III. Kết quả của đề tài

Phó Đức Tài, On intersection number o f plane curves.

+ 03 báo cáo tại Hội thảo 50 nãm Khoa Toán-Cơ-Tin học:

Đào Phương Bắc, Các tiêu chuẩn cho nhóm con quan sát được và nhóm

con Grosshans trên trường bất kỳ.

Phó Đức Tài, Vấn đ ề phân loại đường và mặt cố kì dị.

Lê Quý Thường, Hàm zêta của kì dị suy biến.

+ Và các báo cáo tại các xêmina khác; bài giảng cho sinh viên ở trường

hè Toán học ở Quảng Bình.

- Một bài báo đã được đăng trên tạp chí quốc tế có uy tín:

Phó Đức Tài và Ichiro Shimada, Unirationality of certain supersingular

K3 surfaces in characteristic 5, Manuscripta Mathematics, Vol. 121, 425-435

(2006).

- Hai bài báo khác đã gửi các tạp chí quốc tế:

1. Phó Đức Tài, Dual of smooth quartics (với phần phụ lục được viết bởi

H. Tokunaga).

2. Lê Quý Thường, Zeta function of degenerate plane curve

singularities.

- Viết gói lệnh Ellcurves cho phần mềm toán học Maple 10. Địa chỉ trang web

của gói lệnh:

http://www.math.hokudai.ac.jp/~tai/Ellcurves/

- Các khóa luận đại học và luận văn cao học liên quan đến đề tài đã bảo vệ

thành công trong năm 2006:

Khóa luân tốt nghiẽp:

1. ứng dụng của đại s ố máy tính trong hình học của sinh viên Đỗ Thị

Bích Phượng (lớp K47A1S).

L u â n v ă n c a o ho c:



1. Một s ố thuật toán về đường cong elliptic của học viên Lê Thị Minh

Hải (Khóa 2004-2006).

2. Đường cong hữu tỷ có kỳ dị của học viên Nguyễn Thị Quyên (Khóa

2004-2006).

3. Hàm ĩêta của kỳ dị suy biến học viên Lê Quý Thường (Khóa 2004­

2006).



10



Kết luận

Sau một năm thực hiện đề tài, chúng tôi tự nhận thấy là đã hoàn thành

được các mục tiêu đặt ra ban đầu, đó là:

-



Xây dựng một nhóm nghiên cứu Hình học đại số tại Khoa Toán.



-



Tiến hành nghiên cứu một số vấn đề chọn lọc trong Hình học đại số,

bước đầu đã thu được kết quả. Công bố một số kết quả thông qua việc

đăng bài báo và báo cáo tại các xêmina và hội thảo chuyên ngành.



-



Phát triển đào tạo: Hướng dẫn sinh viên, học viên cao học làm khóa

luận và luận văn tốt nghiệp; Biên dịch sách tham khảo chuyên ngành.



-



Hợp tác nghiên cứu khoa học với các nhóm trong và ngoài nước.



Một số định hướng phát triển nghiên cứu sau khi kết thúc đề tài:

Các hướng nghiên cứu 1,2 và 3 sẽ là ưu tiên hàng đầu: Hướng 1 và

2 đây vẫn là vấn đề lâu dài về mặt lý thuyết, hướng 3 có thể phát triển

thêm bằng cách hướng dẫn sinh viên và học viên làm luận văn.

Chúng tôi tiếp tục xây dựng đội ngũ cho nhóm nghiên cứu về Hình

học đại số, duy trì và phát triển các hợp tác nghiên cứu trong và ngoài

nước.



11



Tài liệu tham khảo

Tiếng Việt

[VI] Lê Thị Minh Hải, Một số thuật toán về đường cong elliptic, Luận văn cao học,

Trường Đ H K H T N H N , 2006.



[V2] Đỗ Thị Bích Phượng, ứng dụng của đại số máy tính trong hình học, Khóa luận

tốt nghiệp đại học, Trường ĐH KHTN HN, 2006.

[V3] Nguyên Thị Quyên, Đường cong hữii tỉ có kỳ dị, Luận văn cao học, Trường ĐH

K H T N H N , 2006.



