Universal polynomials for counts of secant planes to projective curves
aa r X i v : . [ m a t h . AG ] N ov Universal polynomials for counts of secant planes toprojective curves
Mara Ungureanu
Abstract
In this article we provide another method for obtaining explicit formulas yieldingcounts of secant planes to a projective curve. We formulate the problem in terms ofSegre classes of suitable bundles over the symmetric product of the curve and takeadvantage of the result of Ellingsrud, Göttsche, and Lehn concerning the structure ofintegrals of polynomials in Chern classes of tautological bundles over Hilbert schemesof points. We use this to set up a recursion starting from the easy to compute case ofthe projective line.
The motivation for this article is twofold. First of all, our interest stems from aclassical problem in enumerative algebraic geometry, namely that of counting the numberof e -secant ( e − f )-planes to a projective variety, where the parameters e and f are chosensuch that one expects a finite answer. In the case of smooth curves, this question has givenrise to a number of well-known formulas dating back to Castelnuovo and Cayley, but hasalso been the object of more recent interest, as in the work of Cotterill [5], [6], Farkas [8],and Le Barz [2]. The general setup is the following: consider a general curve C of genus g equipped with a linear series l of degree d and dimension r ≥
3. Let e and f be positiveintegers such that 0 ≤ f < e ≤ d and denote by C e the e -th symmetric product of thecurve. Then V e − fe ( l ) = { D ∈ C e | dim( l − D ) ≥ r − e + f } ⊂ C e is called the secant variety of effective divisors of degree e which parametrises the e -secant( e − f − C under the embedding into P r via l . The cycle V e − fe ( l ) is in fact a variety with a degeneracy locus structure and as such it has expecteddimension exp dim V e − fe ( l ) = e − f ( r + 1 − e + f ) . Even better, we know that if l is a general linear series, then V e − fe ( l ) is in fact equidimen-sional and of expected dimension [8]. We shall therefore restrict ourselves to this situation,and moreover to the case when dim V e − fe ( l ) = 0, so that the enumerative problem is well-defined. The class of the cycle V e − fe ( l ) ⊂ C e was computed by MacDonald (see also [1,Chapter VIII]), but unfortunately the general formula is difficult to use in practice, alsoin our case of interest when the dimension of the secant variety is zero. However, when f = 1, the number v e of e -secant ( e − d in P e − has a1articularly convenient formula (see [4] or [1, Chapter VIII §4]): v e = e X α =0 ( − α g + 2 e − d − α ! ge − α ! , and its corresponding generating function P e ≥ v e t e has also been computed (see [3] or[6]): S ( t ) = √ t ! d − − t + (1 + 2 t ) √ t t ! g − . Thus, our task from this first point of view is to find similar formulas for higher values ofthe parameter f .The second aspect we consider, and which will help us tackle our problem, is that ofintegrals of characteristic classes of tautological bundles over Hilbert schemes of points ofvarieties. These objects are not only interesting because they frequently have an enumera-tive interpretation, as we shall see below, but also because they were crucial in establishingthe fact that the cobordism class of the Hilbert scheme of points of a smooth projectivesurface depends only on the class of the surface itself (see [7]). The general setup is asfollows: consider a smooth projective variety X and denote by X [ n ] the Hilbert scheme of n points of X . Given a vector bundle F of rank r on X one has an associated tautologicalbundle F [ n ] of rank nr on X [ n ] whose fibre at [ Z ] ∈ X [ n ] is H ( Z, F | Z ). Our guiding resultin this context is the following theorem due to Ellingsrud, Göttsche, and Lehn on thestructure of such integrals: Theorem 1.1. [7, Proposition 4.1] Let X be a smooth projective surface equipped with avector bundle F whose rank r is constant on the irreducible components of X . Moreover, let P be a weighted homogeneous polynomial of degree n in the Chern classes of the tangentbundle T n of X [ n ] and of F , where each Chern class c i has weight i . Then there is a univer-sal polynomial e P in the variables { R X c ( T X ) , R X c ( T X ) , R X c ( F ) , R X c ( F ) , R X c ( T X ) c ( F ) } and depending only on P and r such that deg( P ∩ X [ n ] ) = Z X [ n ] P = e P . (1)Although the original result was stated and proved for smooth projective surfaces, one canuse the same arguments as in [7] to obtain the analogous statement for smooth projectivecurves. Furthermore, by using significantly different methods, Rennemo [13] has extendedthe theorem to smooth projective varieties of any dimension.Despite the simple form of the equality (1), it can in practice be difficult to computethe universal polynomial e P . A particular case in which such convenient formulas can befound is that of integrals of top Segre classes over Hilbert schemes of points of curves orsurfaces. For example, if X is a K3 surface with a line bundle H , Voisin [14] has recentlyshown that Z X [ n ] s n ( H [ n ] ) = 2 n H + 2 − nn ! , where H denotes the intersection product R X c ( H ). In fact, the formula for the generatingfunction of the above integrals for any surface X is the subject of Lehn’s conjecture210], which was also recently established in a series of papers by Marian, Oprea, andPandharipande ([11], [12]) in which the authors showed that ∞ X n =0 z n Z X [ n ] s n ( H [ n ] ) = (1 − w ) a (1 − w ) b (1 − w + 6 w ) c , (2)for constants a, b, c depending only on H , K X , H.K X and χ ( O X ) in a precise way, andwith a change of variables z ↔ w (for details see for