The number of plane conics 5-fold tangent to a given curve

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by  Andreas Gathmann

THE NUMBER OF PLANE CONICS 5-FOLD TANGENT TO A
GIVEN CURVE
2
0
0
ANDREAS GATHMANN
2

n
a Abstract. GivenageneralplanecurveY ofdegreed,wecomputethenumber
J nd ofirreducibleplaneconics thatare5-foldtangent toY. Thisproblemhas
beenstudiedbeforebyVainsencher[V]usingclassicalmethods,butitcouldnot
1
be solved there because the calculations received too many non-enumerative
3
correction terms that could not be analyzed. In our current approach, we

] expressthenumberndintermsofrelativeGromov-Witteninvariantsthatcan
G then be directly computed. As an application, we consider the K3 surface
given as the double cover of P2 branched along a sextic curve. We compute
A
thenumberofrationalcurvesinthisK3surfaceinthehomologyclassthatis
. the pull-back of conics in P2, and compare this number to the corresponding
h
Yau-Zaslow K3invariant. This gives anexampleofsuch aK3invariantfora
t
a non-primitivehomologyclass.
m
[
Let Y ⊂ P2 be a generic plane curve of degree d ≥ 5. We want to consider
1
smooth plane conics that are 5-fold tangent to Y. As the space of all plane conics
v
2 is5-dimensionalandeachtangencyimposesoneconditiononthecurves,weexpect
0 afinitenumberofsuch5-foldtangentconics. Itwillbeeasytoseethatthisnumber
0 is indeed finite; let us call it n . The goal of this paper is to compute it.
d
2
Of course this is a classical problem, and attempts have been made to solve it
0
using classical methods of enumerative geometry. I. Vainsencher [V] tried to use
2
0 variousblow-upsofthe ordinaryP5 ofconicsasmodulispaces,buttheintersection
/ ofthefivetangencyconditionsinthismodulispacealwaysresultedinaschemewith
h
manynon-enumerativecomponents,mostofwhichhadtoobigdimension,werenon-
t
a reduced or even had embedded components. Their geometry was so complicated
m
that the problem could not be solved that way.
: In this paper we use different moduli spaces, namely moduli spaces of relative
v
i stable maps, to solve the problem. There is a well-defined compact moduli space
X M¯Y (P2,2) ⊂ M¯ (P2,2) that parametrizes rational stable maps to P2 of
(2,2,2,2,2) 0,5
r
degree 2 (i.e. conics) with 5 marked points such that the stable map is tangent to
a
Y at all these points. It comes equipped with a 0-dimensional virtualfundamental
class, whose degree N can be computed explicitly using the methods of [Ga1].
d
We can interpret the number N as the “virtual number” of conics that are 5-
d
fold tangent to Y. It is only virtual because it contains — just as in Vainsencher’s
classical computations — non-enumerative contributions from the “boundary” of
the moduli space. These contributions are quite simple however. It is not hard
to see that the only degree-2 rational stable maps f : C → P2 that satisfy the
tangency conditions at the 5 marked points are all double covers of a bitangent of
Y, and have the marked points distributed in one of the following two ways:
1
2 ANDREAS GATHMANN
C C
f f
bitangent bitangent
Y Y
There are only finitely many stable maps with marked points as in this picture on
the left, so we just have to count them and subtract their number from the virtual
invariant N . The picture on the right however shows a 1-dimensional family of
d
stablemaps(theunmarkedramificationpointoff canmove). Wewilluseequations
from relative Gromov-Witten theory to compute the degree of the 0-dimensional
virtual fundamental class of the moduli space on this 1-dimensional component.
Bysubtractingbothcorrectionterms,wefinallyarriveattheenumerativenumbers
n . They are
d
1
n = d(d−3)(d−4)(d7+12d6−18d5−540d4+251d3+5712d2−1458d−14580).
d
5!