[V4] Lê Quý Thường, Hàm zeta của kỳ dị suy biêh, Luận văn cao học, Trường ĐH

K H TN HN, 2006.



Tiếng nước ngoài

[AC75] A'Campo, N.: La fonction zêta d' une monodromie, Comment. Math. Helv. 50

(1975X 233-248.

[A 0 9 6 ] A'Campo, N. and Oka, M.: G eom etry of plane curves via Tschirnhausen



resolution tower, Osaka J. Math. 33 (1996), 1003-1033.

[AGR95] Alonso, c., Gutierrez, J. and Recio, T.: A rational funtion decomposition

algorithm by nearseparated polynomials, J. Symbolic Computation 19 (1995), 527­

544.

[A75] Artin, M.: Supersingular K3 surfaces. Ann. Sci. Ecole Norm. Sup. 7(4), 543­

567 (1975)

[BS63] Birch B.J. and Swinnerton Dyer, H.P.F.: Notes on elliptic curves I, J. Reine

Angew. Math. 212, 7-25, 1963.

[B79] Bogomolov, F.A.: Holomorphic tensors and vector bundles on projective



varieties, Math. U.S.S.R. Irzvestiya 13, 499-555 (1979).

[B91] Borel, A.: Linear Algebraic Groups. Second enlarged edition. GTM



126.



B erlin-H eidelberg-N ew Y ork: springer 1991.

[CS99] Conw ay, J.H. and Sloane, N.J.A.: S p h e r e p a c k in g s , la ttic e s a n d q r o u p s , 3rd



edn. In: Grundlehren der Mathematischen Wissenschaften. vol. 290. Springer, Berlin

H eidelberg N ew Y ork (1999)



[C77] Cremona, J.E.: Algorithms for modular elliptic curves, Cambridge University

Press, C am bridge, 1977.



[E02] Ebeling, w.: Lattices and codes, revised ed. In: Advanced Lectures in

M athem atics. Friedr. V iew eg & Sohn, Braunschweig (2002)



[G97] Grosshans, F.: Algebraic Homogeneous Spaces and Invariant Theory. Lecture



12



Notes in Mathematics 1673. Berlin-Heidelberg-New York: Springer 1997.

[GRY02] Gutierrez, J., Rubio, R. and Yu, J.-T.: D-Resultant For Rational Functions,

Proc. o f AMS 130 (2002), 2237-2246.

[K78] Kempf, G.: Instability in invariant theory. Ann. Math. 108, 299-316 (1978).

[L095] Lê, D. T. and Oka, M.: On resolution complexity of plane curves, Kodai

M ath.J. 18(1995), 1-36.

[M68] Milnor, J.: Singular Points of Complex Hypersurface, Ann. of Math. Stud. 61,

Princeton Univ. Press, Princeton, 1968.

[M94] Mumford, D., Fogarty, J. and Kirwan, F. : Geometric Invariant Theory.

Ergebnisse der Mathematik und ihrer Grenzgebiete 34. Berlin-Heidelberg-New York:

Springer 1994.

[NB05] Nguyen Quoc Thang, Dao Phuong Bac: Some Rationality Properties of

Observable Groups and Related Questions. Illinois J. Math. 49, n. 2, 431-444 (2005).

[N79] Nikulin, V.V.: Integer symmetric bilinear forms and some of their geometric

applications. Izv. Akad. Nauk SSSR Ser. Men. 43 111-177, 238 (1979). English

translation: Moth. USSR-Izv. 14, (1979) 103-167 (1980)

[096] Oka, M.: Geometry of plane curves via toroidal resolution, Algebraic

Geometry and Singularities, Edited by A. Campillo and L. Narvaez, Birkhauser,

Progress in Math. 134 (1996), 95-118.

[097] Oka, M.: Non-degnenerate complete intersection singularity, Actualites

mathematiques, Hermann, Paris, 1997.

[RS78] Rudakov, A.N.. Shafarevich, I.R.: Supersingular K3 surfaces over fields of

characteristic 2. Iz v . A kacl. N a u k SS S R S er. M a t. 42, 8 4 8 -8 6 9 (1978).



[RS81] Rudakov, A.N. and Shafarevich, I.R.: Surfaces of type K3 over fields of finite

characteristic. Cuưent Problems in Mathematics, vol. 18, Akad. Nauk SSSR,

Vsesoyuz.