example equation (2) of [12]). In thecase of curves, generating functions for the analogous top Segre integral of tautologicalbundles associated to vector bundles of rank r ≥ n (resp. 2 n ) in Segre classes of tautologicalbundles over X [ n ] , where X is a curve (resp. a surface).The connection between the two aforementioned tasks is realised via the enumera-tive interpretation of the Segre integrals. More precisely, the top Segre integrals yieldthe expected number of n -secant ( n − X in P n − (resp. P n − ), i.e. precisely the counts discussed above in the case of curves when f = 1.It is therefore natural to consider the next case, namely f = 2, and investigate to whatextent the methods developed for computing Segre integrals may help in finding explicitenumerative formulas.Hence, the aim of this article is to use these methods to compute the number of e -secant ( e − C in P e − , or equivalently, as shall be explainedin the course of the paper, the universal polynomial corresponding to Z C [ e ] s e ( L [ e ] ) − s e − ( L [ e ] ) s e +1 ( L [ e ] ) , (3)where e ≥ L is the line bundle whose sections give the embeddingof C in projective space. For simplicity we restrict to complete linear systems, so we have h ( C, L ) = e − V e − fe ( L ), where f = 2. Note thatthe parameter choice indeed yields exp dim V e − fe ( L ) = 0. Roadmap
The paper is organised as follows:• In Section 2 we summarise the description of secant varieties as degeneracy loci ofmaps between vector bundles on the symmetric product of the curve. We then recallhow to use it to obtain the expression of the class of V e − fe ( L ) in terms of Segreclasses of L [ e ] in the cases f = 1 and f = 2, thus recovering the top Segre integraland (3), respectively.• The main results of Section 3, namely Theorem 3.2 and Corollary 3.3, are inspiredby the Structure Theorem 4.2 of [7], which, in a simplified formulation, states thefollowing: consider a multiplicative characteristic class φ , i.e. with the property that3 ( F ⊕ F ′ ) = φ ( F ) φ ( F ′ ), for two vector bundles F and F ′ on a variety X . Then, if X is a smooth projective surface with a vector bundle F , the authors prove that ∞ X n =0 Z X [ n ] φ ( F [ n ] ) z n = exp (cid:16) X i =1 C i A i ( z ) (cid:17) , where the A i ( z ) are power series in Q [[ z ]] and the C i denote the entries of the vec-tor ( R X c ( T X ) , R X c ( T X ) , R X c ( F ) , R X c ( F ) , R X c ( T X ) c ( F )). The analogous state-ment for curves also holds. The total Segre and Chern classes are such multiplicativeclasses, and this property, together with the Structure Theorem above have been cru-cial ingredients in finding the formulas for integrals of top Segre classes and hencethe expression (2). However, for f = 2 we shall be dealing instead with products ofadditive classes. Thus, our Theorem 3.2 and Corollary 3.3 provide useful formulasfor dealing with generating functions of integrals of products of additive classes oftautological bundles.• Section 4 is dedicated to the computation of the integral (3). To do so we usethe results of Section 3 to establish recursions starting from the easy case of theprojective line. After first establishing some preliminary facts in Section 4.1 andSection 4.2, this is done in two parts:• In Section 4.3 we first consider a special case of our problem, namely f = 2 and e = 4. A closed expression for this count already exists (see [1, Chapter VIII §4,Example 3]), but we provide here a different way to approach the computation. Moreprecisely, we notice that for any vector bundle F the following holds: s ( F ) − s ( F ) s ( F ) = c ( F ) + s ( F ) − c ( F ) ch ( F ) . Now, the integrals R C [4] c ( L [4] ) and R C [4] s ( L [4] ) can be immediately read off fromthe literature (see [12] or [15]), where, as we mentioned before, one takes advantageof the multiplicativity of the total Chern and Segre classes. Thus, to find (3) it isenough to compute R C [4] c ( L [4] ) ch ( L [4] ) using the additivity of c and of the Cherncharacter and the results of Section 3. This yields the formula: Z C [4] c ( L [4] ) ch ( L [4] ) = 16 (( d + g − − g − g ( g − , for a smooth curve C of genus g with L of degree d ≥ We begin by recalling a few well-known facts regarding secant varieties to projectivecurves or surfaces, and in particular their description as degeneracy loci of vector bundlemorphisms over Hilbert schemes of points. We follow Fulton [9]. Let X be a variety, ϕ : E → F a morphism of vector bundles over X , and k ≤ min rk E, rk F a positiveinteger. Denote by A m ( X ) the m -th Chow group of X . The k -th degeneracy locus D k ( ϕ ) = { x ∈ X | rk( ϕ ) ≤ k } E − k )(rk F − k ) in X , if non-empty. Now set m = dim X − (rk E − k )(rk F − k )and denote by D k ( ϕ ) the class of D k ( ϕ ) in A m ( D k ( ϕ )). By the Thom-Porteous formula,the image of D k ( ϕ ) inside A m ( X ) is given by∆ (rk E − k )rk F − k ( c ( F − E )) ∩ [ X ] , (4)where ∆ ( p ) q ( c ) denotes the determinant of the p × p matrix with entries ( c q + j − i ) ≤ i,j ≤ p .We are interested in the situation where the variety in question is the Hilbert scheme ofpoints X [ e ] , where X is either a smooth projective curve or a smooth projective surface. Inboth cases X [ e ] is a smooth projective variety of dimension e if X is a curve and 2 e if X isa surface. Moreover, if X is a curve we also have that X [ e ] is isomorphic to the symmetricproduct X e . Moreover, let L be a line bundle of degree d on X and with h ( X, L ) = r + 1whose sections give an embedding of X in P r .Secant varieties V e − fe ( L ) of e -secant ( e − f − X can be endowed with adegeneracy locus structure inside the Hilbert scheme X [ e ] of 