Enumerative Geometry for Plane Cubic Curves in Characteristic 2

Compositio Mathematica 111: 123–147, 1998. 123

c 1998 Kluwer Academic Publishers. Printed in the Netherlands.

Enumerative geometry for plane cubic curves in characteristic 2

ANDERS HØYER BERG Department of Mathematics, University of Oslo, PB 1053 Blindern, N-0316 Oslo, Norway, e-mail: [email protected]

Received 14 March 1996; accepted in ﬁnal form 23 December 1996

Abstract. Consider the plane cubic curves over an algebraically closed ﬁeld of characteristic 2. By 9

blowing up the parameter space P twice we obtain a variety B of complete cubics. We then compute

the characteristic numbers for various families of cubics by intersecting cycles on B . Mathematics Subject Classiﬁcations (1991). Primary 14N10; Secondary 14C17, 14H45. Key words: Enumerative geometry, plane cubic curves, positive characteristic, blow-up, characteristic numbers.

1. Introduction One of the major objects of enumerative geometry is to determine the characteristic

numbers for families of plane curves. A characteristic number counts the number

of curves in a family that pass through given points and touch given lines,

+ where equals the dimension of the family. For families of plane cubic curves these numbers were first found by Maillard and Zeuthen in the early 1870’s, but their methods were based on assumptions that were not rigorously justified. More than a century went by before these numbers were confirmed. Kleiman and Speiser [8, 9, 10] and Aluffi [1, 2] both compute the characteristic numbers for smooth, nodal and cuspidal cubics, but by very different means. Kleiman and Speiser’s works are based on the classical degeneration method of Maillard and Zeuthen. They specialize their family to more degenerate ones and then use the numbers already obtained for the special families. In this way the characteristic numbers for smooth cubics depend on the numbers for nodal cubics, which in turn depend on the numbers for cuspidal cubics. Aluffi’s method is more direct. By a sequence of five blow-ups of P9 he con- structs a variety of complete cubics. The characteristic numbers for smooth cubics are then obtained by intersecting certain divisors on this variety. By a closer exam- ination of the space parametrizing the singular cubics, Aluffi also obtains the characteristic numbers for nodal and cuspidal cubics. These papers, like most other papers on enumerative geometry, assume that the characteristic is different from 2 and 3. One exception is Vainsencher’s Conics

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in characteristic 2 [11] determining the number 51 of conics tangent to 5 other, assuming the characteristic is 2. In this paper we will apply the method used by Aluffi [1] and Vainsencher [11, 12] to construct a variety of complete plane cubics in characteristic 2. The strategy is to blow up the parameter space along smooth centers until the proper transforms of the line conditions no longer intersect. The technical difficulties of this method is at each blow-up to determine the intersection of the line conditions and to compute certain Segre classes. In characteristic 2, the intersection of the line conditions has nonsingular support and is reduced at the general point. This makes our case significantly less demanding than the characteristic 0 case, where the intersection of the line conditions has more structure. In our case, two blow-ups

will suffice, giving a smooth space, B , of complete cubics which is relatively easy to handle. The computation of the characteristic numbers follows the lines of Aluffi and Vainsencher. In particular we will rely on Aluffi’s blow-up formula [1, Thm. II], which relates intersection numbers before and after taking proper transforms.

2. Generalities about characteristic numbers

In this section, which is independent of the characteristic of the base ﬁeld k ,we

deﬁne the characteristic numbers and give their basic properties.

N R

Intuitively, a characteristic number ; for a family of plane curves is the

number of curves passing through given points and properly tangent to given

+ = R lines where dim . (We call a tangent proper if it is tangent at a nonsingular point. By just tangent we mean a line intersecting the curve with multiplicity at least 2 at a point.) To determine these numbers it is convenient to work in the

n 1

n = dd + d

P 2 3 parametrizing all plane curves of degree . n

DEFINITION 2.1. A point condition in P is a hyperplane H parametrizing the

curves containing a given point. A line condition is a hypersurface M parametrizing the curves tangent to a given line. The following deﬁnition of characteristic numbers differs slightly from the intu- itive one in that it may count curves with multiplicity greater than one. But as we

shall see, this need not be a big problem.

r DEFINITION 2.2. Suppose R P is an irreducible, -dimensional subvariety

parametrizing a family of curves such that the generic curve is reduced and irre-

+ = r

ducible. Suppose we have points and lines in general position, with .

H M j Let i and be the corresponding point and line conditions in P .Wedeﬁnethe

characteristic numbers for R to be

N = mx;R H H M M ;

; 11

x2Q

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ENUMERATIVE GEOMETRY FOR PLANE CUBIC CURVES IN CHARACTERISTIC 2 125

Q = fx 2 C g C x

where P : x intersects the lines only at smooth points is

x m N

the curve corresponding to ,and is the usual intersection multiplicity. ;

counts the weighted number of curves in R passing through the points and

properly tangent to the lines. The following theorem shows that these numbers are well deﬁned, and that the curves counted by a given characteristic number all appear with the same

multiplicity.

THEOREM 2.3. Let R P be an irreducible, -dimensional family of generically

reduced and irreducible curves, and let and be nonnegative integers such that

+ = r .

(i) There exists an open dense subset U P P and nonnegative

N e p ;::: ;p ;

integers and such that for each conﬁguration of points and lines 1

;::: ;l 2 U N R l

1 there are exactly different curves from passing through the

points and properly tangent to the lines, and such that the multiplicity in the

N p p sense of Deﬁnition 2 :2 at each of the curves is when the characteristic is

and 1 when the characteristic is 0.

= m

(ii) The multiplicity m is a non-decreasing function of . N Proof of (i). The existence of U and is well known and follows from

[5, Sect. 2]. It is also clear (same reference) that the number N remains the same R when R is replaced by any open subset of . We may then assume that all curves

in R are irreducible.

U R

Let T be deﬁned by

T = fp ;::: ;p ;l ;::: ;l x C

1 1 ;:

p l g j

contains i and is properly tangent to

q T U R x 2 R

and let p and be the projections from to and respectively. Let be any

1 2 2

q x C C

point. Then is an open subset of x P P .Since

_ 1

C C q x

x is irreducible, so is the dual . This shows that the ﬁbre is irreducible

x R

and it follows that T is irreducible (since is). s

We know that p is a generically ﬁnite surjective map of integral varieties. Let p

and m be the separable and inseparable degree of . Then it is well known ([11, s

Sect. 7], is one reference) that the general ﬁbre of p has distinct points, and the U multiplicity at each is m. Shrinking if necessary we have this statement for all the ﬁbres. Finally, the argument given in the remark in [11, Sect. 7] shows that the multiplicity in the last paragraph coincides with the intersection theoretic one. (The intersection-theoretic multiplicity can be obtained from an alternating sum of Tor’s. Since the line conditions are smooth, and in particular Cohen-Macaulay, at the

points of intersection, all the higher Tor’s vanish.) 2

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The following two lemmas are used in [1] to prove the characteristic 0 version of the above theorem. We will need the lemmas to prove the second part of the

theorem.

