž mod p, which in turn is equi ¨ alent to ' 0 mod p. y2 ly1
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HEFEZ AND KAKUTA
Case d 0 s 0. If l0 G 1, then we have
ž
py2 dy2 ' l0 y 1 ly1
/
ž
py1 py1 ??? l1 l ty1
/ž / ž /ž
dt y 1 lt
d tq1 ??? l tq1
/ž /
which is not congruent to zero mod p in view of Remark 7. So l0 s d 0 . Case d 0 / 0. From Remark 7 we have that l0 / 0 and l Fp d, hence 0 - l0 F d 0 . So if d 0 s 1, we have that l 0 s 1. If d 0 G 2, then d 0 y 2 l0 y 1, hence d 0 - l0 q 1, and therefore d 0 s l0 . Žii. Suppose that d ' l mod p, and write d s d Ž l . q a p lq1 and l s l q b p lq1. Žl.
Case d Ž l . s lŽ l . / 0. From Lemma 5.1Ži. and from the fundamental congruence we have, for all r with 1 F r F d Ž l ., that
ž
a dyr ' b lyr
/ ž /ž
a lŽ l . y r s . Žl. b d yr
/ ž/
Now, since from Remark 7 we have that l i F d i , for all i G l q 1, then from the definition of l it follows that l lq1 - d lq1 and hence
ž
Ž d lq1 y 1 . q pd lq2 q ??? ay1 s b l lq1 q pl lq2 q ???
/ ž
/
k0
mod p.
It then follows from Stifel’s relation and from Lemma 5.1Ži. that
ž
ay1 a d y dŽ l. y 1 d y dŽ l. ' k ' Žl. b y 1 b lyd y1 l y dŽ l.
/ ž
/ ž / ž
/
mod p,
hence m Ž d, l . s d Ž l .. Case d Ž l . s lŽ l . s 0. From Lemma 5.1Žii. we have that
m Ž d, l . s m Ž a p lq1 , b p lq1 . s p lq1m Ž a , b . . 1 1. . k 0, and from Since Ž ld y k 0, it follows from Lemma 5.1Ži. that Ž ab y y1 y1 the definition of l we have that a k b mod p, so from Ži. we have that m Ž a , b . s 1, from which the result follows.
PROPOSITION 5.3. Let d and l be integers such that 1 F l - d, and Ž . ' 0 mod p. Then we ha¨ e dy 1 ly 1
m Ž d, l . s
½
if l gp d
d Ž m. , d
Ž sy1.
,
if l Fp d
with m s max i < d Ž i. - lŽ i.4 , and p s the highest power of p that di¨ ides l.
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POLAR CURVES
Proof. Case l gp d. Writing d s d Ž m. q a p mq 1 and l s lŽ m. q b p mq 1, it follows from the definition of m that d Ž m. - lŽ m., d mq 1 ) l mq1 , and a Gp b . First of all let us observe that d Ž m. / 0. This is so because otherwise we would have as a consequence of d mq 1 ) l mq1 that dy1 ' ly1
ž
/
ž
p mq 1 y 1
l
Ž m.
y1
ay1 / 0. b
/ž /
Now, for all r such that 1 F r F d Ž m., since d Ž m. - lŽ m., it follows that
ž
d Ž m. y r dyr ' Ž m. lyr l yr
/ ž
a ' 0. b
/ž /
On the other hand,
ž
d y Ž d Ž m. q 1 .
ly Žd
Ž m.
q 1.
/ ž '
p mq 1 y 1
l
Ž m.
yd
Ž m.
y1
ay1 / 0, b
/ž /
hence in this case we have m Ž d, l. s d Ž m.. Case l Fp d. In this case the hypotheses imply that s ) t G 0, hence d Ž sy1. / 0. Write d s d Ž sy1. q a p s and l s b p s. Now, since d Ž dy1. - p s, we have for all r, such that 1 F r F d Ž sy1., that
ž
d Ž sy1. y r dyr ' ps y r lyr
/ ž
/ž
a ' 0. by1
/
On the other hand,
ž
d y Ž d Ž sy1. q 1 .
ly Žd
Ž sy1.
q 1.