0-dimensional subschemes of X of length e . To see this, let E = O X [ e ] ⊗ H ( X, L ) be the trivial vector bundle of rank r + 1 on X [ e ] and L [ e ] := τ ∗ ( σ ∗ L ⊗ O U ) be the tautological bundle of rank e , where U isthe universal family of subschemes parametrised by X [ e ] U = { ( x, [ Z ]) | x ∈ Z and [ Z ] ∈ X [ e ] } ⊂ X × X [ e ] , and σ , τ are the usual projections: X × X [ e ] ⊃ U X X [ e ] σ τ Let Φ : E → L [ e ] be the vector bundle morphism obtained by pushing down to X [ e ] therestriction map σ ∗ L → σ ∗ L ⊗ O U . Then one obtains V e − fe ( L ) as the ( e − f )-th degeneracylocus of Φ. Indeed, fibrewise, the morphism Φ is given by the restrictionΦ Z : H ( X, L ) → H ( X, L | Z ) , while Z ∈ V e − fe ( L ) if and only if rk Φ Z ≤ e − f .From the above considerations we see that V e − fe ( L ) has expected dimensionexp dim V e − fe ( L ) = e · dim X − f ( r + 1 − e + f ) . Our enumerative study thus restricts itself to the case exp dim V e − fe ( L ) = 0 where weexpect to have only finitely many e -secant ( e − f − X . Of course, the numbersobtained from (4) are virtual, unless one is able to prove that the secant variety V e − fe ( l ) isindeed not empty and of expected dimension 0. This has been verified for general curveswith general linear series by Farkas [8].In the remainder of this section we describe the class of V e − fe ( L ) from (4) in moredetail in the cases f = 1 and f = 2. As mentioned in the Introduction, the case f = 1is by now well-known for both curves and surfaces, while the case f = 2 constitutes themain focus of this article. 5 .1 The case f = 1 If f = 1, then we have that e · dim X = r − e + 2, so that if X is a curve, then r = 2 e − X is a surface, then r = 3 e −
2. From now on we write C for curves and S forsurfaces. Hence rk E = r + 1 ∈ { e − , e − } and rk L [ e ] = e , which means that the classof V e − fe ( L ) is given via (4) by∆ ( e )1 ( c ( L [ e ] )) ∩ C [ e ] = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c ( L [ e ] ) c ( L [ e ] ) · · · c e ( L [ e ] )1 c ( L [ e ] ) · · · c e − ( L [ e ] )... ... ... ...0 0 1 c ( L [ e ] ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∈ A ( C [ e ] )for curves and∆ (2 e )1 ( c ( L [ e ] )) ∩ S [ e ] = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c ( L [ e ] ) c ( L [ e ] ) · · · c e ( L [ e ] )1 c ( L [ e ] ) · · · c e − ( L [ e ] )... ... ... ...0 0 1 c ( L [ e ] ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∈ A ( S [ e ] )for surfaces. In both cases we used the fact that E is a trivial bundle so that c ( L [ e ] − E ) = c ( L [ e ] ) c ( E ) = c ( L [ e ] ). Although the above expressions for the degeneracy loci V e − fe ( L ) appearquite complicated, one may obtain a certain degree of simplification by rewriting them interms of determinants in Segre classes of the dual bundle L [ e ] . To see how, we record herethe following useful result, which is a direct consequence of Lemma 14.5.1 of [9]: Lemma 2.1.
Let α , α , . . . and β , β , . . . be two sequences of commuting variables relatedby the identity: (1 + α t + α t + · ) · (1 − β t + β t − β t + · · · ) = 1 . Then ∆ ( p ) q ( α ) = ∆ ( q ) p ( β ) for any positive integers p and q . Substituting c i ( L [ e ] ) for the α i and s i ( L [ e ] ) for the β i yields:∆ ( e )1 ( c ( L [ e ] )) ∩ C [ e ] = s e (( L [ e ] ) ∨ ) ∩ [ C [ e ] ] (5)for curves and ∆ (2 e )1 ( c ( L [ e ] )) ∩ S [ e ] = s e (( L [ e ] ) ∨ ) ∩ [ S [ e ] ] = s e ( L [ e ] ) ∩ [ S [ e ] ] (6)for surfaces.Thus the number of e -secant ( e − P e − and to a surface in P e − is given by Z C [ e ] s e (( L [ e ] ) ∨ ) = deg( s e (( L [ e ] ) ∨ ) ∩ [ C [ e ] ]) , and Z S [ e ] s e (( L [ e ] ) ∨ ) = deg( s e ( L [ e ] ) ∩ [ S [ e ] ]) , respectively.These counts (or their corresponding generating functions) were indepedently com-puted by Cotterill [6], Le Barz [3], and Wang [15] for curves, whereas the surfaces casewas settled in a series of papers by Marian, Oprea, and Pandharipande ([11], [12]), Voisin[14], with some of the earliest partial results dating back to Lehn [10]. In what follows weare interested in carrying out the computations for curves in the case f = 2.6 .2 The case f = 2 If f = 2 we have that e · dim C = 2( r − e + 3) so that rk E = r + 1 = e −
2. The rankof L [ e ] is, as before, e so using (4) again we can write the class of V e − fe ( L ) in this case as∆ ( e/ ( c ( L [ e ] )) ∩ C [ e ] = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c ( L [ e ] ) c ( L [ e ] ) · · · c e +1 ( L [ e ] ) c ( L [ e ] ) c ( L [ e ] ) · · · c e ( L [ e ] )... ... ... ...0 · · · c ( L [ e ] ) c ( L [ e ] ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . To write this in terms of Segre classes we once more use the framework of Section 4. Wehave the partition λ = (2 , , . . . ,
2) of e corresponding to ∆ ( e/ ( c ( L [ e ] )), whose conjugatepartition is µ = (cid:0) e , e (cid:1) . Thus∆ ( e/ ( c ( L [ e ] )) = ∆ (2) e/ ( s (( L [ e ] ) ∨ )) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s e (( L [ e ] ) ∨ ) s e +1 (( L [ e ] ) ∨ ) s e − (( L [ e ] ) ∨ ) s e (( L [ e ] ) ∨ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = s e ( L [ e ] ) − s e − ( L [ e ] ) s e +1 ( L [ e ] ) . Hence the class of V e − fe ( L ) can be written in the case f = 2 in terms of a seconddegree polynomial in certain Segre classes. This is already a simplification compared tothe original expression although less convenient than the top Segre class that appears inthe f = 1 case. In what follows we compute the integrals Z C [ e ] s e ( L [ e ] ) − s e − ( L [ e ] ) s e +1 ( L [ e ] ) . (7)We shall use an inductive argument starting from the simple case of P equipped with theSerre twisting