Inst.



N auchn.



i Tekhn.



Informatsii,



M oscow ,



1981.



Reprinted



in:



Shafarevich, Ị.R.: Collected Mathematical Papers, pp. 657-714. Springer, Berlin

Heidelberg New York (1989)

[S04] Shimada, I.: Rational double points on supersingular K3 surfaces. Math. Comp.

73, 1989-2017 (2004).

[S74] Shioda. T.: An example of unirational surfaces in characteristic p. Math. Ann.

211,233-236 (1974)

[S77] Shioda. T.: On unirationality of supersingular surfaces. M a th . A n n . 225, 155­

159 (1977)



[ST92] Silverman, J.H. and Tate, J.: Rational points on elliptic curves, Undergraduete

Texts in Mathematics. Springer-Verlag, New York, 1992.



13



[S90] Sukhanov, A. A.: Description of the observable subgroups of linear algebraic

groups. Math. U.S.S.R. Sbornik 65, No.l, 97-108 (1990).

[vdE97] van den E ssen, A. and Yu, J.-T.: The D-Resultant, singularities and the



degree of unfaithfulness, Proc. of AMS 125 (1997), 689-695.

[W03] Washington, L.C.: Elliptic curves: Number Theory and Ci-yptography,

Chapman - Hall/CRC, 2003.



14



c. Phụ lục:

1. Photocopy các bài báo, bìa các luận văn Đại học, Thạc sỹ

2. Tóm tắt các công trình NCKH của cá nhân

3. Scientific Project

4. Phiếu đăng ký kết quả nghiên cứu



manuscripta math. 121, 4 2 5 - 4 3 5 (2006)



© Springer-Verlag 2006



Due Tai Pho • Ichiro Shimada



Unirationality of certain supersingular

in characteristic 5



K



3 surfaces



Received: 21 February 2006 / Revised: 13 June 2006

Published online: 30 September 2006

Abstract. We show that every supersingular K 3 surface in characteristic 5 with Artin

invariant < 3 is unirational.



1. Introduction

We work over an algebraically closed field k.

A K Ĩ surface X is called sitpersingular (in the sense o f Shioda [22]) if the

Picard number o f X is equal to the second Betti number 22. Supersingular K 3

surfaces exist only when the characteristic o f k is positive. Artin [3] showed that,

if X is a supersingular K 3 surface in characteristic p > 0, then the discriminant

o f the Néron-Severi lattice N S (X ) o f X is written as —p 2ơ(X\ where ơ ( X ) is a

positive integer < 10. (See also Illusie [9, Sect. 7.2].) This integer ơ { X ) is called

the Artin irn ariant o f X .

A surface s is called imiratioiial if the function field k( S) o f s is contained

in a purely transcendental extension field o f k, or equivalently, if there exists a

dominant rational map from a projective plane p 2 to s . Shioda [22] proved that, if

a smooth projective surface 5 is unirational, then the Picard number o f 5 is equal

to the second Betti number o f s . Artin and Shioda conjectured that the converse is

true for K 3 surfaces (see, for example, Shioda [23]):

Conjecture 1. Every supersingular K 3 surface is unirational.

In this paper, we consider this conjecture for supersingular K 3 surfaces in

characteristic 5.

From now on, we assume that the characteristic o f k is 5. Let Ả

:[.V]6 be the space

o f polynomials in V o f degree 6, and let u c k[x ]6 be the space o f / ( . v ) e k[x]e

such that the quintic equation f \ x ) = 0 has no multiple roots. It is obvious that

u is a Zariski open dense subset o f k[x}(,. For / € u , we denote by C / c p 2 the



D T Plio: D ep a rt me n t of Ma t hemat i c s , Vietnam National University.

^ 3 4 1'



uyen Tr ii Street, Hanoi, Vietnam, e-mail: p h o d u c t a i @ y a h o o . c o m ; ta i pd @v nu . ed u. v n



I S h i m a d a ( IS): D ep ar t me n t o f Mathematics, Faculty o f Science, H o k k ai d o University.