/ ž '
ps y 1 p yd s
Ž sy1.
y1
/ž
ay1 k 0, by1
/
hence in this case we have m Ž d, l. s d Ž sy1.. COROLLARY 5.4. we ha¨ e that
For any pair Ž d, l. of positi¨ e integers with 1 F l - d,
m Ž d, m Ž d, l . . s m Ž d, l . . Proof. This follows from Propositions 5.2 and 5.3 by a direct verification. PROPOSITION 5.5. we ha¨ e that
For any pair Ž d, l. of positi¨ e integers with 1 F l - d,
m Ž d, d y l . s m Ž d, l . .
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HEFEZ AND KAKUTA
Proof. The proof of this result is elementary and may be done using the definition of m Ž d, l. and twice Stifel’s relation. Alternatively, this can be proved using Propositions 5.2 and 5.3. COROLLARY 5.6. Let Z be a plane projecti¨ e cur¨ e of degree d and let l be an integer such that 1 F l - d. Then Ži. Žii.
Z is Ž d, l.-general if and only if Z is Ž d, d y l.-general. If Z has mild singularities, and l is a truncation of d, then
hZ , dy l G hZ , l . Proof. Ži. is a consequence of Theorem 2.5 and Proposition 5.5, while Žii. is a consequence of Corollary 3.4Žii. and of Proposition 5.5.
6. STATIONARY POINTS Let Z : F s 0 be a projective irreducible plane curve of degree d. Let l be an integer such that 1 F l - d and some mixed derivative of F of order l is nonzero. DEFINITION.
A point P g Z will be called l-stationary if I Ž P , Z.DlP Z . ) hZ , l .
By the very definition of hZ, l we have that there are finitely many such points, and the aim of this section is to determine their number. We will compute this number when either Z is Ž d, l.-general, or when l is a truncation of d, and Z has mild singularities with respect to the pair Ž d, l.. We will make this assumption from now on in this section. For any integer r and distinct integers, i, j, k s 0, 1, 2, we define for i - j, SkŽ r . s
r
r
Ý Ž DXr ( DXryr F . Ž FX . Ž yFX . i
j
j
i
r yr
.
rs0
Note that from the arguments we used in the proof of Theorem 2.5, and from Remark 5 and Proposition 3.1, we have that P g Z is a l-stationary point if and only if SkŽh Z, l. Ž P . s 0, for all k s 0, 1, 2. Remark that from the above definition we have that all singular points of Z are stationary points. To establish the main result of this section, namely Theorem 6.7, we will need several auxiliary results.
471
POLAR CURVES
LEMMA 6.1. Let a be an integer. If b is minimal among the integers greater than a but not p-adically greater, then there exists some r G 1 such that b s a y aŽ ry1. q p r . Proof. Let a s a s p s q a sq1 p sq1 q ??? qa n p n, with a s / 0, be the padic expansion of a. It is clear that the first integer greater than a which is not p-adically greater than a is b 0 s Ž a sq1 q 1 . p sq 1 q ??? qan p n s a y aŽ s. q p sq1 . Now we can restrict our search to the integers which are greater than b 0 but not p-adically greater, and the result follows clearly. PROPOSITION 6.2. Let Z be a projecti¨ e irreducible plane cur¨ e of degree d, and let l s d Ž l . for some l. Suppose that Z is not Ž d, l.-general and with mild singularities, then we ha¨ e for some r,
hZ , l s d Ž l . y d Ž ry1. q p r .
Ž 20 .
Proof. Since l is a truncation of d, we have from Remark 5 that
hZ , l s min n G l q 1 < S Ž n . 4 . Now, in view of Proposition 4.4, hZ, l must be an integer greater than h Ž d, l. s l q 1, but not p-adically greater than l q 1, and minimal with respect to the p-adic ordering. So from Lemma 6.1 we have that
hZ , l s Ž l q 1 . y Ž l q 1 .
