Virtual Intersection Numbers

Virtual intersection numbers

Georg Hein

July 31, 2000

Dedicated to H. Kurke

Abstract We attempt to present the intersection theory which is required to understand the work of Kontsevich and Manin. Finally, we repeat their computations of intersection numbers in a concrete example. To do so, we study the moduli stack of stable maps of degree two from rational curves to P1. We show that its Picard group is inﬁnite cyclic. We give an ´etalemap 2 1 from M to P of degree 2 . Eventually, we compute an intersection number which arises in Kontsevich’s computation of the number of rational curves on the quintic threefold.

Introduction

The computation of the (expected) numbers of rational curves on a general quintic threefold in 4 P is an important result in algebraic geometry. One of the main tools for these calculations are moduli stacks of stable maps of curves to a target space. We have to compute certain intersection numbers on these moduli stacks. It turns out that these intersection numbers give only the virtual number of rational curves on a quintic threefold. Using residual intersection theory we can compute the real numbers from these virtual numbers. In section 1 we give the results on intersection theory we need in the sequel. We show how well known numbers can be computed using residual intersection theory in section 2. In the next section we give the results of Kontsevich and Manin. The main object of this article is to study 1 the moduli stack M of stable maps of degree two from rational curves to P . However, in section 4 we ﬁrst study an example of a smooth stack N of dimension one. We will see later on that this stack N shares many features with our moduli stack M. We included this example, because the computations are less technical in this case. Finally, we come in section 5 to the moduli stack 2 M. We compute its Picard group, the canonical line bundle KM , an ´etale map from M to P , and last but not least the intersection number a2. None of the presented results is new. The author is responsible for all mistakes and inaccuracies.

1 Preliminaries

1.1 Residual intersection theory The standard reference for intersection theory we follow here is W. Fulton’s book [1]. Let X be a smooth variety of dimension n over Spec(k). Furthermore, let E be a X-vector bundle with rk(E) = n = dim(X). The vanishing locus V (s) of a global section s ∈ H0(E) is locally given by n equations. Thus, we expect it to be of codimension n which means of dimension zero. If this is the case, the Euler characteristic χ(OV (s)) is the n-th Chern number of the vector bundle E. However, in many situations there arise components of higher dimension in V (s). Since their expected dimension is zero, we say that the virtual dimension of V (s) is zero. Let Z be

1 2 2 FIRST EXAMPLES a connected component of V (s), we want to know what is its contribution (E, s)Z to the n-th R Chern number X cn(E) ? For simplicity, we assume that Z is regularly embedded. The answer is given by residual intersection theory (cf. [1], §9): Z Z −1 (E, s) = c(E|Z ) · c (NZ|X ) Z where NZ|X denotes the normal bundle of Z in X, and c(−) denotes the total Chern class of a `k vector bundle. Suppose we have V (s) = j=1 Zj, then we have

Z k X Zj cn(E) = (E, s) . X j=1

1.2 The normal bundle of smooth substacks We suppose the reader to be familiar with the theory of moduli spaces of stable curves. A good introduction to this subject is the article of W. Fulton and R. Pandharipande [2]. Let Z ⊂ X be a closed subscheme. Let MX and MZ be the moduli stacks of curves with given numerical invariants in X and Z respectively. As usual, we denote the universal curves with CX , and CZ (respectively). We have the following diagram

q CZ / Z F FF FF FF f FF # qX p CX / X

pX ι MZ / MX

In the sequel we need the normal bundle NMZ |MX of the closed subscheme MZ of MX . For simplicity we assume Z is regularly embedded in X and both moduli stacks are smooth of the ∗ expected dimension. From the sequence 0 → TZ → f TX → NZ|X → 0, and the fact that the ∗ tangent bundle TMX of MX is given by (pX )∗qX TX (analogously for TMZ ) we derive the normal bundle equation: ∗ NMZ |MX = p∗q NZ|X .

2 First examples

Here we consider two examples where we can illustrate the above techniques. They provide nice and applicable tools for many problems in intersection theory. To be honest, this section is included as a compensation for the lack of explicit computations in the next section. However, it might be interesting to see how well-known numbers are computed in the virtual way.