Our (as well as Vainsencher’s) motivation for studying this problem came from
a question concerning rational curves in K3 surfaces. If X is a K3 surface and
β ∈H (X,Z)thehomologyclassofaholomorphiccurveinX,thenvariousauthors
2
[YZ], [B], [G], [BL] have shown that the number of rational curves in X of class β
is equal to the qd coefficient of the series
1
G(q)= =1+24q+324q2+3200q3+25650q4+176256q5+···
(1−qi)24
iY>0
with d = 1β2 +1, if the class β is primitive, i.e. not a non-trivial multiple of a
2
smaller integral homology class. There is a well-defined “K3 invariant” (using a
modified obstruction theory on the moduli spaces of stable maps to X) for non-
primitive β too [BL]; it is however not known yet how this invariant relates to the
above series G(q) or to the enumerative number.
Theresultofthispaperallowsustostudythisquestioninanon-trivialexample:
we let X be the double cover of P2 branched along a sextic curve Y, and take β
to be the pull-back of the class of conics in P2. Our work allows us to compute
the enumerative number of rational curves in X of class β, which we can then
compare to the corresponding number of the series G(q). The result is that the
“K3invariant”is equal to the correspondingtermin the G(q) series,plus a double
covercorrectiontermthatisequalto 1 timesthenumberofrationalcurvesinX of
8
class 1β. Notethatthisisthesamesortofcorrectiontermasformultiplecoversof
2
rationalcurvesinCalabi-Yauthreefolds. Weconjecturethatthis patterncontinues
for classes β of higher divisibility.
The paper is organized as follows. In section 1 we will show how to com-
pute the relative Gromov-Witten invariant N . We analyze the moduli space
d
M¯Y (P2,2)anditsvirtualfundamentalclassinsections2and3,respectively,
(2,2,2,2,2)
leadingtothe finalresultforn incorollary3.6. Section4containsthe application
d
to K3 surfaces mentioned above.
We thank J. Kock for introducing us to the problem, as well as for a Maple
program to compute the relative Gromov-Witten invariants using the algorithm
of [Ga1]. The author is also grateful to the Institute for Advanced Study for its
THE NUMBER OF PLANE CONICS 5-FOLD TANGENT TO A GIVEN CURVE 3
hospitalityandstimulatingworkingatmosphere. Thisworkhasbeenfundedbythe
NSF grant DMS 9729992.
1. Relative Gromov-Witten invariants
In this section we will show how to compute the relative Gromov-Witteninvari-
ant that corresponds to the number of conics that are 5-fold tangent to a given
smoothplane curve. We will use the notations andresults from[Ga1], to whichwe
also refer for further details.
Definition 1.1. Let Y ⊂ P2 be a smooth curve of degree d, and let m ,...,m
1 n
be non-negative integers. We denote by M¯ =M¯Y (P2,2) the mod-
(m1,...,mn) (m1,...,mn)
uli space of n-pointed relative stable maps of degree 2 to P2 relative to Y with
multiplicities m ,...,m , as defined in [Ga1] definitions 1.1 and 1.18. Its virtual
1 n
fundamental class is denoted [M¯ ]virt.
(m1,...,mn)
Remark 1.2. The moduli space M¯ can be thought of as a compactifi-
(m1,...,mn)
cation of the space of irreducible plane conics together with n distinct marked
points on them at which the conic has the prescribed local intersection multiplici-
ties m ,...,m with Y. In particular,the moduli space M¯ corresponds to
1 n (2,2,2,2,2)
conics 5-fold tangent to Y.
Remark 1.3. For future reference let us recall the precise definition from [Ga1].
Consider the degree-d Veronese embedding i : P2 → PD with D = D+2 −1. We
2
have i(Y) = i(P2)∩H in PD for a suitable hyperplane H ⊂ PD.(cid:0)The(cid:1)inclusion
morphism i induces an inclusion of moduli spaces M¯ (P2,2) ⊂ M¯ (PD,2d).
0,n 0,n
Moreover,letM¯H (PD,2d)betheclosureinM¯ (PD,2d)ofallstablemaps
(m1,...,mn) 0,n
(C,x ,...,x ,f) such that C is irreducible, f(C)6⊂ H, and the divisor f∗Y on C
1 n
contains the points x with multiplicities m . Then we define
i i
M¯ :=M¯ (P2,2)∩M¯H (PD,2d)
(m1,...,mn) 0,n (m1,...,mn)
(withthe intersectiontakeninM¯ (PD,2d)). The virtualfundamentalclassis the
0,n
corresponding refined intersection product
[M¯ ]virt :=[M¯ (P2,2)]·[M¯H (PD,2d)]∈A (M¯ ).