LEMMA 2.4. Suppose S P is a curve parametrizing generically reduced and

2 S

irreducible curves, and let x be a general point. Then there exist at most

H S x

ﬁnitely many point conditions p tangent to at .

T S x T H p

x p

Proof.Let x be the tangent line to at , and suppose .Then is T

contained in all the curves parametrized by x . Clearly (since the curve parame-

p 2

trized by x is irreducible), only a ﬁnite number of such can exist.

n LEMMA 2.5. Suppose R P is an irreducible family of generically reduced

and irreducible curves. Then a general point condition will intersect R transver-

sally by transversal we always mean that the scheme theoretical intersection has

no nonreduced components.

p H R

Proof. Since the set of points such that p does not intersect transversally H

is closed, it is enough to show the existence of one p that does. Suppose that all

point conditions intersect R in a nonreduced component. Then the union of these

x 2 R S R components will cover R .Let be a general point, and let be any

curve having x as a smooth point. Since the set of point conditions is 2-dimensional x

there will be inﬁnitely many point conditions tangent to R at . These will also be 2

tangent to S , contradicting Lemma 2.4.

H ;::: ;H M ;::: ;M

Proof of Theorem 2.3(ii).Let 1 and 1 be general point and

R \ H \\H \M \\M

line conditions in P . We know that the points in 1 1

N m

counted by ; all appear with the same multiplicity . If we remove one of the point conditions, then by (2.5) the components containing these points will also

have multiplicity m. When these components are intersected with a line condition, 2 all the points in the new intersection must have multiplicity at least m.

Remark. Note that the first part of this theorem seems to be a special case of [5, Thm. 2]. The important difference is the definition of the characteristic numbers. While we intersect in P n , the characteristic numbers in [5] are defined by

2 r

R I intersections in I where P P is the incidence variety.

DEFINITION 2.6. With the same hypotheses as in Deﬁnition 2.2 we deﬁne the

total characteristic numbers for R to be

= H M M ; mx; R H

; 1 1

2 nL x P

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ENUMERATIVE GEOMETRY FOR PLANE CUBIC CURVES IN CHARACTERISTIC 2 127

where is the locus of the nonreduced curves. ; is the weighted number of

reduced curves passing through the points and tangent to the lines (but not

necessarily at smooth points).

In order to compute the total characteristic numbers ; (from which the characteristic numbers will be deduced), we shall need the concept of a variety of

complete curves, which is deﬁned as follows.

B ! DEFINITION 2.7. A variety B together with a surjective morphism : P is

called a variety of complete curves if: L (1) restricts to an isomorphism outside the locus, , of nonreduced curves.

(2) The proper transforms of the line conditions in Pn do not have a common

intersection in B .

PROPOSITION 2.8. Suppose B is a complete nonsingular variety of complete

curves of degree d. Suppose P is a subvariety parametrizing a family of ~

generically reduced and irreducible curves, and let R be its proper transform in

~ ~

B M . Also, denote by H and the proper transforms of point and line conditions

respectively. Then the total characteristic numbers for R are given by

~ ~ ~

= [R ][H ] [M ] + = r = R:

; with dim

H ;::: ;H M ;::: ;M

Proof.Let 1 and 1 be general point and line conditions

n n

= L B !

in P ,andletE .Since : P restricts to an isomorphism

~ ~ ~ ~ ~

B nE !~ nL R \ H \ \ H \ M \ \ M

P it is sufﬁcient to show that 1 1 R

does not intersect E . Since the general curve in is reduced we can assume that

~ ~ ~

R \ E 6 r H M j

dim 1. The result follows if we can show that i and intersect

B V W a given irreducible subvariety V properly. ( and intersect properly if

V + W = V \ W fl 2 V M g

codim codim codim ).) The set P : l is closed,

and since B is a variety of complete curves this set is not all of P . It follows

that the general line condition does not contain V , so the intersection is proper.

H V B 2

Similarly, we can show that i intersects a given properly.

3. Generalities about cubics in characteristic 2 In this section we will discuss some elementary facts about plane cubic curves in characteristic 2. It is essential to get information about the subvarieties of P9 parametrizing special families of cubics. From now on, and for the rest of the paper, we will assume that the characteristic of the ground ﬁeld is 2. The deﬁning polynomial of a plane cubic curve will be written in the following form

com4041.tex; 19/02/1998; 13:54; v.7; p.5 128 ANDERS HØYER BERG

3 3 3 2 2

F x; y ; z = ax + by + cz + dx y + ex z

22 2 2

+fxy + gy z + hxz + iy z + jxyz:

V = fp; l l C g

Let : is tangent to p P P ,andlet 1 and 2 be the two

1 _

p C C

projections. Then the ﬁbre is the ‘total dual’ of p (the union of and 1 p

the possible multiple lines corresponding to each singularity of C ), and 2 is M

the line condition l .

z LEMMA 3.1. If we use x; y and as coordinates on P , we have the following

equation for V

3 3 3 2

bc + gix +ac + ehy +ab + df z +cf + gh + ij x y

2 2 2

z +cd + ei + hj xy +ai + dh + ej y z +fi + bh + gj x

2 22

dg + be + fjxz +ag + ef + dj yz + j xy z = : + 0

2 2

nfy = g m = x=y t = z=y Proof. Assume A = P 0 has afﬁne coordinates and

2 2 2

= V \ l 2 y = mx + tz

and let W P A .Let A be the line in P given by ,

F x; y ; z = C p; l 2 W

and let 0 be the equation for a cubic p .Now if and only

x; z =Fx; mx + tz ; z

if g has multiple factors, and this happens exactly when

g x; = the discriminant 1 0. In characteristic 2 the discriminant of a cubic poly-

3 2 3 3

+ bx + cx + d ad + bc F x; y ; z =ax + by + +jxyz

nomial ax is . Letting ,

F x; mx + t;

and computing 1 we easily obtain (after some elementary, but

V 2 tedious computations) the equation for W , and the equation for follows.

It should be well known that in characteristic 0 any nonsingular plane cubic can

3 3 3

+ y + z + txy z = be linearly transformed into one with equation x 0. This

is also true in characteristic 2. See [4, Sect. 7.3] for a proof that works in all C

characteristics different from 3. Note that in characteristic 2 the curve t given by

3 3 3 3

+ y + z + txy z = t = x 0 is singular if and only if 1.

LEMMA 3.2. The following holds for plane cubic curves in characteristic 2: (1) The dual of a nonsingular cubic is a nonsingular cubic. (2) The dual of a nodal cubic is a nonsingular conic.

(3) The dual of a cuspidal cubic is a line. In particular, a cuspidal cubic is strange there is a point common to all the proper tangents .

3 3 3 3

C x + y + z + txy z = t 6=

Proof. (1) This can be checked on t given by 0( 1).