sheaf. Remark . For surfaces S we have∆ ( e )2 ( c ( L [ e ] )) ∩ S [ e ] = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c ( L [ e ] ) c ( L [ e ] ) · · · c e +1 ( L [ e ] ) c ( L [ e ] ) c ( L [ e ] ) · · · c e ( L [ e ] )... ... ... ...0 · · · c ( L [ e ] ) c ( L [ e ] ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and the same argument as for curves yields∆ ( e )2 ( c ( L [ e ] )) = s e ( L [ e ] ) − s e − ( L [ e ] ) s e +1 ( L [ e ] ) . A very useful tool in computing integrals of certain characteristic classes is the beautifulStructure Theorem 4.2 of [7], which we now recall in a slightly simplified form that sufficesfor our needs. In keeping with the notation in loc.cit., let K ( X ) be the Grothendieckgroup generated by locally free sheaves on X . Moreover, let ψ : K ( X ) → H ∗ ( X, Q ) be7 multiplicative function, i.e. a group homomorphism from the additive group K ( X ) of asmooth projective curve or surface X into the multiplicative group of units in H ∗ ( X, Q ).Examples of such multiplicative functions are the total Chern or the total Segre classes.Now, given x ∈ K ( X ), one defines a power series in Q [[ z ]] as follows: H ψ ( X, x ) := ∞ X k =0 Z X [ k ] ψ ( x [ k ] ) z k . We then have:
Theorem 3.1. ([7, Theorem 4.2]) For each integer r there are universal power series A i ∈ Q [[ z ]] with i = 1 , if X is a curve and i = 1 , . . . , if X is a surface, depending onlyon ψ and r , such that for each x ∈ K ( X ) of rank r on every component of X one has H ψ ( X, x ) = e R X c ( x ) A + R X c ( T X ) A when X is a curve and H ψ ( X, x ) = e R X c ( x ) A + R X c ( x ) A + R X c ( x ) c ( T X ) A + R X c ( T X ) A + R X c ( T X ) A if X is a surface.Remark . We note here that in [7], the proof of Theorem 3.1 is done only for surfaces.However, the same argument can be applied in the case of curves using the elements( P , r ·
1) and ( P , O P (1) + ( r − ·
1) of the space K r = { ( X, x ) | X algebraic curve, x ∈ K ( X ) } , which under the map γ : K r → Q sending ( X, x ) to ( R X c ( x ) , R X c ( T X )) yieldtwo linearly independent vectors (0 ,
2) and (1 , R X [ e ] s top and hence thenumber of e -secant ( e − P e − , unfortunately for different choicesof parameters this is no longer the case. Indeed, if we take f = 2 instead of f = 1, then thecycle classes in (5) and (6) corresponding to e -secant ( e − H φψ ( X, x ) := ∞ X k =0 Z X [ k ] φ ( x [ k ] ) ψ ( x [ k ] ) z k , where φ, ψ : K ( X ) → H ∗ ( X, Q ) are additive functions, or more precisely, group homomor-phisms between the additive groups K ( X ) and H ∗ ( X, Q ). In particular, we are interestedin how the power series H φψ relates to the two power series H φ ( X, x ) := ∞ X k =0 Z X [ k ] φ ( x [ k ] ) z k and H ψ ( X, x ) := ∞ X k =0 Z X [ k ] ψ ( x [ k ] ) z k . To this end we prove: 8 heorem 3.2.
For each integer r there are universal power series B i ∈ Q [[ z ]] with i = 1 , if X is a curve and i = 1 , . . . , if X is a surface, depending only on r , φ , and ψ , suchthat for each x ∈ K ( X ) of rank r on every component of X one has H φψ ( X, x ) − H φ ( X, x ) H ψ ( X, x ) = Z X c ( x ) B + Z X c ( T X ) B (8) for curves and H φψ ( X, x ) − H φ ( X, x ) H ψ ( X, x ) = Z X c ( x ) B + Z X c ( x ) B (9)+ Z X c ( x ) c ( T X ) B + Z X c ( T X ) B + Z X c ( T X ) B for surfaces.Proof. As in the proof of Theorem 3.1 in [7] and as already hinted at in Remark 3.1, westart by considering the space K r = { ( X, x ) | X algebraic curve or surface, x ∈ K ( X ) } and the map γ : K r → Q j , with j = 2 if X is a curve and j = 5 if X is a surface, definedby ( X, x ) (cid:18)Z X c ( x ) , Z X c ( T X ) (cid:19) (10)in the case of curves and by( X, x ) (cid:18)Z X c ( x ) , Z X c ( x ) , Z X c ( x ) c ( T X ) , Z X c ( T X ) , Z X c ( T X ) (cid:19) (11)in the case of surfaces. As shown in Remark 3.1 and [7], in both cases one can find elementsof K r whose image under γ form a basis of Q j , for the appropriate value of j .Now, if X = X ⊔ X and we set x i := x | X , then γ ( X, x ) = γ ( X , x )+ γ ( X , x ) by theadditivity of the entries of the vectors in (10) and (11) under disjoint unions. Furthermore,the Hilbert scheme of a disjoint union satisfies X [ k ] = F k + k = k X [ k ]1 × X [ k ]2 , and the classof the corresponding tautological sheaf can be decomposed as x [ k ] | X [ k × X [ k = p ∗ x [ k ]1 ⊕ p ∗ x [ k ]2 , where the p i : X [ k ]1 × X [ k ]2 → X [ k i ] i with i = 1 , φ and ψ we obtain H φψ ( X, x ) = H φψ ( X , x )+ H φψ ( X , x )+ H φ ( X , x ) H ψ ( X , x )+ H φ ( X , x ) H ψ ( X , x ) . We now reinterpret the power series H φψ ( X, x ), H φ ( X, x ), and H ψ ( X, x ) as three functions H φψ , H φ , H ψ : K r → Q [[ z ]]. We then deduce from Theorem 1.1 that these functionsall factor through γ and three additive maps h, f, g : Q j → Q [[ z ]], respectively. Byconstruction we see that f and g are also additive functions. Since the image of γ isZariski dense in Q j , we conclude that h ( y + y ) = h ( y ) + h ( y ) + f ( y ) g ( y ) + f ( y ) g ( y ) , ( f g )( y + y ) = f ( y ) g ( y ) + f ( y ) g ( y ) + f ( y ) g ( y ) + f ( y ) g ( y ) , for all y , y ∈ Q j , with j = 2 when X is a curve and j = 5 when X is a surface. Hence( h − f g )( y + y ) = ( h − f g )( y ) + ( h − f g )( y ), i.e. h − f g is a linear function, whichproves the theorem. 9t is not difficult to extend the above result to the product of several additive functions φ , . . . φ n : K ( X ) → H ∗ ( X, Q ). To do so, we introduce the following power series: H φ i ...φin := ∞ X k =0 Z X [ k ] φ i ( x [ k ] ) · · · φ i n ( x [ k ] ) z k . Using induction on n with base case given by the statement of Theorem 3.2, one gets Corollary 3.3.