(m1,...,mn) 0,n (m1,...,mn) ∗ (m1,...,mn)
The virtual dimension of M¯ is 5− (m −1). By abuse of notation, we
(m1,...,mn) i i
will always drop the superscript virt from thPe notation of the virtual fundamental
class from now on, as we do not need the ordinary fundamental classes of these
spaces.
Remark 1.4. From remark 1.3 we get immediately the following statement: let
C =(C,x ,...,x ,f)∈M¯ beanautomorphism-freestablemapsuchthat
1 n (m1,...,mn)
C ∼=P1 is irreducible and f(C)6⊂Y. Then, locally around this point, M¯
(m1,...,mn)
is scheme-theoretically the subscheme of M¯ (P2,2) given by the m equa-
0,n i i
tions that describe the vanishing of the mi-jets of evi∗Y at the poinPts xi, where
ev : M¯ (P2,2) → P2 are the evaluation maps. If moreover M¯ has the
i 0,n (m1,...,mn)
expected dimension at this point C, then this point lies on a unique irreducible
component of M¯ , and the virtual fundamental class on this component is
(m1,...,mn)
just the ordinary scheme-theoretic fundamental class, i.e. the length of the scheme
M¯ at this irreducible component.
(m1,...,mn)
4 ANDREAS GATHMANN
Remark 1.5. ThereisaneasierdescriptionofM¯ asaset. Namely,M¯
(m1,...,mn) (m1,...,mn)
isthesubspaceofM¯ (P2,2)ofalln-pointedrationalstablemaps(C,x ,...,x ,f)
0,n 1 n
of degree 2 to P2 such that the following two conditions are satisfied:
(i) f(x )∈Y for all i such that m >0,
i i
(ii) f∗Y − m x ∈A (f−1(Y)) is effective.
i i i 0
As mentioPned above, the moduli space M¯ has virtual dimension zero
(2,2,2,2,2)
and corresponds to conics 5-fold tangent to Y (together with a labeling of the five
tangency points). Hence we define:
Definition 1.6. The number
1
N := ·deg[M¯ ]∈Q
d 5! (2,2,2,2,2)
will be called the virtual number of conics 5-fold tangent to Y.
The number N is only virtual because it receives correction terms from double
d
covers of lines (see sections 2 and 3). In the rest of this section we will show how
to compute the number N . Obviously, we can assume that d≥5.
d
The computationis doneusing the maintheorem2.6of[Ga1] thattells us “how
to raise the multiplicities of the moduli spaces”: it says that
(ev∗Y +m ψ )·[M¯ ]=[M¯ ]+correction terms, (1)
n n n (m1,...,mn) (m1,...,mn−1,mn+1)
where ev : M¯ → P2 is the evaluation map at the last marked point,
n (m1,...,mn)
and ψ is the first Chern class of the cotangent line bundle L , i.e. of the bundle
n n
whose fiber at a stable map (C,x ,...,x ,f) is the cotangent space T∨ . The
1 n C,xn
correctiontermsareasfollows. Everycorrectiontermcorrespondstoamodulispace
of reducible curves with r+1 components C ,...,C , where C is contracted1 to
0 r 0
a pointofY, andthe other components C intersectC in a pointwhere they have
i 0
local intersection multiplicity µ to Y. We get such a correction term for every
i
r, every choice of the µ , and every splitting of the total homology class and the
i
markedpoints onto the components C , such that the following two conditions are
i
satisfied:
(a) the last marked point x lies on the component C ,
n 0
(b) the sum of all the µ is equal to the sum of those m such that x ∈C .
i i i 0
These correction terms appear in the above equation with multiplicity r µ .
i=1 i
Q
Example 1.7. Here is an example of equation (1). In the case
(ev∗Y +ψ )·[M¯ ]=[M¯ ]+correction terms
5 5 (2,2,2,2,1) (2,2,2,2,2)
we want to figure out the correction terms. As this is an equation in (virtual)
dimension 0, the contracted component C must have exactly 3 special points (it
0
wouldnotbestableifithadfewer,anditwouldhavemoduliifithadmore). Hence
the correctionterms fall into these two categories:
(i) r =1(intheabovenotation),C isaconic,andC isacontractedcomponent
1 0
with three special points x , the intersectionpoint with C , andone other x
5 1 i
for i=1,...,4,
(ii) r = 2, C and C two lines, and C is a contracted component with three
1 2 0
special points x and the two intersection points with C and C .