_ 3 3 3 2

x + y + z + t xy z =

By Lemma 3.1 we see that C is given by 0whichis t nonsingular. (2) It is an easy exercise to check that all nodal cubics are projectively equiva-

3 3

+ y + xy z = lent, so we only need to consider the nodal cubic given by x 0. By

com4041.tex; 19/02/1998; 13:54; v.7; p.6 ENUMERATIVE GEOMETRY FOR PLANE CUBIC CURVES IN CHARACTERISTIC 2 129

+ xy = z = (3.1) the dual curve is z 0 (we ignore the line 0inP corresponding to the node) which is a nonsingular conic.

3 2

+ y z = y = (3) The dual of the cuspidal curve x 0 is the line 0. The argu-

ments are similar to the nodal case. 2

Remark. In characteristic 0, it is well known that the dual of a nonsingular plane

dd d = C

curve C has degree 1 ,where deg . What happens in characteristic _

2 is that the deﬁning polynomial of C reduces to the square of a polynomial of

d degree 2 d 1 .

3 3 3

C x + y + z + txy z =

If t is a nonsingular cubic given by 0 we see from the

_ _

C = C

proof of the ﬁrst part of the lemma that 4 so that in general biduality

t t does not hold. We have biduality only for a special class of cubics characterized by the following proposition.

PROPOSITION 3.3. The following are equivalent for a nonsingular cubic

C P :

_ _

=C (1) C .

3 3 3

x + y + z =

(2) C is projectively equivalent to the curve with equation 0. = (3) j 0 in the equation for C.

(4) C has Hasse-invariant 0. j

(5) C has -invariant 0.

_ _

, C C = C 4

Proof. 1 2 is a trivial consequence of the fact that when t

t t

3 3 3

x + y + z + txy z = C D j =

given by 0. If (projective equivalence) and D 0

j = , : ,

then it is easy to verify that C 0, so we have 2 3 3 4 is a special

, 2 case of [7, IV Prop. 4.21] 4 5 follows from [7, IV 4.23] (note to corollary).

12 3 3

j C t =t +

Remark. It can be shown that the -invariant of t equals 1 ,which

, j

gives another proof of 3 5 . When we use the notation C we do not mean

j C the j -invariant, but simply the coefﬁcient in an equation for .

The cubics described in Proposition 3.3 we call j -curves. We next show that the

cuspidal cubics are degenerate j -curves. First we need some lemmas.

C j = H

LEMMA 3.4. Let be a cubic with C 0, and let be the matrix

0 1

a f h

B C

d b i

@ A

e g c

Then:

, H = (1) C is nonsingular rk 3

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130 ANDERS HØYER BERG

, H =

(2) C is singular and reduced rk 2

, H =

(3) C is nonreduced rk 1

x; y ; z Proof. The singular locus is precisely the set of points such that all the

2 22

x ;y ;z

partial derivatives are zero, or equivalently: belongs to the nullspace of 2

H . The lemma now follows by elementary linear algebra.

LEMMA 3.5. Let C be a singular cubic. Then C is cuspidal possibly degenerate =

if and only if j 0 in the equation for C.

2 C ; ;

Proof. Choose a B PGL 3 mapping a singularity of to 0 0 1 .Let

0 0

= B C x = x=z y = y=z

D and introduce afﬁne coordinates , . The afﬁne equa-

0 0

D f x ;y = f = f + f + f + f f

tion of can be written as 0. Let 3 2 1 0 where i is

D ; f = f =

homogeneous of degree i.Since is singular at 0 0 we have 1 0 0, and

0 0 0

0 2 2

f = e x + g y + j x y D

D D 2 D is the equation of the tangent cone. is cuspidal

exactly when the tangent cone is a double line, and that happens exactly when

j = j = , j = 2

C D

D 0. Since by (3.3) 0 0 the lemma follows.

PROPOSITION 3.6. A plane cubic curve C is cuspidal possibly degenerate

H =j =

if and only if det 0.

j = H =

Proof.If C is cuspidal, then 0by(3.5)anddet 0 by (3.4).

H =j = C 2 det 0 implies that is singular by (3.4) and cuspidal by (3.5).

3 3

F x; y ; z = ax + by +

Let C be a non-degenerate cuspidal cubic given by

+ jxyz H =

. By (3.4) we have that rk 2. It follows that the cofactor matrix,

H H cof , has rank 1, so that nonzero rows (resp. columns) of cof deﬁne the

same point in P2.

1 0

bc + gi cd + ei dg + be

C B

cf + gh ac + eh ag + ef

: H =

cof

A @

fi + bh ai + dh ab + df Q

Let P be the point deﬁned by the columns, and let be the point deﬁned by the

; ; 6= ; ; Q = ; ; square root of the rows: If 000is a row, then .

This is well deﬁned since there is only one square root in characteristic 2.

P Q Q C P

PROPOSITION 3.7. Let C , and be as above. Then is the cusp of , and C is the only ﬂex of C. Also, P is a strange point, that is: every proper tangent of

contains P .

p p

= bc + gi; cd + ei ; dg + be

Proof. Suppose Q is given by the ﬁrst row

H H = F Q=F Q=F Q=

y z

of cof . Using that det 0 we easily see that x

C u ;u ;u 2 C 0, so Q must be the cusp of . Now the last part: The tangent at 0 12 is given by

2 22 2 2 2 2 22

au + fu + hu x +du + bu + iu y +eu + gu + cu z = 0 12 0 1 2 0 12 0

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222t

xy zHu u u = :

or equivalently 0120We must show that this equation is

=xy z H H

satisﬁed when P is a column in cof or a row in adj . But from

H H = I H = xy zH = the identity adj det 0wehavethat 0, and the result follows.

222t

2 C F =xy zHx y z

To prove that P just note that and use the same

S P argument. If the tangent at P meets the curve at another point , then (since is

a strange point) this tangent would be a bitangent which is impossible for cubics. P This proves that P is a ﬂex. The tangents at other nonsingular points all contain

so there cannot be more ﬂexes. 2

The set of j -curves (including degenerate curves) is parametrized by P when we to

8 3 3 2

a;b;:::;i ax + by + +iy z a point in P associate the curve with equation . We have seen that the cuspidal cubics are parametrized by a hypersurface of degree

3inP8. PROPOSITION 3.8. The cuspidal cubics with cusp resp. ﬂex on a given line are

parametrized by a 6-fold of degree 3 in P8. =

Proof. Assume the line is given by x 0. By the ﬁrst part of (3.7) we ﬁnd the

+ gi = cf + gh = fi + bh = desired locus to be given by bc 0 which the computer program Macaulay tells us has degree 3 and codimension 2 in P8. The case with

the ﬂex is similar. 2

2 2

Let : P P P be the map given by

a; b; c; d; e; f = ad; be; cf ; ae; af ; bd; bf ; cd; ce; ; 0

j = ( is the Segre embedding on the P given by 0). We claim that the image of is

l m 2 L exactly L, the locus of the nonreduced curves. Indeed, the curve can easily 2

2 22

x ;y ;z ;x ;y ;z l =x ;y ;z 2 be seen to be the image of 0 00111where 000P

2 2 2

=x ;y ;z 2 L ' and m 111P. This shows that we have an isomorphism P P .