Ž ry1 .
q pr
for some r, and the proposition follows. COROLLARY 6.3. Let Z: F s 0 be a plane projecti¨ e irreducible cur¨ e of degree d and l s d Ž l . for some l. If Z is not Ž d, l.-general, with mild singularities, then hZ, l ' 0 mod p. LEMMA 6.4. ha¨ e
ž
If Ž d, l. is a pair of integers such that 1 F l - d, then we
dys ' 0 mod p h Ž d, l . y s
/
;s; s s 2, . . . , h Ž d, l . y 1.
Proof. From Stifel’s relation we have that
ž
d y Ž s y 1. dys dys q s . h Ž d, l . y s m Ž d, l . y s m Ž d, l . y Ž s y 1 .
/ ž
/ ž
/
Ž 21 .
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HEFEZ AND KAKUTA
Now, from Corollary 5.4 it follows that
ž
d y m Ž d, l . dy1 dy2 ' ' ??? ' . m Ž d, l . y 1 m Ž d, l . y 2 m Ž d, l . y m Ž d, l .
/ ž
ž
/
/
From this and from Ž21., we get that
ž
dys ' 0 mod p, h Ž d, l . y s
s s 2, . . . , h Ž d, l . y 1.
/
LEMMA 6.5. Let d and l be integers such that 1 F l - d. If l is a truncation of d, and hZ, l is as in Ž20., then
ž
dys hZ , l y s ' 0 mod p,
s s 2, . . . , l .
/
Proof. Suppose that l s d Ž l . for some l G 0, and let hZ, l s d Ž l . y d q p r , for some r with 1 F r F l q 1. We will initially suppose that r - l q 1. From the fundamental congruence we get, for all s s 0, . . . , d Ž l ., that Ž ry1.
ž
d Ž l . q a p lq1 y s dŽ l. y s dys s ' ' 0. Ž l . hZ , l y s d y d Ž ry1. q p r y s d Ž l . y d Ž ry1. q p r y s
/
ž
/ ž
/
Suppose now that r s l q 1, then hZ, l s p r , and from the fundamental congruence we have for s G 1 that
ž
d Ž ry1. q b p r y s dys d Ž ry1. y s s ' s 0. r r p ys pr y s p ys
/ ž
/ ž
/
PROPOSITION 6.6. Let Z : F s 0 be a projecti¨ e irreducible plane cur¨ e of degree d and let l be an integer such that 1 F l - d. Suppose that either Z is Ž d, l.-general, or Z has mild singularities and l is a truncation of d, then we ha¨ e for h s hZ, l , and for all integers i, j s 0, 1, 2, that X jh Si Žh . ' X ih Sj Žh .
mod F.
Proof. If i s j, there is nothing to prove. We will only prove the case j s 0 and i s 2, since the other cases are similar.
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POLAR CURVES
Let P s Ž a0 , a1 , a2 . g K 3. From Lemma 2.4Žii. we have X 0h Ž DhP F . 1,
ž
h s ay 0
X1 X0
,
X2 X0
/
h
Ý X 0hyr rs0
dyr yr
žh /
a Ý Ž D 0, a , b F . Ž P . Ž a0 X1 y a1 X 0 . Ž a0 X 2 y a2 X 0 . b ,
=
aq b sr
and X 2h Ž DhP F . s
ž
h ay 2
X0 X2
,
X1 X2
,1
h
Ý X 2hys ss0
=
/ dys ys
žh /
g Ý Ž Dg , d , 0 F . Ž P . Ž a2 X 0 y a0 X 2 . Ž a2 X1 y a1 X 2 . d .
gq d ss
Now, since obviously X 0h Ž DhP F . 1,
ž
X1 X0
X2
,
X0
/
s X 2h Ž DhP F .
ž
X0 X2
,
X1 X2
/
,1 ,
we get by comparing the coefficients of X 1t X 2hyt of both expressions that h D 0, t , hyt F Ž P . s ay 2
h
Ý ss0
=
dys ys
žh /
Ý Ž Dg , d , 0 F . Ž P . Ž ya0 .
g
gq d ss
žd/ t
a2t Ž ya1 .
d yt
.