2.1 Quadrics in P2 tangent to 5 lines 2 We want to answer the following question: How many quadrics in P are tangent to 5 lines 2 {li}i=1...5 in general position (i.e., through any point in P there pass at most two of them)? See also [1], Example 9.1.8. 5 We easily see that the space of all quadrics is a P . The discriminant of a quadric on a line ∼ 1 li = P is a quadratic form in the coeﬃcients. Hence, the quadrics tangent to li form a divisor 5 Di in P of degree 2. Thus, our task is just to determine the intersection D1 ∩ ... ∩ D5. Bezout’s 3 2.2 Virtual number of lines on a cubic in P 3

5 theorem persuades us to expect that there are 32 = 2 quadrics tangent to all li. Unfortunately, the family Z of double counting lines is contained in the intersection D1 ∩ ... ∩ D5. Obviously, 2 5 Z is the Veronese embedding of P into P . The five divisors Di define a global section of ⊕5 the vector bundle E := O 5 (2) . From Bezout’s theorem, we obtain its fifth Chern number R P X c5(E) = 32. Using the residual intersection theory, we can compute the contribution of Z to the intersection number 32:

Z R −1 (E, s) = c(E|Z ) · c(NZ|X ) Z

= R (1 + 4h)5 · (1 + h)3(1 + 2h)−6 Z

= R 1 + 11h + 31h2 = 31. Z ∼ 2 Here h denotes the hyperplane class on Z = P . Hence, there is only one smooth quadric tangent to ﬁve lines in general position. It should be remarked that considering the dual situation this can be easily derived.

2.2 Virtual number of lines on a cubic in P3 Here we try to compute the number of lines on a cubic. We took this example because the reader presumably knows the answer. 3 Let X be the grassmannian G(1, 3), i.e., the space of all lines in P . Let U(1, 3) be the universal curve over G(1, 3). We look at the following diagram:

p q 3 G(1, 3) o U(1, 3) / P .

∗ We have to consider the vector bundle E = p∗q O(3) on G(1, 3). Since rk(E) = dim(G) and E 1 R is globally generated a general global section s of E vanishes in G(1,3) c4(E) diﬀerent points. 3 At the other hand, a global section s of E corresponds to a global sections ˜ of the P -line bundle 3 O(3). The vanishing locus V (s) consists of those lines which are on the divisor V (˜s) ⊂ P . We R could compute the number G(1,3) c4(E) directly, but we want to practice residual intersection 3 theory. We regard a concrete cubic in P by taking a union of a plane H with a smooth quadric Q intersecting each other in a smooth conic. The lines contained in Q ∪ H are in three disjoint families. The family of dimension two of lines ∼ 1 1 in H, and the two one-dimensional families of lines in Q = P × P . 1 ∼ 1 1 Let us compute the contribution of P ⊂ X parameterizing one ruling of Q = P × P . To do so, we have to consider the following situation:

f g 1 3 P o Q / P . It turns out that we need only the following facts:

∗ ⊕3 f g O 3 (2) ∼ O 1 (2) ∗ P = P ∗ ⊕4 f g O 3 (3) ∼ O 1 (3) . ∗ P = P 1To see that E is globally generated we consider the exact sequence 0 3 ∗ 0 → ker(ψ) → H (O(3)) → O(3) → 0 on P . Applying the functor p∗q to this sequence yields an exact sequence 1 ∗ on G(1, 3) as long as R p∗q ker(ψ) vanishes. To see that this coherent sheaf vanishes it is enough to show that the restriction of ker(ψ) to any line l does not contain a direct summand of type Ol(a) with a ≤ −2. This is an easy computation. 4 3 THE COMPUTATIONS OF KONTSEVICH AND MANIN