For all integers r and n ≥ there are universal power series B i ∈ Q [[ z ]] with i = 1 , if X is a curve and i = 1 , . . . , if X is a surface, depending only on r , φ , . . . , φ n such that for each x ∈ K ( X ) of rank r on every component of X one has that H φ i ...φ n − H φ H φ ...φ n − · · · − ( n − H φ H φ · · · H φ n is linear in the B i , i.e. it is equal to the right-hand-side of (8) or (9) if X is a curve or asurface, respectively.Proof. The proof is essentially the same as the one for Theorem 3.2. The idea is oncemore to view the power series H φ i ...φ il with i , . . . , i l ∈ { , . . . , n } and 1 ≤ l ≤ n asfunctions K r → Q [[ z ]] that each factor through the map γ : K r → Q j from above and acorresponding map f i ...i l : Q j → Q [[ z ]], with j = 2 if X is a curve and j = 5 if X is asurface. Consider the map F : Q j → Q [[ z ]] defined by F ( y ) := f ...n ( y ) − n X i =1 f i ( y ) f ... ˆ i...n ( y ) − n X i,j =1 f i ( y ) f j ( y ) f ... ˆ i... ˆ j...n ( y ) − · · ·− ( n − f ( y ) f ( y ) · · · f n ( y ) , for all y ∈ Q j and where ˆ i indicates the removal of the index i . Using induction on n and the fact that γ has a dense image in Q j we get F ( y + y ) = F ( y ) + F ( Y ) for all y , y ∈ Q j .Thus, the remainder of the paper is dedicated to applications of Theorem 3.2 andCorrolary 3.3 to the problem of counting the number of certain secant planes to algebraiccurves. This section is dedicated to the calculation of the Segre integral Z C [ e ] s e ( L [ e ] ) − s e − ( L [ e ] ) s e +1 ( L [ e ] ) , (12)where C is a smooth curve with a line bundle L , and e ≥ V e − e ( L ) = 0. The general strategy we follow here is that of reducing thecomputation to cases that are easily understood. One of the main ingredients of thereduction is Theorem 1.1 that allows us to parametrise the sought-after integrals by asmall number of characteristic numbers that are well-behaved with respect to disjointunions. We begin by describing our reduction procedure in general terms in Section 4.1.Sections 4.2 and 4.3 are dedicated to implementing this reduction to a particular choiceof parameters yielding a very nice form of the above integral, but which unfortunately isnot generalisable. In Section 4.4 we give a structure formula for (12).10 .1 Reduction to the projective line As mentioned before, for a curve C of genus g equipped with a line bundle L of degree d it follows from Theorem 1.1 that the integral P g,d = R C [ k ] P , where P is a polynomial inthe Chern classes of L [ k ] , is itself a polynomial in R C c ( C ) = χ ( C ) = 2 − g and R C c ( L ) =deg( L ). Thus in order to compute P g,d we attempt to find, or construct a suitable pair( C, L ) with the correct characteristic numbers g and d for which the calculation is easy;for characteristic numbers g ′ and d ′ for which such a convenient pair ( C ′ , L ′ ) cannot easilybe found, we instead relate the pair ( C ′ , L ′ ) to another pair ( C ′′ , L ′′ ) with g ( C ′′ ) = g ′ anddeg( L ′′ ) = d ′ via a geometric construction and use this to obtain P g ′ ,d ′ . We now explainthis procedure in more detail.For the first step we compute P ,d for any polynomial P in the Chern classes of L [ k ] using the very convenient pair ( P , O P ( d )). This is done in Section 4.2. With this resultin hand, we then proceed as follows: fix two positive integers g and d . Let C = C ⊔ C be the disjoint union of a curve C of genus g of with another curve C of genus g . Equip C with a line bundle L so that L = ( L ) | C has degree d and set L = L | C . Suppose L and L are line bundles of degree d and d , respectively. As mentioned before, the Hilbertscheme of k points of C is given by C [ k ]1 = F k + k = k C [ k ] × C [ k ]2 , while the tautologicalbundle corresponding to L has the following property:( L ) [ k ] | C [ k × C [ k = p ∗ L [ k ] ⊕ p ∗ L [ k ]2 , where the p i denotes the projection to C [ k i ] i for i = 1 ,