S a p p o ro 0 60 - 08 10 , Japan, e-mail: shi mad a @ma t h . s ci . ho kud ai .ac . j p



DOI: 10.10 07 / s 0 0 2 2 9 - 0 0 6 - 0 0 4 5 - 3



426



D. T. Pho, I. Shimada



projective plane curve o f degree 6 whose affine part is defined by

y5 - fix) = 0.



Let Y f — IP" be the double covering of p 2 whose branch locus is equal to c f , and

>

let X f - > Y f be the minimal resolution o f Yf .



Theorem 1. I f f is a polynom ial in Li, then X f is a supersingular K 3 surface with

f ) < 3 . Conversely, i f X is a supersingulcir K 3 surface with ơ ( X ) < 3, then



there exists f € Z such that X is isom orphic to X f .

-/

The affine part o f Y f is defined by w 2 = >’5 — f ( x ) . Hence the function field

k ( X f ) is equal t o k ( w , X , y) , and it is contained in the purely transcendental exten­

sion field fc(uí1/ 5, X 1/ 5) o f k. Therefore w e obtain the following corollary:



Corollary 1. Every supersingular K 3 surface in characteristic 5 w ithA rtin invari­

ant < 3 is unirational.

The unirationality o f a supersingular K 3 surface X in characteristic p > 0 with

Artin invariant a has been proved in the following cases: (i) p = 2, (ii) p — 3 and

Ơ < 6, and (iii) p is odd and Ơ < 2. In the cases (i) and (ii), the unirationality

was proved by Rudakov and Shafarevich [15. 16] by showing that there exists a

structure o f the quasĩ-elIiptic fibration on X. The case (iii) follows from the result

o f Ogus [13, 14] that a supersingular K 3 surface in odd characteristic with Artin

invariant < 2 is a Kummer surface associated with a supersingular abelian sur­

face, and the result o f Shioda [24] that such a Kummer surface is unirational. The

unirationality o f X in the case ( p , ơ ) = (5, 3) proved in this paper seems to be

new.

In [19], w e have shown that a supersinoular K 3 surface in characteristic 2 is

birational to a normal K 3 surface with 2 1 AI -singularities, and that such a normal

K 3 surface is a purely inseparable double cover o f p 2. In [20], we have proved that

'a supersinsular K 3 surface in characteristic 3 with Artin invariant < 6 is birational

to a normal K 3 surface with lO/b-sinsularities, and it is also birational to a purely

inseparable triple cover o f p 1 X p 1. These yield an alternative proof to the results

o f Rudakov and Shafarevich [15, 16] in the cases (i) and (ii) above.

In this paper, we show that a supersingular K 3 surface in characteristic 5 with

Artin invariant < 3 is birational to a normal K 3 surface with 5Aa-singularities that

is a d o u b l e c o v e r o f p : . a n d t h e n p r o v e t h a t s u c h a n o r m a l K 3 s u r f a c e is i s o m o r ­

phic to Y f for s o m e /



e u . T h e first s t e p f o l l o w s f r o m t he s t r u c t u r e t h e o r e m o f t he



Néron-Severi lattices o f supersingular K 3 surfaces due to Rudakov and Shafarevich [16]. For the second step, we investigate projective plane curves o f degree 6

with 5 / \ 4-sineulanties in Sect. 2.



2. Projective plane curves Iwith 5 /i4-singularitie_s

D efinition 1 Í ger m o f a curve singularity in characteristic Ỷ- - ' V cal led an

An -singularity i f it is f o r m a l l y i somorphic to



r - . v " - 1 =0,



U nirationality o f certain supersingular



K 3 surfaces in characteristic 5



427



(see A rtin [4], a n d G reuel and Kroning [8].)



We assume that the base field k is of characteristic 5 until the end of the paper.

Proposition 1. Let c c p 2 be a reduced projective plane curve o f degree 6. Then

the fo llo w in g conditions are equivalent to each other.



(i) The singular locus o f c consists o ffiv e A ị-singuìar points.

(ii) There exists f e W such that c = C f.

For the proof, w e need the following result due to Wall [26], which holds in any

characteristic. Let D c p 2 be an integral plane curve of degree d > 1, and let

I d c p 2 X (P2) v be the closure o f the locus of all (j:, /) 6 p 2 X (P 2) v such that X

is a smooth point o f D and I is the tangent line to D at X . Let D v c (P 2) v be the

image o f the second projection

7TD : I D - + (P 2) v .