5 1 2
1ThisusesthefactthatthecurveY haspositivegenus,andthatthereforeeveryrationalstable
maptoY mustbeconstant. Ingeneral,C0 canbeacurvewithanyhomologyclassinY.
THE NUMBER OF PLANE CONICS 5-FOLD TANGENT TO A GIVEN CURVE 5
Actually, case (ii) cannot occur, because condition (b) above cannot be satisfied:
thesumµ +µ isatleast2,whereasm isonly1. Hencetheonlycorrectionterms
1 2 5
are of type (i). We get four of them: one for each choice of the point x that is to
i
lie on the contracted component C . We have µ = m +m = 3 in each of these
0 1 i 5
cases by condition (b). All four correction terms appear with multiplicity µ = 3.
1
Pictorially, the equation reads
(ev5*Y+ψ5). x1x2 x3 x4 x5 C = x1x2 x3 x4 x5 +3. x2x3x4 xx15
Y
x5 x5 x5
+3. x1x3x4 x2 +3. x1x2 x4 x3 +3. x1x2 x3 x4
Here, the dotted curve is the fixed curve Y, and the solid curve is the moving
conic C. In the four correction terms, the component with x on it is meant to be
5
contracted. Written down as an equation of virtual fundamental classes of moduli
spaces, the formula reads
(ev∗Y +ψ )·[M¯ ]=[M¯ ]+3[M¯ ] (2)
5 5 (2,2,2,2,1) (2,2,2,2,2) (3,2,2,2)
+3[M¯ ]+3[M¯ ]+3[M¯ ].
(2,3,2,2) (2,2,3,2) (2,2,2,3)
Remark 1.8. Thecorrectiontermsinequation(1)arethemselvesproductsofmod-
uli spaces of relative stable maps of the form M¯Y (P2,d′) with d′ ≤ 2 and
(m′1,...,m′k)
m′ +···m′ ≤ m +···m . In other words, this equation expresses invariants
1 k 1 n
(i.e. intersection products of ev∗H and ψ classes, where H is a line in P2) on the
i i
relative moduli space M¯ in terms of other invariants on relative mod-
(m1,...,mn+1)
uli spaces M¯Y (P2,d′) whose “total multiplicity” m′ is smaller than the
(m′1,...,m′k) i i
total multiplicity 1+ m of M . Hence, apPplying equation (1) recur-
i i (m1,...,mn+1)
sivelym1+···+mn tPimes, we canexpresseveryinvarianton M¯(m1,...,mn) in terms
ofinvariants onM¯ , which are just ordinaryGromov-Witteninvariants ofP2.
(0,...,0)
As the Gromov-Witten invariants of P2 are well-known, we can thus compute all
relativeGromov-Witteninvariantsrecursively,inparticularN . Example1.7isthe
d
firststep in this recursionprocess;it expressesthe invariantM¯ (with total
(2,2,2,2,2)
multiplicity 10) in terms of invariants with total multiplicity 9.
Without actually carrying out the recursion, we can see the following.
Lemma 1.9. The function d7→N is a polynomial of degree 10 with leading coef-
d
ficient 1.
5!
Proof. Using equation (1) it is easy to show by induction that every invariant (i.e.
intersection product of ev∗H and ψ classes) on a moduli space M¯ is a
i i (m1,...,mn)
polynomial in d of degree (at most) m +···+m . In fact, this is obvious for
1 n
m +···+m =0, as we then just have ordinary Gromov-Witten invariants of P2
1 n
(that do not depend on Y). Equation (1) reads
[M¯ ]=(dev∗H +m ψ )·[M¯ ]−correction terms.