2. Furthemore, one immediately seesthat since χ ( C ) = χ ( C ) + χ ( C ), and if g denotes the genus of C , then g = g + g − L ) = d + deg( L ).Thus, if we take C ≃ P and L = O P (1), then C has genus g − L is of degree deg( L ) = d + 1. Hence, one can compute P g,d by induction on g with base case P ,d and induction step going from P g − ,d +1 to P g,d . One of course needsto understand for each choice of P how P g − ,d +1 and P g,d relate to each other. We shalluse this strategy in Section 4.3 in order to calculate a special case of the integral in (12). P We now compute the Chern and Segre classes of the tautological bundle associated to O P ( d ). Note first that we have the following concrete description of the Hilbert schemeof points of the projective line: ( P ) [ k ] ≃ P k . From the proof of Theorem 2 in [12] we havethat ch (cid:16) O P k ( d ) [ k ] (cid:17) = ( d + 1) − ( d − k + 1) · exp( − h ) , (13)where h denotes the hyperplane class on P k . We therefore have that ch (cid:16) O P k ( d ) [ k ] (cid:17) =rk (cid:16) O P k ( d ) [ k ] (cid:17) = k and for i ≥ ch i (cid:16) O P k ( d ) [ k ] (cid:17) = ( d − k + 1) i ! h i = 1 i ! p i , p i is the i -th power sum symmetric polynomial in the Chern roots of O P k ( d ) [ k ] , i.e. p i = α i + · · · + α ik . To find the Chern classes of O P k ( d ) [ k ] , recall that the following expression of the totalChern class in terms of elementary symmetric polynomials in the Chern roots: c (cid:16) O P k ( d ) [ k ] (cid:17) = 1 + e ( α , . . . , α k ) + · · · + e k ( α , . . . , α k ) . Moreover, we also have the following relation between the e j and the p j : e l = 1 l l X j =1 ( − j − e l − j p j , (14)where we omit the argument ( α , . . . , α k ) to ease notation. Using this, we can prove Lemma 4.1.
For all positive integers l , we have: c l (cid:16) O P k ( d ) [ k ] (cid:17) = d − k + ll ! h l ,s l (cid:16) O P k ( d ) [ k ] (cid:17) = ( − l d − k + 1 l ! h l . Proof.
We show the computation for the Chern classes. The Segre classes are then ob-tained in a similar fashion. The proof goes by induction. The base case is confirmedimmediately: c (cid:16) O P k ( d ) [ k ] (cid:17) = ( d − k + 1) h, as expected from (13). The induction step follows from (14): e l = 1 l l X j =1 ( − j − d − k + l − jl − j ! h l − j · ( − j − ( d − k + 1) h j = d − k + 1 l l X i =1 d − k + l − jl − j ! h l = d − k + 1 l d − k + ll − ! h l = d − k + ll ! h l , where in the third equality we used the hockey-stick identity.Thus, it follows that if P is a monomial c j i · · · c j l i l of degree k in Chern classes of L [ k ] on C [ k ] for a rational curve C , then P ,d = (cid:0) d − k + i i i (cid:1) · · · (cid:0) d − k + i l i l (cid:1) . In this section we restrict ourselves to the case e = 4, which, as mentioned before,has nice properties that are unfortunately not generalisable to higher values of e . It doeshowever offer an interesting application of Theorem 3.2 and provides us also with sometechnical results needed in the proof of the general case in Section 4.4.12hus, the purpose of this section is to compute the integral Z C [4] s ( L [4] ) − s ( L [4] ) s ( L [4] )yielding the number of 4-secant lines to a curve of degree d ≥ P . An elementarycomputation shows that s ( L [4] ) − s ( L [4] ) s ( L [4] ) = c ( L [4] ) + s ( L [4] ) − c ( L [4] ) ch ( L [4] ) . (15)As discussed in the Introduction, the generating functions of the integrals R C [4] c ( L [4] )and R C [4] s ( L [4] ) have already been computed. Using the Segre generating functions fromSection 4 we can read off the coefficients to obtain (see for example Paragraph 5) in [3]): Z C [4] s ( L [4] ) = d ! − d ! + d ! (6 − g ) + 5 d ( g −
2) + g ! − g + 15 . To find a similar expression for c ( L [4] ), one could read them off the coefficients of thegenerating functions in [15], or one could use an elementary recursive argument based onthe multiplicativity of the total Chern class. We show how to do this in the following Lemma 4.2.
Given a smooth curve C with a line bundle L of degree d . We have thefollowing closed form formulas for the top Chern and Segre classes of the tautologicalbundle L [ k ] on C [ k ] : Z C [4] c ( L [4] ) = d + g ! − g − X m =0 d + g + m ! − g − X n =0 d + m + n !! . Proof.
Let C = C ⊔ C , where C is a smooth curve. Equip C with a line bundle L anddenote by L and L its restrictions to C and C , respectively. Let p, p be the projectionsfrom C [ k ] × C [ k ]2 to C [ k ] and C [ k ]2 , respectively. Then we have that Z C [ k ]1 c ( L [ k ]1 ) = X k + k = k Z C [ k × C [ k c (cid:0) p ∗ L [ k ] ⊕ p ∗ L [ k ]2 (cid:1) = X k + k = k Z C [ k c ( L [ k ] ) Z C [ k c ( L [ k ]2 )= X k + k = k Z C [ k c k ( L [ k ] ) Z C [ k c k ( L [ k ]2 ) , where we used the fact that the only contribution to the integral comes from the respectivetop Chern classes. We let now C ≃ P and C be a smooth curve of genus g . Moreover,let L = O P (1) and let L be a line bundle of degree d on C from which it follows thatdeg L = d + 1. Furthermore, as explained in Section 4.1, g ( C ) = g ( C ) + g ( C ) − g − R C [ k ] c ( L [ k ] ) by C k,g,d . We then rewrite the above equality as C k,g − ,d +1 = X k + k = k C k , , C k ,g,d .