We equip D v with the reduced structure, and call it the dual curve o f D. Note that

the first projection I d - > D is birational. Therefore, by the projection 7ĨD, we can

regard the function field k ( D ) as an extension field of the function field k ( D v ).

The corresponding rational map from D to D v is called the Gauss map. We put

d e g 7tD : = [ k ( D) : k ( D v )].

We ch oose general h om ogen eou s coordinates [it'o : U : U 2 ] o f IP2, and let

!\

J

F( wo , W\ , W2 ) = 0 be the defining equation of D. We denote by D q c ip2 the

curve defined by



Ề L = o.

dw 2



which is called the p o l a r curve o f D with respect to Q = [0 : 0 : 1].

P ro p o sition 2 (Wall [26]). For a singular point s o f D, we denote by ( D . D q ) s the

local intersection multiplicity o f D a nd D q at s. Then we have

deg7TD ■ 2 D v = d{d — 1) —

de



( D . D q )j.

jESing(D)



Remark 1. If s e D is an ,4 n-singular point, then the polar curve D q is smooth at

i' and the local intersection multiplicity ( D . D q ) s is n + ] .

P r o o f (Proof o f Proposition 1). Suppose that c has 5 A 4-singular points as its only

singularities. Since an /44-sin?ular point is unibranched, c is irreducible. By Prop­

osition 2 and Remark 1, w e have



deg 7 C

Ĩ

S u p p o s e t ha t ( d e g



c



■ deg



=



5 .



. d e g c v ) = ( 1, 5). L e t r : c - > c b e t h e n o r m a l i z a t i o n o f



Since digTTc = u we can consider c as a normalization o f c

I*v : c



C'



We denote by



428



D. T. Pho, I. Shim ada



the morphism of normalization. Let s be a singular point of c , and let J € c be the

point o f c that is mapped to s by V. We can choose affine coordinates (.t, y) o f p 2

With the origin s and a formal parameter t o f c at 1 such that V is given by

t t~> ( x , y ) =



( t2 , t5 +



c 6 t6 +



C jt1 +



■■• ) .



Let (u, v) be the affine coordinates o f (P2) v such that the point (u, V) € (P 2) v

corresponds to the line o f p 2 defined by > = ux + V. Then Vv is given at f by

'

t



(m, 1») = ( 3 c 6 ?4 H

------ , / 5 H

--------).



(See, for example, Namba [10, p. 78].) Therefore L (?) is a singular point o f c v

'v

with multiplicity > 4. We choose distinct two points S | . s~ € S ing(C ). There exists

>

a line o f (P 2) v that passes through both o f

€ c ~ and

€ c v . This

contradicts Bezout s theorem, because d e g C v = 5 < 4 + 4 . Therefore we have

(deg 7ĨQ, deg c v ) = (5, 1). Then there exists a point P e P 2 such that we have



( 1)



/ € c v <=> p € /.



We choose hom ogeneous coordinates [u;o : VJ\ : u>2] of IP2 in such a way that

p = [0 : 1 : 0], Let Ljjc be the line u>2 = 0, and let (.r v) be the affine coordinates

on A 2 : = p 2 \ Lỵ, given by X

W()/W2 and ỵ := Wị / u> 2 . Suppose that c is

d e f i n e d b y h(. x, v) = 0 in A 2 . F r o m (1), w e h a v e



dh



h { a , b) = 0 = > ~ ( a , b) = 0 .



(2 )



dy



Let U c c A 1 be the image o f the projection (C \ Sing(C )) n A 2 -> A 1 given by

(a. b) (-+ a. Note that L’c is Zariski dense in A 1. Let (ao- bo) be a smooth point o f

c n A 2. By (2), we have

dh

--(<30.



dx



bo) 7~ 0.