3 2 Thus, we have c(N 1 ) = (1 + 2h) with h the divisor class of a point. Since h = 0, we P |G(1,3) −1 1 have c(N 1 ) = 1 − 6h. Eventually, we compute the contribution of the component of P |G(1,3) P G(1, 3) to be 1 Z Z (E, s)P = (1 + 3h)4 · (1 − 6h) = (1 + 6h) = 6 . 1 1 P P 2 3 Now we compute the contribution of the lines on a plane P ⊂ P . These lines are parameterized by the so called dual projective plane which we denote by P2. The universal family is denoted by C. Thus, we have the following diagram:

f g 2 3 P2 o C / P / P From the Euler sequence we see that

∗ −1 f g O 2 (1) = T (−1) and c(T (−1)) = 1 − h , ∗ P P2 P2 2 with h the hyperplane class in P2. Thus, we have c(TP2 (−1)) = 1 + h + h . The P2-vector ∗ 3 bundle f g O 2 (3) is isomorphic to the threefold symmetric product S (T (−1)). This allows ∗ P P2 an direct computation of its total Chern class using Chern roots. Indeed, let α and β be the 2 Chern roots of TP2 (−1), i.e., c(TP2 (−1)) = (1 + α)(1 + β). This implies α + β = h and α · β = h . 3 Consequently, the Chern roots of S (TP2 (−1)) are given by 3α, 2α + β, α + 2β, and 3β. Hence, 3 we compute the total Chern c(S (TP2 (−1))) class to be 3 c(S (TP2 (−1))) = (1 + 3α)(1 + 2α + β)(1 + α + 2β)(1 + 3β) = 1 + 6(α + β) + 11(α2 + β2) + 32α · β = 1 + 6(α + β) + 11(α + β)2 + 10α · β = 1 + 6h + 21h2

Finally, we are able to compute the contribution of P2: Z Z (E, s)P2 = (1 + 6h + 21h2) · (1 − h) = (1 + 5h + 15h2) = 15 . P2 P2 This gives us the virtual number of lines on our cubic 27 = 6 + 6 + 15. (We have two rulings on the quadric surface.)

3 The computations of Kontsevich and Manin

3.1 The virtual number of rational curves on a quintic threefold 4 Kontsevich considers in [5] the following situation: Let M¯ (P , d) be the compactiﬁed moduli 1 4 space of degree d-morphisms of P to P , and C(d) the universal curve over the moduli space 4 4 M¯ (P , d). We see that the moduli space M¯ (P , d) is a stack of dimension 5d + 1. For d > 1 this stack is no variety. We have the following universal diagram

q 4 C(d) / P p 4 M¯ (P , d)

¯ 4 We now deﬁne the M(P , d)-vector bundle Ed by:

∗ E := p q (O 4 (5)) . d ∗ P 3.2 The real number of rational curves on a quintic threefold 5

¯ 4 If Ed were globally generated, and M(P , d) were a smooth scheme, then the number Z Nd := c5d+1(Ed)

¯ 4 M(P ,d)

4 would be the number of degree d rational curves on a general quintic in P . In particular, this number would be ﬁnite. Since this does not hold, the number Nd is the virtual number of degree d rational curves on a quintic. However, Kontsevich computes these virtual numbers. We give from [5] the ﬁrst of them:

d 1 2 3 4

4876875 8564575000 15517926796875 N 2875 d 8 27 64

3.2 The real number of rational curves on a quintic threefold 0 We assume from now on that the Clemens conjecture holds, i.e., the number Nd of rational curves of degree d on the general quintic is finite and all these curves are irreducible. We will (still following Kontsevich) derive a formula which relates the virtual numbers Nd with the true 2 0 numbers Nd . 4 We consider a global section s of Ed corresponding to a smooth quintic X(s) in P . We assume the quintic to be general in the sense of the Clemens conjecture, i.e., the number of rational 4 curves of degree less or equal d on X(s) is finite. The vanishing locus V (s) in M¯ (P , d) consists 1 4 ∗ of all those points corresponding to a mapping ψ : → with ψ O 4 (1) ∼ O 1 (d) with P P P = P 1 ψ(P ) ⊂ X(s). If the degree of ψ is 1, then the Clemens conjecture implies that ψ is an isolated point in V (s). 0 1 If ψ is a morphism of degree k, then of course all morphisms ψ form P to the image of ψ belong to the zero set V (s). 1 1 1 Let M0,0(P , k) be the compactified moduli space of mappings from P to P of degree k. Then d 4 1 ¯ 4 any rational curve of degree k in P defines a natural inclusion ιk,d : M0,0(P , k) ,→ M(P , d). Using the results of 1.1 we find the connection between the virtual and the true numbers