2 2

L l 2 Now, T , the locus of the triple lines, is isomorphic to P P corresponds

i T ! L to the triple line l ), and the map : is via these isomorphisms given by 2

l =l ;l l 2 l i (the coordinates of P are the squares of the coordinates of ).

L !

Let j be the embedding P . We record for later use: h LEMMA 3.9. Let h1 and 2 denote the pullbacks of the hyperplane classes on the

2 2 2

' h t T '

factors of L P P , and let and be the hyperplane classes on P and P

h = h + h i h = t i h = t 2 respectively. Then j 1 2, 1 2 and 2 .

com4041.tex; 19/02/1998; 13:54; v.7; p.9 132 ANDERS HØYER BERG

4. The two blow-ups We now follow the strategy of blowing up the parameter space, P9, along nonsingular varieties supported on the intersection of the line conditions, until we have a variety of complete curves. This strategy was successfully employed by Alufﬁ in [1], where ﬁve blow-ups were needed. We shall only need two blow-ups, and the

varieties and maps involved in this process appear in the following diagram

D B

? ?

- - -

F E B

S 1

q p

? ?

2 9

- - -

L :

P T P

i j

By Section 3 we know that L is nonsingular, so that will be the centre for our ﬁrst blow-up.

L N 4.1. The ﬁrst blow-up.LetB1 be the blow-up of P along ,let be the normal

E = N bundle of L in P ,andlet P be the exceptional divisor with maps as

shown in the diagram. ~

9 ~

M H M

If H and are point and line conditions on P , denote by 1 and 1 their prop-

~ ~

H M

er transforms in B1. (We reserve the notation and for the proper transforms

~ ~

H M B in B .) We call 1 and 1 point and line conditions in 1.

To determine the intersection of the line conditions in B 1 we need to examine the tangent hyperplanes of the line conditions in P9.

M l

LEMMA 4.1. Let l be the line condition in P corresponding to the line , and

r 2 M l C l p q r let l be a cubic not containing . Suppose is tangent to at and that

is the other point of intersection.

p 6= q M r X p

(1) If the tangent hyperplane of l at equals the linear span of and

Y X l p Y q

p q q ,where is the cubics tangent to at and is the cubics through and

tangent to l at another point.

p = q M r H p

(2) If the tangent hyperplane of l at equals the point condition .

l x = C l q = ; ;

Proof. Assume that is given by 0, and that r is tangent to at 010

p = ; ; b = c = g =

and also meets l at 001.Thismeansthat 0 in the equation for

C M bc + gi = r . We know from the beginning of Section 3 that l is given by 0, and

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ENUMERATIVE GEOMETRY FOR PLANE CUBIC CURVES IN CHARACTERISTIC 2 133

M C r

a simple computation shows that the tangent hyperplane of l at is given by

g = X b = g = Y c = g = q

0. p 0 and 0 are both contained in this hyperplane,

H b = 2

and (1) follows. (2) follows by a similar argument p is given by 0 .

LEMMA 4.2.

2 T l (1) Suppose x corresponds to the line . The intersection of the tangent

hyperplanes of the line conditions at x is 5-dimensional and consists of the

cubics having l as a component.

2 LnT

(2) If x , the intersection of the tangent hyperplanes of the line conditions

L x at x is 4-dimensional and thus equals the tangent space of at .

l l (3) The tangent space of L at a triple line consists of the cubics having and a touching conic as components. Proof. (1) follows directly from the second part of Lemma 4.1. To prove (2) one

needs to compute (using the ﬁrst part of 4.1) tangent hyperplanes to 5 sufﬁciently

2 LnT l x = general line conditions at a point x . For (3) assume is given by 0.

3 9

; ;::: ; 2 Then the triple line l corresponds to the point 1 0 0P. By (3.4) we know

j = H = that L P is given by 0andrk 1. A simple computation shows that the

l b = c = g = i = j = 2

tangent space of L at is given by 0 and the result follows. B

PROPOSITION 4.3. The intersection S of the line conditions in 1 is a 2-

S = L L

dimensional subvariety of E . More precisely, P ,where is a sub line

bundle of i . E Proof. Obviously, S must be contained in the exceptional divisor . Also, since

the intersection of the line conditions is ‘sufﬁciently transversal’ at points in L

M B outside T (the second part of Lemma 4.2) the line conditions in 1 can only

intersect in the ﬁbres over T . 2

! j i l Let v3: P P be the composition , sending a line to the point corre- 2

sponding to the triple line l . (This is the third Veronese embedding of P projected 2

= v ! l into the hyperplane j 0.) Similarly, we let 2: P P be the map sending to 2 5

l ,whereP parametrizes the conics.

' = Q

We have the following exact sequences of vector bundles on T P P

Q O ! ! v T ! ; 0 !OP2Sym 33P90

! Q O ! v T ! ;

0!OP2Sym 22P50

! i T T ! i N ! : v ! 9

0L 3P0

3 2

Q Q The ﬁrst two are the pullbacks of the Euler sequences on PSym and P Sym ,

and the last is the pullback from L of the standard sequence relating the normal

bundle with the tangent bundles. From the composition

2 2 3

Q O Q Q ! Q; ! Sym 1 Sym Sym

com4041.tex; 19/02/1998; 13:54; v.7; p.11

134 ANDERS HØYER BERG

v T ! v T we get (by tensoring with O 3 ) an induced map 2 P5 3 P9 . In the ﬁber over

2 T l l the image of this map can be identiﬁed with the cubics containing .By

the third part of (4.2) we see that the map f in the third sequence above factors

v T L = v T =i T i N

through 2 P5 . The quotient 2 P5 L is then a sub line bundle of .

= L 2

That S P can now be checked at each ﬁbre using Lemma 4.2 (1).