Now, from the above expression, Lemmas 6.4 and 6.5, and Theorem 3.2, we have that d X 2h D 0, t , hyt F s h
dy1 yt 0 X 2t Ž yX 0 . F q hy1 t
ž /ž / ž /Ž =
0 t
q
Ý
ž
yX 0 . Ž yX1 .
gq d s h
žd/Ž t
yt
X 2t FX 0 q g
/ 1 t
ž /Ž
yX1 .
Dg , d , 0 F . Ž yX 0 . X 2t Ž yX1 .
1yt
d yt
.
X 2t FX 1
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HEFEZ AND KAKUTA
Using Euler’s identity and the above one, we get the following equalities and congruences mod F, X 0h S 2Žh . s '
( D Xd 1 F . X 0g Ž yX 0 FX 0 .
Ý Ž DXg
0
( D Xd 1 F . X 0g Ž X 1 FX 1 q X 2 FX 2 .
gq d s h
h
Ý Ý
gq d s h ts0
žd/Ž t
= X 2t Ž yFX 1 .
hyt
Ž FX . 1
g
Dg , d , 0 F . Ž yX 0 . Ž yX1 .
Ž FX .
d
hyt
1
Ž FX .
t
2
y
ts0
d yt
ž
dy1 hy1
h
= Ž yX 0 . FX 0Ž yFX 1 . q Ž yX1 . FX 1Ž yFX 1 . qX 2 FX 1Ž yFX 1 . h
Ý Ž DXt ( DhXyt F . Ž yFX . 1
g
1
t
Ý Ž X 2h D 0, t , hyt F . Ž yFX .
' X 2h
Ž FX .
2
h
'
g
0
gq d s h
s
d
Ý Ž DXg
2
hyt
1
hy1
Ž FX . 2
t
/
h
FX 2
s X 2h S 0Žh . .
ts0
Remark 8. The above proposition tells us that the data
½ž
Ua ,
SaŽh . Xah
/5
, a s0, 1, 2
where Ua s Ž X 0 ; X 1; X 2 .< Xa / 04 , define a section of the fiber bundle OZ Ž dŽh q 1. y 3h ., whose zeros are the stationary points of Z with respect to Ž d, l.. DEFINITION. We define the l-weight of P as being the order of vanishing w Ž P . at P of the above section. So P g Z is a l-stationary point if and only if w Ž P . ) 0. THEOREM 6.7. Let Z : F s 0 be a projecti¨ e irreducible plane cur¨ e of degree d and let l be an integer such that 1 F l - d. Suppose that either Z is Ž d, l.-general, or Z has mild singularities and l is a truncation of d; then we ha¨ e, for h s hZ, l , that
Ý w Ž P . s d d Ž h q 1. y 3h PgZ
.
POLAR CURVES
475
Proof. This follows immediately from the above remark, and from the definition of w Ž P .. Remark 9. The method of proof of the above theorem is inspired from w4x, where the result is proved for l s 1. Observe that if a curve Z of degree d is reflexive, then it is Ž d, 1.-general. And if Z is smooth and Ž d, 1.-non-general, then d ' 1 mod p Žsee for example w3x or w7x., then 1 is a truncation of d, so that Theorem 6.7 gives us the classical formula for the weighted number of flexes of a non-singular plane curve Z of degree d. Namely, f s d d Ž « 2 q 1. y 3« 2 , where « 2 is the intersection multiplicity of the curve with its tangent line at a general point, or alternatively, the inseparable degree of the dual map of the curve. Žsee for example w10x and w6x or w9x.. Remark also that in characteristic zero, a point is l-stationary if and only if it is a flex, so in this case we are just counting the flexes.
7. POLAR MORPHISMS Let Z : F s 0 be an irreducible smooth plane curve of degree d, and let l be an integer such that 1 F l - d, and suppose that some mixed derivative of F of order l is nonzero. Let Nl s lŽ l q 3.r2. DEFINITION.
The l th polar morphism is the morphism defined by pl : Z ª PKNl P ¬ Ž Di 0 , i 1 , i 2 F Ž P . ; i 0 q i 1 q i 2 s l .