Hence there exists a formal power series y { q ) € £[[/}]] such that c is defined by

ỵ ( y — bo) locally around (ao. bo)- By (2) asain, y' {rj ) is constantly

equal to 0 , and hence there exists a formal power series Ị3(ri) e Ả

'[[77]] such that

Y {rj) = P ( ĩ ì Ý . Therefore the local intersection multiplicity o f the line X - C = 0

IQ

and c at (< bo) is > 5. Thus w e obtain the following:

30,



X — ao =



If a € ƠC, then the equation h( a, v) = 0 in V has a root o f multiplicity > 5.

(3)

We put

/i(.v,



v )



=



c v



6



+



g i U ) } ' 5



+



• • •



+



g 5 U ) y



+



8b(x).



where c is a constant, and ^„(.v) € Ả;[.v] is a polynomial o f degree < V. Suppose

that c- 7^ 0. We can assum e c = 1. By (3), we have f>zUn = g}U>) = § 4 Ìà)



0 and



5 i(rt)5 5 (a) = gỏ( a) for any Í7 e Uc - Since U c is Zariski dense in L ' , we have g 2 =

gỊ = £ 4 = 0 and g I

= % Then we have /i(.v, v) = (>'■’ + t,'j(.v))(y + ^i(.v)),

().



Unirationality o f certain supersingular A 3 surfaces in characteristic 5

T



429



which contradicts the irreducibility of c . Thus c = 0 is proved. Then, by (3), we

have g ỉ

0 and g 2 = g 3 = g 4 = g5 = 0- We put gi = A x + B , and define a new

homogeneous coordinate system [zo : 2 1 : z i\ of p 2 by

U o . z 1 , z i ) : = ( w o , W\ , A



wq



+ B w i)



( z o , z i , Z2) ■= ( W2 , W\ , A w o )



if B ỹí 0;

if B = 0.



Then c is defined by a homogeneous equation of the form

Z2 Z\ -



F ( z q , Z 2) = 0 ,



where F( z o, z i ) is a hom o gen eou s polynomial o f degree 6. We put L' ^ : = {C2 = 0}.

Defining the affine coordinates (;c, y ) on p 2 \ L'x by (.V, >•) := { 7. ữ/ z 2 , z \ / z ì ) , we

see that the affine part o f c is defined by >’5 — / ( x) for some polynomial f i x )

o f degree < 6 . If deg / < 6 , then L'x would be an irreducible component o f c

because d e g C = 6 . Therefore we have deg f — 6. Then c n / , ^ consists o f a

single point [ 0 : 1 : 0], and c is smooth at [0 : 1 : 0]. Therefore we have

S i n g ( C ) = { ( a , / ( c 0 1/5) I / ' ( at) = 0 }.



Since c has five singular points, we have f e U .

Conversely, suppose that / € u . We show that Sing(C f ) consists o f 5/\«-singular points. Let Loo c p 2 be the line at infinity. It is easy to check that c f n L-x

consists o f a single point [ 0 : 1 : 0], and c / is smooth at this point. Therefore

we have S in g ( C f ) = {{a. f ( ữ ) 1/ 5 ) I f ' ( a ) = 0}. In particular, c f has exactly five

singular points. Let {a, /3) be a singular point o f c / . Since ư is a simple root of

the quintic equation f ' ( x ) = 0 , there exists a polynomial g ( x ) with g (or) 7 0 such

^

that

/ (.v) = / ( a ) +



(x - a ) 2s(.v).



Because p 5 = f (a ), the defining equation o f c is written as

(y - P ) 5 -



(.V - ư ) 2g ( . r ) = 0.



Therefore (a, P) is an i4 4 -singular point o f c f .



□



3. P r o o f o f th eo r em 1

First we show that if f e u , then X f is a supersingular K 3 surface with Artin

invariant < 3. Since the sextic double plane Yj has only rational double points as

its singularities by Proposition I, its minimal resolution A'/ is a A'3 surface by the

results o f Artin [1 2]. Let £ / be the sublattice o f the Néron-Severi lattice N S (X f )

o f X f that is generated by the classes o f the (- 2 ) - c u r v e s contracted by X f

Yf .

Then £ / is isomorphic to the ncsativc-dclinilo roul lattice of type 5 A 4 by Propo­

sition 1 In particular, E f is o f rank 20, and its discriminant is 5-\ Let H f c X f

be the pull-back o f u line o f P “, and put

h f := [ H f \ € N S(A '/).