B S D 4.2. The second blow-up.Let B be the blow-up of 1 along ,andlet be

the exceptional divisor. To show that the intersection of the line conditions on B B is empty, we will introduce local coordinates on 1 and compute the tangent hyperplanes of the line conditions there.

a = U

Let U P be the open set where 1. Afﬁne coordinates for are then

b; c; d; e; f ; g ; h; i; j L

, and afﬁne equations for are

+ df = ; c + eh = ; g + ef = ; i + dh = ; j = :

b 0 0 0 0 0

b; c; d; e; f; g; h; i; j V B We may now choose afﬁne coordinates on an open 1

such that

bj = b + df; d = d; j = j:

gj = g + ef ; e = e;

cj = c + eh; f = f;

ij = i + dh; h = h;

= b; c; g; i

Now j 0 is the exceptional divisor, and are coordinates for a ﬁber in

E l x + y + z = M

.Let be the line given by 0, and let l be the corresponding

9 ~

M V

line condition on P . One may now calculate the equation of l in ,whichturns

out to be

1 0

2 2

j f + h +

C B

= d + b i +

det 0:

A @

e + g + c

= ; ; ; ; ; ; ; ; 2 V

Since O 0 0 0 0 0 0 0 0 0 obviously lies on all the line conditions,

2 S

we have O . We need to show that the intersection of the tangent hyperplanes

S O

of the line conditions at O equals the tangent plane of at . By expanding the

T M O

above determinant we ﬁnd that l is given by

2 2 2 323

j + f + h + d +g+ b+ e + i+ c= :

; 6= ;

The intersection of these planes as and varies 00is given by

=c=f =h=j = ; d=g; e = i; b 0

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ENUMERATIVE GEOMETRY FOR PLANE CUBIC CURVES IN CHARACTERISTIC 2 135

T S

which is 2-dimensional and thus equal to O . This proves that the intersection of

the line conditions on B is empty. In other words

THEOREM 4.4. The variety B , obtained by the sequence of two blow-ups of P ,

is a variety of complete cubics. 2

5. The intersection ring of B The ﬁrst aim of this section is to give a complete multiplication table of the divisor

classes on B . Our main tool will be the following intersection formula.

V l n

LEMMA 5.1. Suppose L are nonsingular varieties of dimensions and .Let

V L E V be the blowup of along , and let be the exceptional divisor with maps as

shown in the diagram

- ~

E V

? ?

V: L

L V sN

Suppose further that N is the normal bundle of in with total Segre class .

> x 2 A V V

Assume 1, and let be the class of a -dimensional cycle on .

Then the following formula holds.

Z Z

E ] x = sNi x:

[ 1

V L

j [E ] = c O sN = p c O

r > E Proof. Recall that 1 E 1 and that 0 1 1 by

deﬁnition. Now we have

Z Z Z

1 1

E ] x = j [E ] j x = c O p i x

[ 1 1

V E E

sN i x

= 1 L

where the last equality is by the projection formula. 2

In view of this lemma, what we need in order to compute intersection numbers on

N N h B 9

are the total Segre classes of the normal bundles and S=B .Let and L=P 1 1

2 2

L ' h2 be the pullbacks of the hyperplane classes of each of the factors of P P ,

S ' and let t be the class of a line on P .

com4041.tex; 19/02/1998; 13:54; v.7; p.13

136 ANDERS HØYER BERG

S B e =[E]

First we will examine the embedding f of in 1.Let be the class of

e f

the exceptional divisor in B1. We would like to know the pullback of by .We

c O E = N c O know that e pulls back to 1 on P 9 andthenalsoto 1

1 L=P 1

S = L L O =L

on P . But since is a line bundle, S 1 . It follows that

f e = c L = c v T =i T =v c T i c T L

1 1 2 P5 L 21P51

v hi h + h = t t = t: = 26313212 9 3 PROPOSITION 5.2.

2 2

N = h h + h + h + h h 9 1 7 7 28 28 59 L=P 1 2 1 2 1 2

2 2 2 2

h h h h + h h ; 276 1 2 276 1 2 1479 1 2

= t + t : sN

S=B1 1 15 120

2 2

r Proof. Recall that L is the image of a Segre embedding, ,ofP P in a hyperplane in P9.Thenwehave

cT +h +h

r P9112

N = = ; c 9

L=P 3 3

cT + h + h 221 1 1 2 PP

N = cN

s 9 9 can now be obtained by expanding and inverting the L= L=P P above expression. The second part is more complicated. The following exact sequence of vector

bundles on E is well known [6, Lem. 15.4]

!p N O ! T ! p T ! :

!O 9 E L 0 E 10 L=P The total Chern class of the tensor product is given by (see [6, Remark 3.2.3])

p N O = cO c N c 9 9

11 i L=

L=P P = i 0

5 4

e + e h + h = 1 17172

3 2 2

e h + h + h h + : +1 21 1 21 2 39 1 2

= T h h t t Restricting to F we get (recall that 1 and 2 pull back to 2 and

respectively)

cT = cq i T cq i N O

k 9 L E 1 L=P

3 3 2 2

= t + t + t e + t te + e + 1 + 2 1 1 21 5 183 84 10

2 2

+ t e + t te + e + ; = 130 5 405 129 10

com4041.tex; 19/02/1998; 13:54; v.7; p.14 ENUMERATIVE GEOMETRY FOR PLANE CUBIC CURVES IN CHARACTERISTIC 2 137

+ t + t e t which pulls back on S to 1 15 108 (remember that pulls back to 3 ). Finally

we have (we omit the pullbacks)

cN cT

E E=B

B1 1

= cN =

S=B1

cT cT

S S

e + t + t

1+ 1 15 108 2

= + t + t

= 3 1 15 105

+ t 1

and the result follows by inverting this expression. 2

= [D ]

Denote by d the class of the exceptional divisor of the second blow-up.

h B [E ] [H ] Also, denote by e and the classes of the pullbacks to of and respec-

tively. We omit pullback and integral signs when no confusion is likely to occur.

d e h

PROPOSITION 5.3. The group of divisor classes on B is generated by , and , and the multiplication table is as follows all other terms are zero

9 9

e ; d = ; = 1479 120

8 8 8

= ; d e = d h = ; he 552 45

2 7 7 2 7 7 2

e = ; d e = d eh = d h = ; h 174 9

3 6

e = ; h 42

4 5 9

e = ; h = : h 6 1

Proof. The ﬁrst assertion follows from the general theory of blowing-up [6, Sect. 6.7], and the numbers can be computed using (5.1) and (5.2) above. For

example

Z Z

2 7 2 7 2 7

e = [H ] [E ] = [H ] [E ]

h 2 1 2 1 B

B 1 Z

sN h + h = 9

L=P 1 2

L Z

2 2 2 2

h + h + h h h + h + h h

= 28 1 28 2 59 1 2 1 2 2 1 2

+ + = ; = 28 28 118 174

2 2 3

h = h = AL 2 where the last ‘integral’ was evaluated using that h 1and 0in . 1 2 i

W B B

Suppose W P is a hypersurface. The proper transforms of in and 1

~ ~ W will be denoted by W and 1 respectively. We shall need a formula for computing

~ 1

W ] 2 A B h e d [ in terms of , and . The following lemma follows directly from

com4041.tex; 19/02/1998; 13:54; v.7; p.15 138 ANDERS HØYER BERG

the general theory of blowing-up [6, Sect. 6.7].

W L m

LEMMA 5.4. Let m1 be the multiplicity of along , and let 2 be the multiplicity

~ S

of W1 along .Then

W ]= W h m e m d [ deg 1 2

B : 2 in A

We shall also need to compute the classes of proper transforms of subvarieties of higher codimension. This can be done with Fulton’s blow-up formula [6, Th. 6.7], but for our purposes it is more convenient to use a different version (Theorem 5.5

below).