(m1,...,mn−1,mn+1) n n n (m1,...,mn)
Allcorrectionterms havetotalmultiplicity atmostm +···+m ,so by induction
1 n
hypothesistheycontributeapolynomialindofdegreeatmostm +···+m . The
1 n
sameis true for the ψ summand onthe righthand side. Hence, as everyinvariant
n
6 ANDREAS GATHMANN
onM¯ isapolynomialindofdegreeatmostm +···+m byassumption,
(m1,...,mn) 1 n
it follows that every invariant on M¯ is a polynomial in d of degree at
(m1,...,mn+1)
most m +···+m +1.
1 n
It can be seen from the same recursive formula that the d10 coefficient of the
invariant deg[M¯ ] is just
(2,2,2,2,2)
5
ev∗H2·[M¯ (P2,2)],
i 0,5
iY=1
i.e. the number of conics through 5 general points in the plane. This number is 1,
proving the statement of the lemma about the leading coefficient.
The precise form of the polynomial N is quite complicated and can only be
d
obtained by carrying out the full recursion as described above. We only give the
result here; a Maple program to compute it can be obtained from the author on
request.
Proposition 1.10. For d≥5, the virtual number of conics 5-fold tangent to Y is
1
N = d(d−3)(d−4)(d7+12d6−18d5−540d4+311d3+5457d2−2133d−12690).
d
5!
Remark 1.11. The first few values of N are given in the following table.
d
d 5 6 7 8 9 10
N 1985 71442 687897 3893256 16180398 54679380
d
Remark 1.12. In this section we have only used equation (1) in the Chow ring of
the moduli space of stable maps M¯ (P2,2). In fact, there is a refined version of
0,n
this equation that we will need in section 3. If P denotes the rank-(k+1) bundle
k
of (relative) k-jets of ev∗O(Y), there is a section σ of the line bundle P /P
n mn mn−1
on M¯ whose vanishing describes precisely the condition that the map f
(m1,...,mn)
of a stable map (C,x ,...,x ,f) has multiplicity (at least) m +1 to Y at x .
1 n n n
The first Chern class of this line bundle is ev∗Y +m ψ . (In fact, this is the idea
n n n
how equation (1) is proven.) The refined version of equation (1) now states that
this equation also holds in the Chow group of the zero locus of the section σ on
M¯ .
(m1,...,mn)
2. The components of M¯
(2,2,2,2,2)
HavingjustcomputedtheinvariantN ,wewillnowstudyitsenumerativesignif-
d
icance. To do this, we have to identify the components of M¯ and compute
(2,2,2,2,2)
their virtual fundamental classes. We will assume from now on that Y is generic
of degree d ≥ 5. The equation of Y is F = a zI = 0, where I runs over all
I I
multi-indices (i0,i1,i2) with i0 +i1 +i2 = d,Pand z0,z1,z2 are the homogeneous
coordinateson P2. We start with irreducible stable maps whose image is a smooth
conic.
Lemma 2.1. Let m ,...,m be non-negative integers such that m ≤ 10, and
1 n i
let C ∈M¯(m1,...,mn) be an irreducible stable map that is not a doublePcover of a line.
Then:
(i) The moduli space M¯ is smooth of dimension 5− (m −1) at C
(m1,...,mn) i i
(which is the expected dimension). P
THE NUMBER OF PLANE CONICS 5-FOLD TANGENT TO A GIVEN CURVE 7
(ii) If this expected dimension is negative, then there is no such point C.
Proof. The plane degree-d curves are parametrized by a projective space PD with
D = 1d(d+3),whosecoordinatesarethecoefficientsa ofF. LetZ ⊂M¯ (P2,2)×
2 I 0,n
PD be the closed substack of pairs ((C,x ,...,x ,f),Y) such that the pull-back
1 n
by f of the equation of Y vanishes at the points x to order m for all i. We claim
i i
that Z is smooth of the expected dimension at every point ((C,x ,...,x ,f),Y)
1 n
such that C is irreducible and f is not a double cover of a line.