13e easily see that C , ,d = d and it follows that C ,g,d = C ,g − ,d +1 − C , , = C , ,d + g − gC , , = d. Furthermore, C ,g,d = C ,g − ,d +1 − C ,g,d C , , − C , , . Since C , , = 0, we obtain C ,g,d = C ,g − ,d +1 − d = C , ,d + g − ( d + ( d + 1) + · · · + ( d + g − d + g ! − (cid:18) dg + g ( g − (cid:19) = d ( d − . We now write C ,g,d = C ,g − ,d +1 − C ,g,d C , , − C ,g,d C , , − C , , . We have that C , , = 0 which yields C ,g,d = C ,g − ,d +1 − C ,g,d = C ,g − ,d +1 − d ( d − C , ,d + g − g − X m =0 d + m ! = d + g ! − g − X m =0 d + m ! . We now finally reach the desired term: C ,g,d = C ,g − ,d +1 − C ,g,d C , , − C ,g,d C , , − C ,g,d C , , − C , , . We again use that C , , = 0 to obtain C ,g,d = C ,g − ,d +1 − C ,g,d = C , ,d + g − g − X m =0 C ,g,d + m = d + g ! − g − X m =0 d + g + m ! − g − X n =0 d + m + n !! . The aim of the remainder of this section is therefore to find R C [4] c ( L [4] ) ch ( L [4] ) byusing Theorem 3.2, or more precisely the proof thereof. In fact we prove the following: Proposition 4.3.
Let C be a curve of genus g equipped with a line bundle L of degree d .The closed formula Z C [ k ] c ( L [ k ] ) ch k − ( L [ k ] ) = ( − k ( k − A ( k, g, d ) , here A ( k, g, d ) = ( d + g − k + 1) − g (2 − k )(2 − k − g − ) − (4 − k ) ( g − d + g − holds forall k ≥ and its corresponding generating function is ∞ X k =0 z k Z C [ k ] c ( L [ k ] ) ch k − ( L [ k ] ) = ( B ( g, d ) z + C ( g, d ) z + D ( g, d )) ze − z , where B ( g, d ) = ( g − (cid:16) − g (cid:17) + ( g − d + g − ,C ( g, d ) = g − d + 2 + ( g − d + g − ,D ( g, d ) = − ( d + g ) + 4 g − g ( g − . Proof.
To apply the setting of Theorem 3.2 to our situation, we first remark on a slightchange of notation intended to make computations easier to follow: from Theorem 1.1 weknow that, if φ is a polynomial function in Chern classes of line bundles x on C , then thecoefficients of z k in H φ ( C, x ) depend only on g and d . We thus write H φ ( g, d ) instead of H φ ( C, x ). We then take φ = c and ψ = ch k − and set the following generating functions: H φ ( g, d ) := ∞ X k =0 z k Z C [ k ] c ( L [ k ] ) = 1 + dzH ψ ( g, d ) := ∞ X k =1 z k Z C [ k − ch k − ( L [ k − ) H φψ ( g, d ) := ∞ X k =1 z k Z C [ k ] c ( L [ k ] ) ch k − ( L [ k ] ) . We proceed by induction. Consider the curve C = C ⊔ C where C ≃ P and C isa smooth curve of genus g . Let L be a line bundle on C such that ( L ) | C = O P ( d ),for some positive integer d , and L = ( L ) | C has degree d . As usual it follows thatdeg L = d + d and that g ( C ) = g −
1. From Theorem 3.2 applied to the pair ( C , L )and setting d = 1 we therefore obtain the following equality: H φψ ( g − , d + 1) = H φψ (0 ,
1) + H φψ ( g, d ) + H ψ (0 , H φ ( g, d ) + H φ (0 , H ψ ( g, d ) . (16)Thus the induction step decreases the genus of the curve and increases the degree of theline bundle at every step whence we obtain H φψ ( g, d ) = H φψ ( g − , d + 1) − H φψ (0 , − H ψ (0 , H φ ( g, d ) − H φ (0 , H ψ ( g, d )= H φψ (0 , d + g ) − gH φψ (0 , − H ψ (0 , − H φ (0 , , (17)where Σ = H φ ( g, d ) + H φ ( g − , d + 1) + · · · + H φ (1 , d + g − , Σ = H ψ ( g, d ) + H ψ ( g − , d + 1) + · · · + H ψ (1 , d + g − . H φ (0 ,
1) = 1 + z and moreover thatΣ = g + ( d + ( d + 1) + · · · + ( d + g − z = g + ( g − d + g − z. (18)We now compute the remaining unknown terms in (17). Step 1.
Finding H ψ ( g, d ).We consider first the auxiliary generating function F g,d ( z ) = ∞ X k =0 z k Z C [ k ] ch k ( L [ k ] )which, by shifting the coefficients has the property that z F g,d ( z ) = H ψ ( g, d ). Denote by a k,g,d the coefficient of z k in F g,d ( z ). From Lemma 4.1 we have a k, ,d = Z P k ch k ( O P ( d ) [ k ] ) = ( − k − k ! ( d − k + 1) , To find a k,g,d for g = 0, we again take advantage of the additivity of the Chern characteras follows: as before let C = C ⊔ C be the disjoint union of two smooth curves and L, L , L and p, p as before. Then, for all k ≥ a k,g ( C ) , deg( L ) = Z C [ k ]1 ch k ( L [ k ]1 ) = X k + k = k Z C [ k × C [ k ch k (cid:0) p ∗ L [ k ] ⊕ p ∗ L [ k ]2 (cid:1) = X k + k = k Z C [ k ch k (cid:0) L [ k ] (cid:1) + Z C [ k ch k (cid:0) L [ k ]2 (cid:1) = Z C [ k ] ch k (cid:0) L [ k ] (cid:1) + Z C [ k ]2 ch k (cid:0) L [ k ]2 (cid:1) . (19)Setting as usual C = P and L = O P (1) and taking C to be a smooth curve of genus g and L a line bundle of degree d and plugging into (19) yields: a k,g,d = a k,g − ,d +1 − a k, , . Using induction on g we obtain a k,g,d = a k, ,d + g − ga k, , = ( − k − k ! ( d + ( k − g − k + 1) . (20)To find the generating function F g,d ( z ) = P k =0 a k,g,d z k , we start from the recurrencerelation for the coefficients a k,g,d : − ( k + 1) a k +1 ,g,d = a k,g,d + ( − k − k ! ( g − . This recurrence relation yields the following differential equation for F g,d ( z ): ddz F g,d ( z ) + F g,d ( z ) = ( g − e z , a ,g,d = − ( d − g + 1), we can easily solve to obtain: F g,d ( z ) = (( g − z − d + g − e − z . Finally, we have that H ψ ( g, d ) = (( g − z − d + g − ze − z . Note that this gives an alternative way of obtaining the generating function for the Cherncharacter of tautological bundles on symmetric products for curves. For a different ap-proach using Grothendieck-Riemann-Roch, see [1, Lemma 2.5, Chapter VIII §2].