X 0 Nd = ak,dNd/k where k|d Z ∗ −1 1 4 ak,d = c(ιk,dEd) · c(NM0,0( ,k)|M¯ ( ,d)) . 1 P P M0,0(P ,k)

Thus, we have to study the numbers ak,d. To do so, we consider the following morphisms.

f g ψ 1 1 4 M0,0(P , k) o C0,0 / P / P

d Here ψ is an embedding of degree k , and C0,0 is the universal curve. By deﬁnition, we have

∗ ∗ ∗ ∗ 5d ι E =∼ f g ψ O 4 (5) =∼ f g O 1 ( ) . k,d d ∗ P ∗ P k Using the normal bundle equation (cf. 1.2), we obtain

∗ N 1 ¯ 4 = f g N 1 4 . M0,0(P ,k)|M(P ,d) ∗ P |P 2It should be remarked that the Clemens conjecture is still open. Thus, the true numbers are a little bit virtual too. 1 6 4 A STRANGE P

5d The normal bundle N 1 4 is of degree − 2, and of rank 3. We have a short exact sequence P |P k

⊕2 5d 0 → O 1 (−1) → N 1 4 → O 1 ( ) → 0 . P P |P P k ∗ ⊕2 Since f g O 1 (−1) = 0, we obtain the following short exact sequence ∗ P

∗ 5d 1 ∗ ⊕2 0 → f g N 1 4 → O 1 ( ) → R f g O 1 (−1) → 0 . ∗ P |P P k ∗ P Since the total Chern class is multiplicative for short exact sequences, we conclude the formula Z 1 ∗ ⊕2 ak,d = c(R f∗g O 1 (−1) ) . 1 P M0,0(P ,k) 1 ∗ ⊕2 Thus, the numbers a do not depend from the integer d. The vector bundle R f g O 1 (−1) k,d ∗ P 1 is the pull back of the normal bundle of P in the quintic X(s). Manin gives a formula for the −3 numbers ak = ak,d which says ak = k (cf. [6]). This gives us the connection between the 0 virtual number (Nd) and the number (Nd ) of degree d rational curves on a general quintic in 4 P :

0 X Nd/k N = . d k3 k|d This provides us with a list of the ﬁrst true numbers of rational curves of degree d on a quintic: d 1 2 3 4 0 Nd 2875 609250 317206375 242467530000

4 A strange P1 Let k be a field of characteristic different from two. We consider the following étalecover of the Deligne-Mumford stack N.

−1 U01 k[z, z ] C 2 u: dHH −2 {{ CC Where the open subsets x7→z uu HHy7→z {{ CC uu HH {{ CC Ui are the spectra of the uu HH }{{ C! uu H U0 U1 coordinate rings on the k[x] k[y] CC { dII u: CC { right hand side. II uu C {{ II uu CC {{ II uu C! }{{ I uu N ON A line bundle on N is given by a unit on U01 modulo the units on Uj. Thus the line bundle L corresponding to z−1 generates the Picard group of N. The transition function of the canonical dx dx d(z2) bundle KN is calculated by computing −dy on U01. We directly compute, that −dy = −d(z−2) = 2zdz 4 ∼ −4 n 2z−3dz = z . Thus, KN = L . To compute the space of global sections of L , we consider the Cechˇ complex

k[x] ⊕ k[y] → k[z, z−1] (f, g) 7→ z−n · α(f) − β(g) where α and β are the ring homomorphisms deﬁned by α(x) = z2 and β(y) = z−2. We see that the cokernel H1(N,Ln) is inﬁnite dimensional, whereas n + 1 for even non-negative n ; h0(N,Ln) = 2 0 else. 7

2 1 The global sections of L deﬁne a morphism from N to P . The next diagram gives a local description of this morphism.