To prove this, we have to show that the matrix of derivatives of the equations
describing Z has maximal rank at the given point ((C,x ,...,x ,f),Y). By a
1 n
projective coordinate transformation of P2 and choosing homogeneous coordinates
on C ∼= P1, we can assume that the map f is given by (s : t) 7→ (s2 : st : t2), and
the n marked points are (1:λ ) with pairwise distinct λ .
i i
Let us now write downthe derivatives ofthe multiplicity equations with respect
to the first m := m of the variables a , a , a , a ,
i i (d,0,0) (d−1,1,0) (d−1,0,1) (d−2,1,1)
a(d−2,0,2), a(d−3,1,2)P, a(d−3,0,3), a(d−4,1,3), a(d−4,0,4), a(d−5,1,4) (remember m ≤ 10
and d ≥ 5). These coordinates are chosen to be the coefficients of s2d−iti for
i=0,...,9 when we substitute the map f into F.
Multiplicity m at the point (s : t) = (1 : λ ) means that F| has
i i s=1,t=λi+ǫ
no ǫ terms of order less than m . So the rows of the derivative matrix are just
i
( i λi−k) , for 0≤k <m and 1≤j ≤n. For example, for m = m =
k j i=0,...,m−1 j 1 2
m(cid:0)3(cid:1)=m4 =m5 =2 we get the matrix
1 λ λ2 ··· λ9
1 1 1
 0 1 2λ1 ··· 9λ81 
. . . . .
. . . . . .
 . . . . . 
 
 1 λ λ2 ··· λ9 
 5 5 5 
 0 1 2λ ··· 9λ8 
 5 5 
Bysubtractingλ timesthei-thcolumnfromthe(i+1)-stcolumnfor1≤i<mand
1
using induction, we see that the determinant is i<j(λi−λj)mimj. In particular,
it is not zero, so Z is smooth of the expected dimQension at ((C,x1,...,xn,f),Y).
By remark 1.4, the statement of the lemma is just that the fiber of Z over a
generalpointofPD issmoothoftheexpecteddimensionaroundapointconsidered
above. This follows now from the Bertini theorem.
Using remark 1.4 again, the following two corollaries are immediate.
Corollary 2.2. Every irreducible stable map in M¯ whose image in P2 is
(m1,...,mn)
asmoothconic lies inauniqueirreducible component ofM¯ of theexpected
(m1,...,mn)
dimension. The virtual fundamental class of this component is equal to the usual
one.
Corollary 2.3. The number of smooth plane conics 5-fold tangent to Y is finite.
We denote it by n .
d
We will now study the additional non-enumerative contributions to the virtual
invariant N .
d
Lemma 2.4. ThemodulispaceM¯ hasthefollowingconnectedcomponents:
(2,2,2,2,2)
(A) 5! points for every smooth conic 5-fold tangent to Y.
8 ANDREAS GATHMANN
(B) 5!·(d−4)(d−5)pointsforeverybitangentofY,correspondingtodoublecovers
of thebitangent, withmarked points as in thepicturebelow ontheleft, i.e. the
map is ramified over two transverse intersection points of the bitangent with
Y, and the five marked points are the two ramification points, both inverse
imagepointsofonebitangencypoint,andoneinverseimagepointoftheother
bitangency point.
C C
f f
(B) (C)
bitangent bitangent
Y Y
(C) 1·5!·(d−4)smoothrational curves for every bitangentof Y, corresponding to
4
double covers of the bitangent, with marked points as in the picture above on
the right, i.e. (for a general stable map in this smooth rational curve) the map
is ramified over one transverse intersection point of the bitangent with Y and
one other arbitrary point, and the five marked points are the first ramification
point, and the four inverse image points of the two bitangency points.
Proof. Case 1: the image of the stable map is a smooth conic. If the five marked
points are distinct in P2, we get the components (A) by lemma 2.1, with the 5!
correspondingto the labeling ofthe markedpoints. If two ofthe points coincide in
P2 (i.e. lie on a contracted component of the stable map), then by the description
of M¯ in remark 1.5 the conic must have contact of order (at least) 4 to Y
(2,2,2,2,2)
at this point, i.e. it lies in M¯ . But this space is empty by lemma 2.1.
(4,2,2,2)
Case 2: the image of the stable map is a union of two (distinct) lines. It is easy
to see that the conditions of remark 1.5 cannot be satisfied in this case.