We will denote the image of pl by Zl . So for l s 1, we have that pl is the Gauss map and Zl is the dual curve of Z. We will also denote by ¨l the Veronese l-uple embedding, U
¨l : PK2 ª Ž PKNl .
Q ¬ Ž Q0i 0 Q1i1 Q2i 2 ; i 0 q i1 q i 2 s l . . THEOREM 7.1. Let Z be a smooth cur¨ e of degree d, and let l be an integer such that 1 F l - d. Then we ha¨ e deg Ž pl . deg Ž Zl . s d Ž d y l . .
476
HEFEZ AND KAKUTA
Proof. We will denote by deg s Ž pl . Žrespectively, deg i Ž pl .. the separable Žrespectively, inseparable . degree of pl. Let P1 , . . . , Pr g Z be the points such that U s Zl R pl Ž P1 . , . . . , pl Ž Pr . 4 is the open set of points R for which aply1 Ž R . s deg s Ž pl . . Let Q be a point of PK2 , and consider the hyperplane HQ in PKNl whose coefficients are the coordinates of the point ¨lŽ Q . g ŽPKNl .U . So HQ :
Ý
i 0qi 1qi 2s l
Q0i 0 Q1i1 Q2i 2 Zi 0 , i1 , i 2 s 0,
where Zi 0 , i1 , i 2 are the coordinate functions of PKNl. For Q general in PK2 we have that HQ l Zl ; U, because the above condition means that the general point Q of PK2 is not a zero of any of the nonzero polynomials DlP i F, for i s 1, . . . , r. Now, from standard ramification theory and from Lemma 2.1Ži. we have that l deg i Ž pl . I Ž pl Ž P . , HQ l Zl . s I Ž P , Z l Dly F Ž Q . . s I Ž P , Z l Ddy Q Z.,
where Dly F Ž Q . . s
Ý
i 0qi 1qi 2s l
Q0i 0 Q1i1 Q2i 2 Di 0 , i1 , i 2 F Ž X 0 , X 1 , X 2 . .
Now, from Bezout’s Theorem we have ´ d Ž d y l. s
l Ý I Ž P , Z l Ddy Q Z . s deg i Ž pl . Ý I Ž pl Ž P . , HQ l Zl .
PgZ
PgZ
s deg i Ž pl . deg s Ž pl .
Ý I Ž R , HQ l Zl . s degŽ pl . degŽ Zl . . RgZl
The above theorem is a generalization of Plucker’s formula, which was ¨ known for l s 1, relating the degree of a curve with that of the Gauss map and of the dual curve.
REFERENCES 1. E. Ballico and A. Hefez, Non-reflexive projective curves of low degree, Manuscripta Math. 70 Ž1991., 385]396.
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2. J. L. Coolidge, ‘‘A Treatise on Algebraic Plane Curves,’’ 1931; Dover, New York, 1959. 3. A. Hefez, Non-reflexive curves, Compositio Math. 69 Ž1989., 3]35. 4. A. Hefez, ‘‘Introduc¸˜ ao ` a geometria projetiva,’’ Monografias de Matematica, Vol. 46, ´ IMPA, 1991. 5. A. Hefez and N. Kakuta, New bounds for Fermat curves over finite fields, in ‘‘Proc. Zeuthen Symp.,’’ Contemporary Math., Vol. 123, pp. 89]97, Amer. Math. Soc., Providence, RI, 1991. 6. D. Laksov, Wronskians and Plucker formulas for linear systems on curves, Ann. Sci. ¨ ´ Ecol. Norm. Sup. 17 Ž1984., 45]66. 7. R. Pardini, Some remarks on plane curves over fields of finite characteristic, Compositio Math. 60 Ž1986., 3]17. 8. G. Salmon, ‘‘A Treatise on the Higher Plane Curves,’’ 3rd ed., Chelsea, New York. 9. K. O. Stohr, ¨ and J. F. Voloch, Weierstrass points and curves over finite fields, Proc. London Math. Soc. Ž 3 . 52 Ž1986., 1]19. 10. R. Walker, ‘‘Algebraic Curves,’’ Princeton Univ. Press, Princeton, NJ, 1950.