L X

Let V be a nonsingular variety, a nonsingular closed subvariety, and let be X

any pure-dimensional subscheme of V . Deﬁne the full intersection class of by V

L in by

L X = cN \ sL \ X; X :

L=V

[X] X V L X = L+i L X

Note that if has codimension 1 in , X where L is the multiplicity of X along (see [1, Sect. 2]). Applied to our ﬁrst blow-up we

have for example

H = h + h ;

L 1 2

M = + h + h :

L 1 2 1 2 2

X = X L L X Another convenient result is that L when both and are nonsingular

[2,Lem.A.1].

V L :

THEOREM 5.5. [1, Thm. II]. Suppose that V is the blowup of along as in 5 1 ,

X ;::: ;X V

and that 1 r are pure-dimensional subschemes of whose codimensions

add to the dimension of V . Then the following formula holds.

Z Z Z

L X

~ ~

[X ] [X ]= [X ] [X ] : r

1 r 1

V V L L=V

Proof. Details may be found in [1]. 2

6. Characteristic numbers of nonsingular cubics N

We will now determine the characteristic numbers ; for the family of all

~ ~

H ] [M ]

nonsingular cubics. By (2.8) this amounts to compute the intersections [

in A . The following lemma is an application of (5.4).

B LEMMA 6.1. In the intersection ring A we have the following relations:

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ENUMERATIVE GEOMETRY FOR PLANE CUBIC CURVES IN CHARACTERISTIC 2 139

H ]=h:

(1) [

M ]= hed: (2) [ 2

Proof. (1) is obvious as L is not contained in any point condition. From (3.1)

+ gi =

we know that bc 0 is the equation for the line condition corresponding to

= M L

the line x 0. It follows that has degree 2 and is generically smooth along .

~ S

We also have that M1 is generically smooth along (in Section 4.2 we computed

S 2 the tangent spaces of M1 at points in ). Hence (2) follows from Lemma 5.4.

By this lemma the characteristic numbers are given by

N = h h e d :

; 2

This can be evaluated by (5.3), and the result is:

THEOREM 6.2. The characteristic numbers for nonsingular cubic curves in char-

acteristic 2 are

N ;N ;::: ;N = ; ; ; ; ; ; ; ; ; ;

; ;

9;0 8109124816 26 34 29 13 2 N

where the last number, 0;9, counts one curve with multiplicity 2, and the other numbers count each curve once.

Proof. We only need to justify the multiplicities. Since by (2.3) all multiplicities

N =

must be powers of 2, the second last number, 1;8 13, must count each curve

with multiplicity one. By the second part of (2.3) so must the 8 preceding numbers.

N C C

We will now show that 0;9 only counts one curve. Assume that 1 and 2 are 2 different nonsingular cubics tangent to 9 given lines in general position. The dual

_ _

curves, C1 and 2 , then contain the 9 corresponding points in P . Since there is

_ _

C = C

only one cubic D passing through the 9 given points, we must have that 1 2 .

N = C C D

Since 0;9 2, 1 and 2 are the only cubic curves having as dual.

3 3 3 2

x + y + z + t xy z = If we assume that the equation of D is in normal form, 0,

3 3 3

+ y + z + txy z =

we have from Section 3 that the curve given by x 0isthe C

only cubic in normal form having D as dual. We may then assume that 1 is in

F x; y ; z =

normal form, and that C2,givenby 0, is not. Obviously, the 6 curves D

given by F and permutations of the variables must all have as dual. It follows

x; y ; z

that these 6 curves must be equal, so we may assume that F is a symmetric

6= polynomial (with d 0)

33322

F x; y ; z =ax +y +z +dx y + + yz +jxyz:

By (3.1) the dual of C2 is given by

2 2 3 3 3 2 22

a + d x + y + z +da +d+jx y + + yz +j xy z :

com4041.tex; 19/02/1998; 13:54; v.7; p.17

140 ANDERS HØYER BERG

+ d = j Since this polynomial is assumed to be in normal form we must have that a .

2 2 2 _ 3 3 3

+ d = j C x + y + z + xy z =

Then a , and the equation for 2 reduces to 0,

_ C which implies that C2 is singular. This is a contradiction since 2 was assumed

nonsingular. 2

Remark.Let :P Pbe the rational map associating to each nonsingu-

M =H l

lar cubic its dual. Then we obviously have that l , but since biduality

does not hold we cannot expect p to be a line condition. In fact, the degree of

H N N

; ; p P must equal 1 8. (The duals of the curves counted by 1 8 is the inter-

section of p and 8 point conditions in P .) Computing the degree of the image of a variety by a rational map can be done with the help of Macaulay. Doing this,

we ﬁnd that p P is given by a polynomial of degree 13 (with 303 terms), N

conﬁrming our computation of 1;8. (By the same argument, Macaulay should

N = N = N =

; ; in principle be able to compute 2 7 29, 3 6 34 and 4;5 26 as well. These numbers are the degrees of the images of lower-dimensional linear spaces. The complexity of these computations seems to be more than our installation of Macaulay can handle within reasonable time.)

7. Characteristic numbers of nodal cubics The computation of the characteristic numbers for nodal cubics is considerably more difﬁcult than in the nonsingular case. Here we will take advantage of Alufﬁ’s results and methods in [1], [2] and [3]; in particular Theorem 5.5 and the results about the full intersection classes. Many of our intermediate results are similar to

Alufﬁ’s and some of the proofs carry over. r

Suppose that R is an -dimensional family of singular curves where the generic l

curve is reduced and irreducible. Denote by R the curves with a singularity on

R p a given line l ,and those with singularity at a given point . The following

deﬁnition will be useful when we consider nodal and cuspidal curves.

+ = r DEFINITION 7.1. Suppose 1. Deﬁne the following numbers associated

to the family R

l l

= R ;

the total characteristic numbers for

;

l l

R : =

N the characteristic numbers for

;

+ = r

Suppose 2. Then deﬁne

= R ;

the total characteristic numbers for

;

= R :

N the characteristic numbers for ;

Note that when the generic curve in R has exactly one singularity we clearly have

p p =

N .

; ;

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ENUMERATIVE GEOMETRY FOR PLANE CUBIC CURVES IN CHARACTERISTIC 2 141

Letting in the above deﬁnition be the nodal cubics, the numbers ; ,

;

N G P

and N are the total characteristic numbers for the three families , and ;

where

= ;

N nodal cubics

l; =

G cubics with singularity on a given line

= p:

P cubics with singularity at a given point l

Also, let F be the nodal cubics properly tangent to a given line . The characteristic N

numbers ; follow from the total characteristic numbers by the following lemma. (Compare with [3, Thm. I].)