Case 3: the stable map is a double cover of a line. There are six possible points
of tangency to Y: the four inverse image points of the bitangency points, and the
two ramification points if they are mapped to points of Y. For the stable map in
M¯ wecanpickanyfiveofthesesixpoints. Ifweleaveoutoneofthepoints
(2,2,2,2,2)
over the bitangency points, we arrive at the components (B), otherwise we get the
components (C).
In case (B) we get a factor of 5! for the choice of labeling of the marked points,
a factor of d−4 for the choice of two transverse intersection points of Y with the
2
bitangent, a(cid:0)nd a(cid:1) factor of 2 for the choice of bitangency point over which we take
only one inverse image point to be marked. In case (C) the second ramification
point is not fixed, so we get one-dimensional families of such curves. Every such
familyhasa2:1maptothebitangentgivenbytheimageofthemovingramification
point; the two stable maps in a fiber of this map differ by exchanging the marked
points over one bitangency point. The map is simply ramified over the two stable
maps where the moving ramification point is one of the bitangency points. Hence
every such family is a P1. The number of such families is d−4 (for the choice of
transverse intersection point of the bitangent with Y) times 1 ·5! (for the labeling
4
of the marked points, taking into account that exchanging the marked points over
the bitangency points does not give us a new family).
THE NUMBER OF PLANE CONICS 5-FOLD TANGENT TO A GIVEN CURVE 9
Of course, the virtual fundamental class of M¯ splits naturally into a
(2,2,2,2,2)
sum of virtual fundamental classes on each of the connected components that we
have just identified. As it is well-known that the number of bitangents of Y is
1d(d+3)(d−2)(d−3)(seee.g.[H]exerciseIV.2.3f),wegetthefollowingcorollary.
2
Corollary 2.5. We have
1 1
N =n + d(d+3)(d−2)(d−3)(d−4)(d−5)b + d(d+3)(d−2)(d−3)(d−4)c ,
d d d d
2 8
where b is the degree of the part of the virtual fundamental class of M¯
d (2,2,2,2,2)
supported on the point M¯ ∈ M¯ below, and c is the corresponding degree
B (2,2,2,2,2) d
supported on the smooth rational curve M¯ ⊂M¯ below.
C (2,2,2,2,2)
x x x C x x x C
1 4 5 1 3 4
x x x x
2 3 2 5
_ f _ f
M M
B C
bitangent bitangent
Y Y
3. Computation of the virtual fundamental classes
In this section we will do the necessary computations to determine the numbers
b andc ofcorollary2.5. Mostofthemaresimplecalculationsinlocalcoordinates,
d d
so we will only sketch these parts and leave the details to the reader.
The computation of b is quite simple, as the component M¯ of M¯ has
d B (2,2,2,2,2)
the expected dimension.
Lemma 3.1. b =1 for all d.
d
Proof. Byremark1.4wejusthavetoshowthatthe10equationsofvanishingofthe
1-jets ofev∗F for i = 1,...,5 locally cut out the point M¯ in M¯ (P2,2) scheme-
i B 0,5
theoreticallywithmultiplicity1. Letusstartwiththe1-jetsatthepointsx ,x ,x ,
1 2 3
i.e. with the space M¯ . We can choose the coordinates on P2 such that the
(2,2,2)
bitangent is {z =0}⊂P2 and the bitangency points are (1:0:0) and (0:1:0).
2
This means that
a =a =0, a ,a 6=0 (tangency at (1:0:0)),
(d,0,0) (d−1,1,0) (d−1,0,1) (d−2,2,0)
a =a =0, a ,a 6=0 (tangency at (0:1:0)).
(0,d,0) (1,d−1,0) (0,d−1,1) (2,d−2,0)
Moreover, we can choose coordinates on the source P1 such that the stable map
M¯ is given by (s :t) 7→(s2−t2 :st : 0), and the marked points are x = (1 : 0),
B 1
x = (0 : 1), x = (1 : 1). Local coordinates of M¯ (P2,2) around this point are
2 3 0,3
then ǫ ,...,ǫ , where the stable map is given by
1 8
(s:t)7→(s2−t2+ǫ s2+ǫ st+ǫ t2 :st+ǫ s2+ǫ t2 :ǫ s2+ǫ st+ǫ t2).