X7→x k[X] / k[x] J I JJ II 2 JJX7→Z IIx7→z JJ II JJ II % Z7→z2 $ k[Z,Z−1] / k[z, z−1] −1 t9 −2 u: Y 7→Z tt y7→z uu tt uu tt uu tt Y 7→y uu k[Y ] / k[y]

1 The morphism ψ : N → P is an ´etalemorphism of stacks. Using the fact that the map U0 → N 1 is ´etale of degree two, we obtain that ψ is of degree 2 . We directly see from the above diagram ∗ 2 that ψ O 1 (1) ∼ L . Thus, we obtain the equalities of Chern numbers P = Z Z Z 1 2 deg(ψ) 1 c1(L) = c1(L ) = c1(O 1 (1)) = . 2 2 1 P 4 N N P

1 5 The moduli stack M = M0,0(P , 2) 5.1 Why do we need a stack? We ﬁx a ground ﬁeld k with char(k) 6= 2. Let M be the space of all stable maps of degree two 1 to P . It is easy to see that M is the disjoint union of the following two sets:

1 M0 the set of mappings of a smooth rational curve to P ramiﬁed over two diﬀerent points;

M1 the curves we consider here are two rational curves identiﬁed in one point and a degree one 1 map of each one to P .

1 At the first glance it seems that our space M is the linear system of two points on P . Further- more, to give an algebra structure on O 1 ⊕ O 1 (−1) we need to give a section of O 1 (2). P P P There exists a universal reason why a moduli scheme does not suffice for this situation: There are non trivial automorphisms. 0 1 ∗ The point is that our moduli space is H ( , O 1 (2)) \ 0 modulo the natural action of k = P P −2 −2 −2 ∗ Iso(O 1 (−1)). In coordinates we identify (x, y, z) with (xt , yt , zt ), for any t ∈ k . The P short exact sequence x→x2 0 / µ2 / Gm / Gm / 0 from étale cohomology yields that our stack is an étalecover of 2 with fibre 1 . This shows P µ2 that M can not be an ordinary scheme. 2 If we take C as our ground field, then we see that as a topological space M is the same as P . 2 Whereas taking a finite field k we see that M(k) is twice as big as P (k). Thus, the existence of the moduli scheme would contradict the Weil conjectures.

5.2 Computing the Betti numbers using the Weil conjectures

l We assume here, that k is a finite field with q elements. By AM (q ) we denote the number of Fql - 1 points of M each counted with multiplicity #Aut . The divisors of degree two are parameterized 2 by Pk. To such an divisor correspond two different curves ramified over that divisor. However, 1 8 5 THE MODULI STACK M = M0,0(P , 2)

number of number description picture different number of of auto- contribution l curves markings morphisms to AC(q ) ...... a smooth point on ...... a smooth curve ...... l l l l ramified over two ql(ql + 1) q −1 1 q (q +1)(q −1) r ? 2 2 Fql -points a ramification ...... point on a ...... l l l l smooth curve ...... q (q + 1) 2 2 q (q + 1) r ramified over ? two Fql -points a point on ...... a smooth curve ...... l l l l ...... q +1 q (q +1)(q −1) ...... l l ramified over two ...... q (q − 1) 2 1 2 Fq2l -points which r ? are conjugated a smooth point on aa !! a!! a reducible curve ! aa ! a l l l l consisting of two r q + 1 q 1 q (q + 1) ? projective lines a smooth point on a contracted curve connecting 2(ql + 1) 1 2 ql + 1 r two projective ? lines

l l 2l l both have two automorphisms. Hence, we conclude that A (q ) = A 2 (q ) = q + q + 1. This M P implies for the ζ-function ζM

l ! X l t 1 ζ (t) = ζ 2 (t) = exp A (q ) = . M P M l (1 − t)(1 − qt)(1 − q2t) l>0

We conclude that the Betti numbers of M are b0(M) = b2(M) = b4(M) = 1 and zero otherwise. To compute the Betti numbers of the universal curve C over M, we use Knudsen’s result (see 1 [4]) that C is the moduli stacks of stable 1-pointed degree two maps from a rational curve to P (see also [3]). Summing up the last column of the above table we obtain the number of Fql -points in the moduli 1 space M0,1(P , 2) which is the universal curve C over M.

l 3l 2l l AC(q ) = q + 2q + 2q + 1

As before we calculate the ζ-function ζC of C and obtain 1 ζ (t) = . C (1 − t)(1 − qt)2(1 − q2t)2(1 − q3t)

Eventually, we can give the Betti numbers of the universal curve C to be b0(C) = b6(C) = 1, b2(C) = b4(C) = 2, and bk = 0 otherwise. 5.3 Construction of M 9