LEMMA 7.2. We have the following relations between the characteristic numbers

G P

and the total characteristic numbers for the families N , and

= N + N + N ;

; ;

;

1 2 ; 2

l l

= N + N :

; ;

; 1

[N \ M ] = [F ]+[G] N

Proof. We will prove that l ,inotherwordsthat and

M x N \ M l

l intersect transversally. Let be a general point of . All we need to

T N T M

x x

show is that the tangent hyperplanes and l are different. It is sufﬁcient to

l C T M l

x x

show this for a general line tangent to . But (4.1) tells us that l varies as

T M 6= T N

x x varies, so in general we must have that l . The lemma now follows by

arguments similar to those following Lemma 1.3 in [3]. 2 M

Remark. That N and intersect transversally is special for characteristic 2. In all

[N \ M ]=[F]+ [G]

other characteristics we have l 2as shown in [3, Lem. 1.3]. The rest of Alufﬁ’s proof is characteristic free. The only difference is that the multiplicity 2 appears as a coefﬁcient in the formulas.

9 L Consider the following blow-up diagram where B1 is the blowup of P along

as before.

2 - 2

E B

P P 1

g f

? ?

2 - 2 9

L :

P P P i

2 9

Let D P P be given by the vanishing of the three partial derivatives of

3 3

F = ax + by + + jxyz p; t 2 D C

. This means that if and only if t is

com4041.tex; 19/02/1998; 13:54; v.7; p.19 142 ANDERS HØYER BERG

2 k singular at p.Let be the class of a hyperplane in P .

L D LEMMA 7.3. The full intersection class P regarded as a class in

L k P is a quadratic polynomial in , and the coefﬁcients are given by

N = k ;

L coefﬁcient of

G = k;

L coefﬁcient of

P = : L the constant term Proof. This follows from the birational invariance of Segre classes [5, Sect. 4.2].

See [2, Prop. 2.1] and [3, Lem. 2.2] for details of the proofs of similar results. 2

P L

PROPOSITION 7.4. The full intersection classes of N; G and by are

N = + h + h ; L 8 12 1 2

+ h + h + h + h ; G = L 2 8 1 6 2 6 1 2

2 3

P = h + h + h h +h +h : L 1 2 1 3 1 2 12

2 2

L D W = L\D Proof. By (7.3) this amounts to computing P .Let P. (That the intersection is transversal can easily be checked by pulling the equations

2 2 2

L W = fp; l m p 2 l g L for D back to P .) This means that : P,sothat

W ]=h + k 2A L p W L

[ 12P. Letting be the inclusion of in P we have

+ k

2 1h12

: p sW; L=p =

N + h + k

c 2 1 2

L W=P 1

2 9

cN =

Since D P P is given by 3 equations of bidegrees (2,1) we have 29 D=PP

32 3

+h+ k L + h + h + k 12which pulls back on P to 1 1 2 2 .

2 i 2 9

P P P

6 6

W D: As all the varieties in the above diagram are nonsingular, the full intersection classes

2 L commute, and we get (as classes in P )

p [ L D ]

2 2

p [D L] = p [cN sW; L]

= 29

P P D=PP

+h +h + k h + k 122

2 121

p sW; Li cN = = 29

D=PP

h + k 1+ 12

com4041.tex; 19/02/1998; 13:54; v.7; p.20 ENUMERATIVE GEOMETRY FOR PLANE CUBIC CURVES IN CHARACTERISTIC 2 143

and the result follows by expanding the last expression. 2

N = + h + h m N L m

Since L 8 12 1 2 , the multiplicity, 1,of along is 8. Let 2

~ ~

S [N ]= h e m d

denote the multiplicity of N1 along . By (5.4) we have that 12 8 2 .

To determine 2 we will compute 0;8 in two different ways. Since the dual of a nodal cubic is a conic, no nodal cubic can be tangent to more than 5 given lines in

general position. Also, at most two of the lines can pass through the node, so we

see that 0;8 0. On the other hand by (2.8) and (5.3) we have Z

= h e m d h e d = m ;

0;8 12 8 2 2 60 12 2

B = so we must have m2 5.

THEOREM 7.5. The following table gives the complete list of characteristic num-

G P

bers and total characteristic numbers for the families N , and

l l

; N N N

; ;

; ; 1 1 ; 2

8 ; 01212

7; 124186 6

6; 2482512111

5; 3963024202

4; 4 144 24 36 24 4

3; 5 168 8 42 22 5

2; 6 123 0 33 8 5

1; 7420 120 2

0; 80 0 0 0 0

Proof. The total characteristic numbers for N are given by

~ ~ ~

= [N ][H ] [M ] = h h e d h e d;

; 2 12 8 5 B

which can be evaluated by (5.3).

N 6

The numbers and with 4 follow by applying (5.5) and (7.4) to

; ;

the ﬁrst blow-up. For example

Z Z

~ ~ ~ ~ ~

~ 3 4 3 4

= [G][H ] [M ] = [G ][H ] [M ]

3;4 1 1 1 B B 1

since the intersection of 3 general point conditions in

B1 does not meet

Z Z

3 4 3 4

[G][H ] [M ] L GL H L M sN = 9 9 L=P

P L

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+ h + h + h + h = 96 2 8 1 6 2 1 2

+ h + h h h + 1 2 1 2 2 1 7 1 7 2

h + h + h + h + = = : = 96 1 2 2 10 1 8 2 96 54 42 All the zeros follow from the fact (3.2) that the dual of a nodal cubic is a nonsingular conic, so that a nodal cubic can be properly tangent to at most 5 lines in general position. The remaining numbers can now be computed using the relations

in (7.2) 2

Remark. Some of the arguments above (in particular the ‘zero-arguments’) could

have been replaced by the computation of the full intersection classes

N = + t;

S 1 5 12

~ 2

= + t + t G

S 1 1 7 6

~ 2

P = t + t ; S 1 2 combined with another application of Theorem 5.5. See [2, Sect. 2.1] for a similar computation in characteristic 0.

8. Other characteristic numbers

In Section 3 we mentioned the very special class of curves called j -curves. These

curves are parametrized by an open subset of the hypersurface J P given by = j 0. The computation of the characteristic numbers for this family is now an

easy consequence of our previous results.

J ]=he B

LEMMA 8.1 [ as a class in the intersection ring of .

= L J

Proof. We know that deg J 1, and that with multiplicity one, so by

~ J (5.4) we only need to prove that S is not contained in 1. From Section 4 we know

= L L l

that S P where the ﬁber of over a triple line can be identiﬁed with

T M =T L \ T M

\ 3 3 3 l

l .Now,by(4.2), is just the 5-dimensional space of cubics

l l l

\ T M 6 J S 6 J 2 l 3 containing the line . Obviously, l , and it follows that . l 1

THEOREM 8.2. The characteristic numbers for the family, J , of cubics with

j -invariant 0 are

N ;N ;::: ;N = ; ; ; ; ; ; ; ;

; ; 8;0 7108124810 8 4 2 1

and all the numbers above count curves with multiplicity 1.

R = J N = h h e d h e

Proof. We apply (2.8) with to obtain ; 2

N =

which can be evaluated by (5.3). Since the last number, 0;8 1, clearly counts

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curves with multiplicity 1, it follows from (2.3) that all the other numbers will also

count curves with multiplicity 1. 2

Remark. The symmetry of the numbers in (8.2) reﬂects the fact that the dual

_ _

j C = C j C of a j -curve is also a -curve, and that for a -curve . This is similar to the case of smooth conics and cuspidal cubics in characteristic 0. We now proceed to the characteristic numbers for families of cuspidal cubics.