1 2 3 4 5 6 7 8
The three tangency equations are that F| , F| , and F| have
s=1,t=ξ s=ξ,t=1 s=1,t=1+ξ
no constant and linear ξ terms. It is an easy computation to see that these 6
equations, linearized in the ǫ , give
i
ǫ +ǫ +ǫ =ǫ =ǫ =ǫ =ǫ =ǫ =0,
1 2 3 4 5 6 7 8
10 ANDREAS GATHMANN
so M¯ is smooth of dimension 2 at the point M¯ (with the points x and x
(2,2,2) B 4 5
forgotten).
Now let us consider the two other tangency conditions at the points x and x .
4 5
As the four coordinates of M¯ around M¯ we can choose the images of the
(2,2,2,0,0) B
ramificationpointsandapointinthedomainofthestablemapintheneighborhood
ofeachramificationpoint. Consideringonlyoneramificationpointfornow,thetwo
correspondinglocalcoordinatesareǫ˜ andǫ˜ ,wherethestablemapisgivenlocally
1 2
in affine coordinates as t 7→ t2 +ǫ˜ , and the marked point is t = ǫ˜ . Tangency
1 2
means that the constant and linear ξ terms of (ǫ˜ +ξ)2+ǫ˜ vanish, so linearly in
2 1
the ǫ˜ we get ˜ǫ =ǫ˜ =0. The same is true for the other ramification point, so we
i 1 2
see that M¯ is a smooth point of M¯ .
B (2,2,2,2,2)
To study the space M¯ , we need a lemma that tells us how the stable maps in
C
M¯ can be deformed if we relax some of the multiplicity conditions.
C
Lemma 3.2. Let H be a line in P2, and let P ∈ H be a point where Y is simply
tangent to H. Let C = (C,x ,x ,f) ∈ M¯ be a (possibly reducible) double
1 2 (2,2)
cover of H, such that f−1(P) = {x ,x }. Then every stable map in M¯ in a
1 2 (2,2)
neighborhood of C is also a double cover of a (maybe different) line.
Proof. It is obviousthat C cannotbe deformed into a union of two distinct lines in
M¯ . So we have to show that C cannot be deformed to an irreducible smooth
(2,2)
conic in M¯ .
(2,2)
Weusetheclassicalspaceofcompleteconics(whichisisomorphictoM¯ (P2,2)).
0,0
Recall that this space is the closure in P5 ×(P5)∨ of the set (C,C∨), where C is
an irreducible conic and C∨ its dual. For our given point C (with the two marked
pointsforgottenforthemoment),C isthedoublelineH,andC∨ istheunionofthe
two lines in P∨ that correspond to the two ramification points of f in P2. Assume
that we can deform (C,C∨) in the space of complete conics to an irreducible conic
that is still tangent to Y at two points in the neighborhood of P. In particular,
we would then deform C∨ to an irreducible conic that is tangent to the dual of Y
(or more precisely if H is a bitangent of Y: to the branch of the dual of Y that
correspondsto the point P) at two points in this neighborhood. By the continuity
of intersectionproducts this means that both lines of C∨ must actually be the line
corresponding to the point P. Hence both ramification points of f would have to
be P. This means that C must have a contracted rational component over P, in
contradiction to the assumption f−1(P)={x ,x }.
1 2
We want to reduce the computation of c to spaces that have the expected
d
dimension. To dothis, weuseequation(2)fromexample1.7. Note thatbyremark
1.12 this equation is true in the Chow group of the geometric zero locus of the
section σ (that describes the tangency condition and whose zero locus has class
ev∗Y +ψ ), so it makes sense to restrict the equation to a connected component
5 5
of this zero locus. Using lemma 3.2 it is easy to see that M¯ is such a connected
C
component, so we will restrict equation (2) to M¯ and denote this restriction by
C
|M¯C.
Note first that on M¯ the point x can only come close to x , but never to the
C 5 4
other three marked points. Hence equation (2) restricted to M¯ reads
C
(ev5∗Y +ψ5)·[M¯(2,2,2,2,1)] |M¯C =cd+3[M¯(2,2,2,3)]|M¯C. (3)
(cid:0) (cid:1)
Let us first compute the virtual fundamental classes occurring in this equation.