5.3 Construction of M

2 Let k be a ﬁeld not of characteristic two. We assume that k∗ x−→7→x k∗ is surjective. Furthermore, let V be a k-vector space of dimension two with base vectors X and Y . We denote the projective 1 space P(V ) by P . To construct M we add a marking to get rid of the automorphisms. One possibility is to consider the following moduli spaces:

 1  Stable 2 : 1 ﬁnite maps φ : X → P from a rational curve 1 Uj := X to P for which j is no critical value plus the choice of a 1 point in the ﬁbre of j ∈ P . 

It is obvious that M is covered by the open sets U0, U1, and U∞. Furthermore, we see that these Uj are ´etale neighbourhoods of M of degree two. 1 1 Next we will construct U0 and the universal family over U0. Let V0 = P \ ∞ and V1 = P \ 0. Let e and e be trivializing sections of O 1 (−1) restricted to V respectively to V with e = xe 0 1 P 0 1 0 1 on V = V ∩ V . We obtain an algebra structure on O 1 ⊕ O 1 (−1) by setting 01 0 1 P P

2 2 e0 = A(1 − x) + Bx + C(x − 1)x and e1 = A(y − 1)y + By + C(1 − y) .

1 The requirement that 0 ∈ P is not a critical value is equivalent to A 6= 0. The marking corresponds to a surjection π of O 1 ⊕ O 1 (−1) to k = O . This is given by 1 7→ u and e 7→ a. P P 0 0 However, since π has to be an algebra homomorphism, we must have u = 1 and a2 = A. Thus, a point in U0 can be parameterized by a triple (a, B, C). If we replace e0 and e1 by λe0 and −1 −2 −2 λe1 respectively, then we have to pass from the triple (a, B, C) to (λ a, λ B, λ C). Thus in ∼ 2 every equivalence class there exits a unique representative with a = 1. We end up with U0 = A .

Now we are able to describe our moduli space U01∞ GG xx GG M. All arrows are ´etalemorphisms of degree xx GG xx GG two. We have {xx G# ∼ ∼ ∼ 2 U01 U0∞ U1∞ U0 = U1 = U∞ = A , F G FF xx GG ww U =∼ U =∼ U =∼ 1 × k∗, FFxx G w 01 0∞ 1∞ A FF wwG ∼ ∗ ∗ xx F ww GGG and U01∞ = k × k . {xx F# {ww # U0 U1 U∞ In the next diagram we give the corresponding GG v GG vv coordinate rings and ring homomorphisms. GG vv GG vv G# {vv M

k[b, b−1, c, c−1] C7→c2 jj5 O iTTTc7→b−1c jj TT −2 b7→b jjj 2 TTT A7→b jjjj B7→b TTTT jjjj c7→c TTT k[b, b−1,C] k[B, c, c−1] k[A, c, c−1] L −2 L O e LCL7→b C rr9 e LL A7→A rr9 O LL −2 rr LL C7→c2 rr LLA7→b rr A7→c−2 LL rr B7→b2 LL r LL rr A7→c−2A LL rr B7→c−2B L r LrL rrLr C7→C rr L r LL B7→c−2 rr LLL rr LL rr B7→B LL rrr LL rr 2 LL r LL rr C7→c L rrr L k[B,C] k[A, C] k[A, B] jTTTT O jjj4 TTTT jjjj TTTT jjjj TTTT jjjj TTT jjjj OM 1 10 5 THE MODULI STACK M = M0,0(P , 2)

5.4 The Picard group and the universal family

A line bundle on M is given by units ϕ01, ϕ1∞, and ϕ∞0 satisfying the cocycle condition k l ϕ01 · ϕ1∞ · ϕ∞0 = 1 on U01∞. Using the above coordinates we can write ϕ01 = b , ϕ1∞ = c , and m k cl m ϕ∞0 = c . The cocycle condition reads b bl c = 1. This implies l = k and m = −k. Thus, the line bundle L corresponding to k = −1, l = −1, and m = 1 generates the Picard group Pic(M) of M. Analogously to section 4 we can 2 n +6n+8 , for even n ≥ 0; compute the space of global sections h0(M,Ln) = 8 of Ln and obtain: 0 else. ∼ ∼ −6 We conclude that Pic(M) = Z. Repeating the computations of section 4 we obtain Km = L . 2 2 1 As before the global sections of L deﬁne a morphism ψ : M → P of degree 2 . The corresponding ∗ 2 morphism of Picard groups is given by ψ O 2 (1) = L . Eventually, we obtain P Z Z Z 2 1 2 2 deg(ψ) 2 1 c1(L) = c1(L ) = c1(O 2 (1)) = . 4 4 2 P 8 M M P