K; G P k

Let k and be the subvarieties of P deﬁned by

= ;

K cuspidal cubics

l G =

k cubics with cusp on a given line

P = p:

k cubics with cusp at a given point

LEMMA 8.3. We have the following relations in the intersection ring of B

~ ~ ~

N ][J ] = [K ];

[ 4

~ ~ ~

[P ][J ] = [P ]:

k J

Proof. We know from Section 3 that K has degree 3 and is contained in .From

K L

the equation for K we can easily check that is singular along of multiplicity

~ ~

K ]= h e AJ

2. It follows that [ 32regarded as a class in (we omit the pullback

\ J = ; i d =

signs). Note that the proof of (8.1) also shows that S 1 ,sothat 0,

J B AB

where i is the inclusion . Now, we have (as classes in )

~ ~ ~ ~

[N ][J ]=i i [N]=i i h e d=i h e= [K]:

12 8 5 12 8 4

~ ~

~ 9

[P ][J ] = [P ] P J

That k is clear since and are both linear subspaces of P inter-

P 2

secting transversally along k .

= K

We now use the notation from Deﬁnition 7.1 with R , the cuspidal cubics.

N N N

This means that ; , and are the characteristic numbers for the three

;

K G P

k ;

families , k and ,and , are corresponding total characteristic num-

;

F l

bers. Also, let k be the cuspidal cubics properly tangent to a given line .From F

(3.7) we know that k can also be described as the cuspidal cubics with a ﬂex on

the given line l . The following lemma is the cuspidal version of Lemma 7.2.

LEMMA 8.4. We have the following relations between the characteristic numbers

K G P k

and the total characteristic numbers for the families , k and

= N + N + N ;

; ;

;

1 2 ; 2

l l

= N + N :

; ; ; 1

com4041.tex; 19/02/1998; 13:54; v.7; p.23 146 ANDERS HØYER BERG

Proof. This is similar to the proof of (7.2). All we need to show is that

[K \ M ] = [F ]+[G ] K = M =

k k l

l . Recall that deg 3 and deg 2. From

G F K M

k l (3.8) we have that k and both have degree 3, and it follows that and

intersect transversally. 2

Before we prove our ﬁnal result, we note that

L J = L+j [J]= +h +h ;

J 112

~ ~

SJ = S+f [J ]=f he= t t= ; 1~ 1330

S B S J = where f is the embedding of in 1. The last relation, 1 0, basically tells

us that the second blow-up is superﬂuous when our family is contained in J .

THEOREM 8.5. The following table gives the complete list of characteristic num-

K G P k bers and total characteristic numbers for the families , k and , and all the

numbers count curves with multiplicity 1.

l l

; N N N

; ;

; ; 1 1 ; 2

7 ; 03 3

6; 16 3 3 3

5; 2121 6 5 1

4; 3120 6 2 2

3; 46 0 3 0 1

2; 50 0 0 0 0

1; 60 0 0 0 0

0; 70 0 0 0 0

K P

Proof. The total characteristic numbers for the families and k follow from

J = (5.5) and (8.3). Recall that S 1 0 so that we only need to apply (5.5) on the

ﬁrst blow-up. For example Z

Z 1

~ ~ ~ ~ ~ ~

~ 3 4 3 4

= [K ][H ] [M ] = [N ][J ][H ] [M ]

3;4 B

B 4

1 3 4

= [N ][J ][H ] [M ]

4 P9

3 4

L M sN L N L J L H 9

L=P L

= = :

4 192 168 6

l l

G = =

Since k has degree 3 we clearly have 3and 6. Also, since a cuspidal ; 6 0 5;1 cubic is strange, it can not be properly tangent to 3 general lines. It follows that

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ENUMERATIVE GEOMETRY FOR PLANE CUBIC CURVES IN CHARACTERISTIC 2 147

N = >

; 0when 3. The remaining numbers in the above table now follow by

Lemma 8.4. N

It remains to prove that the multiplicities are 1. By (2.3) this holds for ; and

l N

N . If we can show that counts two different curves, the result will be true ;

; 4 2

l l

for andthenby(8.4)alsofor ; and .

; ; l

N counts the curves passing through 4 given points, with the ﬂex at another

4;2 l

given point p (the intersection of the two lines) and with cusp on a given line .The p

curves counted by N have the same description with cusp and ﬂex interchanged.

4;1

= ; ; l z = C N Suppose p 001and is given by 0. Let be a curve counted by .

4;1

t t

C H

Let H be its matrix and let be the cuspidal cubic given by the transpose .

t t t

H = H C p

Since cof cof it follows from (3.7) that has a ﬂex at and a cusp

t l l

C N N N

on l .So is counted by .Since counts different curves then also

; ; 4 2 4;1 4 2

must count different curves. 2

Acknowledgements Some of the material in this paper is part of the authors cand. scient. thesis written under the guidance of Ragni Piene. It is a pleasure to thank Ragni Piene for proposing the problem and for many helpful suggestions. I would also like to thank Carel Faber, William Fulton and Steve Kleiman for helpful and inspiring conversations. The Maple package Schubert (by S. Katz and S. A. Strømme) has been of some help with the computations.

References 1. Aluffi, P.: The enumerative geometry of plane cubics I: Smooth cubics, Transactions of the AMS 317 (1990) 501–539. 2. Aluffi, P.: The enumerative geometry of plane cubics II: Nodal and cuspidal cubics, Math. Ann. 289 (1991) 543–572. 3. Aluffi, P.: Some characteristic numbers for nodal and cuspidal plane curves of any degree, Manuscripta Mathematica 72 (1991) 425–444. 4. Brieskorn, E. and Knorrer,¨ H.: Plane Algebraic Curves,Birkhauser,¨ 1986. 5. Fulton, W., Kleiman, S. and MacPherson, R.: About the enumeration of contacts, Lecture Notes in Mathematics 997, (1983) 156–196. 6. Fulton, W.: Intersection Theory, Springer, 1984. 7. Hartshorne, R.: Algebraic Geometry, Springer, 1977. 8. Kleiman, S. and Speiser, R.: Enumerative geometry of cuspidal plane cubics, Canad. Math. Soc. Conf. Proc. 6 (1986) 227–268. 9. Kleiman, S. and Speiser, R.: Enumerative geometry of nodal plane cubics, Lecture Notes in Mathematics 1311, (1988) 156–196. 10. Kleiman, S. and Speiser, R.: Enumerative geometry of nonsingular plane cubics, Cont. Math. 116 (1991) 85–113. 11. Vainsencher, I.: Conics in Characteristic 2, Compositio Mathematica 36 (1978) 101–112. 12. Vainsencher, I.: Schubert Calculus for Complete Quadrics, Progress in Mathematics 24 (1982) 199–235.

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