1 Now we glue the universal families of 5.3 to obtain a ﬁnite map ρ : C → M × P . It turns out that we do not need the marking we introduced before. However, we need that we have s square ∗ root of the ample line bundle ψ O 2 (1). We deﬁne C locally as before by putting P

2 e00 = (1 − x) + Bx + C(x − 1)x on U0 × V0; 2 e01 = (y − 1)y + By + C(1 − y) on U0 × V1; 2 e10 = A(1 − x) + x + C(x − 1)x on U1 × V0; 2 e11 = A(y − 1)y + y + C(1 − y) on U1 × V1; 2 e∞0 = A(1 − x) + Bx + (x − 1)x on U∞ × V0; 2 e∞1 = A(y − 1)y + By + (1 − y) on U∞ × V1.

These local data glue well if we set ej0 = x · ej1, for all j ∈ {0, 1, ∞}, and

−1 e0k = b · e1k e1k = c · e∞k e∞k = c e0k for k ∈ {0, 1} .

p p 1 M 1 P 1 Considering the morphisms M o M × P / P . We see that this deﬁnes an algebra ∗ −1 ∗ structure on the direct sum OM× 1 ⊕ p L ⊗ p 1 O 1 (−1) . P m P P

1 5.5 Drawing the conclusion: a2 = 8 ∗ Now we are able to compute the direct image bundles p q O 1 (m) for the morphisms ! P p q 1 M o C / P .

∗ ∗ ∗ −1 ∗ p q O 1 (m) = p p O 1 (m) ⊕ (p L ⊗ p O 1 (m − 1)) ! P M! 1 P M 1 P ∗ P1 ∗P 1 −1 = H ( , O 1 (m)) ⊗ O ⊕ H ( , O 1 (m − 1)) ⊗ L P P M P P

1 ∗ −1 In particular, this yields R p q O 1 (−1) ∼ L . Coming back to our formula from 3.2 we ∗ P = conclude Z Z −1 −1 2 1 a2 = c(L ⊕ L ) = c1(L) = . M M 8 Thus, we were able to reproduce a very small part of an important calculation. REFERENCES 11

References

[1] Fulton, W., Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 3, Band 2, Springer-Verlag, Berlin, 1984.

[2] Fulton, W. and Pandharipande, R., Notes on stable maps and quantum cohomology, alg- geom/9608011, in Koll´ar, J. (ed.) et al., Algebraic geometry, Proceedings of the Summer Research Institute, Santa Cruz, CA, USA, July 9–29, 1995. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 62 (part 2), 45-96 (1997).

[3] Keel, S., Intersection theory of moduli spaces of stable N-pointed curves of genus zero, Trans. of the AMS 330 (1992, 545-574.

[4] Knudsen, F., Projectivity of the moduli space of stable curves II, Math. Scand. 52 (1983), 161-199.

[5] Kontsevich, M., Enumeration of rational curves via torus action, hep-th/9405035, in Dijk- graaf, R. H. (ed.) et al., The moduli space of curves, Proceedings of the conference held on Texel Island, Netherlands, April 1994, Birkh¨auser, Basel, Prog. Math. 129, 335-368 (1995).

[6] Manin, Yu. I., Generating functions in algebraic geometry and sums over trees, in Dijkgraaf, R. H. (ed.) et al., The moduli space of curves, Proceedings of the conference held on Texel Island, Netherlands, April 1994, Birkh¨auser, Basel, Prog. Math. 129, 401-417 (1995).

Georg Hein Humboldt-Universit¨atzu Berlin Institut f¨urMathematik Rudower Chaussee 25 12489 Berlin

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