Step 2.
Finding Σ .Having computed H ψ ( g, d ), we now easily getΣ = H ψ ( g, d ) + H ψ ( g − , d + 1) + · · · + H ψ (1 , d + g − g − z − d + g −
1) + · · · + ( − d − g + 1)) ze − z = (cid:18) g ( g − z − Σ + g ( g − (cid:19) ze − z . Step 3.
Finding H φψ (0 , d + g ) and H φψ (0 , z k in H φψ (0 , d )is given by Z P k c ( O P ( d ) [ k ] ) ch k − ( O P ( d ) [ k ] ) = ( − k ( k − d − k + 1) , where we again used Lemma 4.1. We shall compute the auxiliary generating function G d ( z ) = ∞ X k =0 b k,d z k := ∞ X k =0 ( − k +1 k ! ( d − k ) z k with the property that H φψ (0 , d ) = z G d ( z ). To do so, we observe that the recurrencerelation − ( k + 1) b k +1 ,d = b k,d + ( − k k ! (2 d −
1) + 2 ( − k − ( k − z k in G d ( z ) gives rise to the differential equation ddz G d ( z ) + G d ( z ) + (2 d − e − z + ze − z = 0 . With the initial condition b = − d we find that the solution is G d ( z ) = − (cid:18) z d − z + d (cid:19) e − z , so that we finally obtain H φψ (0 , d ) = − (cid:18) z d − z + d (cid:19) ze − z . H φψ ( g, d ) in the statement of theProposition.To get the closed form of the coefficient of z k in H φψ ( g, d ), one could either read itoff from the generating function itself or just apply the same induction procedure: for adisjoint union C = C ⊔ C of smooth curves equipped with the usual line bundles L, L , L ,the following holds: Z C ′ [ k ] c ( L ′ [ k ] ) ch k − ( L ′ [ k ] ) = Z C [ k ]1 c ( L [ k ]1 ) ch k − ( L [ k ]1 ) + Z C c ( L ) Z C [ k − ch k − ( L [ k − )++ Z C [ k − ch k − ( L [ k − ) Z C c ( L ) + Z C [ k ]2 c ( L [ k ]2 ) ch k − ( L [ k ]2 ) . Let b k,g,d := Z C [ k ] c ( L [ k ] ) ch k − ( L [ k ] ) , With the same choices for C , C , L , L as above we have that b k,g − ,d +1 = b k, , + 1 · a k − ,g,d + a k − , , · d + b k,g,d . This in turn yields b k,g,d = b k,g − ,d +1 − ( − k ( k − − k ) − ( − k ( k − d + ( k − g − k + 2) − d ( − k ( k − − k )= b k,g − ,d +1 − ( − k ( k − − k ) + (4 − k ) d + ( k − g − b k, ,d + g − ( − k ( k − (cid:18) g (2 − k ) + (4 − k ) ( g − d + g − k − g ( g − (cid:19) . Finally, we have that b k, ,d + g = ( − k ( k − d + g − k + 1) , which concludes the proof.We therefore have Z C [4] c ( L [4] ) ch ( L [4] ) = b ,d,g = 16 (( d + g − − g − g ( g − . f = 2 case Unfortunately we were not able to find a similar expression to the one in (15) forhigher values of e . Therefore, to deal with the general situation we take a different ap-proach, although we will still rely on a similar induction step and Theorem 3.2 for thefinal computation.Thus, the goal of this section is to find an expression for the integral Z C k s k ( L [ k ] ) − s k − ( L [ k ] ) s k +1 ( L [ k ] ) , C and L are as before and k ≥
2. To proceed, recall that the total Segre classof the dual vector bundle ( L [ k ] ) ∨ may be written in terms of the complete homogeneoussymmetric polynomials h i in the Chern roots as follows: s (( L [ k ] ) ∨ ) = 1 + h ( α , . . . , α k ) + · · · + h k ( α , . . . , α k ) , and the Segre power series s t = 1 + s t + s t + · · · can therefore be written as: s t (( L [ k ] ) ∨ ) = ∞ Y i =1 − α i t We now obtain the generating function with coefficients given by s i ( L [ k ] ) = s i (( L [ k ] ) ∨ ).To do so, note that for two generating functions F ( z ) = P ∞ i =1 a i z i and G ( z ) = P ∞ i =1 b i z i ,their Hadamard product is given by( F ∗ G )( z ) = ∞ X i =1 a i b i z i . Lemma 4.4. (i) For k ≥ we have: s t (( L [ k ] ) ∨ ) ∗ s t (( L [ k ] ) ∨ ) = exp X n ≥ t n n ( n !) ch n ( L [ k ] ) . Moreover, the coefficient s n (( L [ k ] ) ∨ ) of t n is given by X n ,n ,... ≥ n ! n ! · · · (1! ch ) n (2! ch ) n · · · n n · · · , (21) where n + 2 n + · · · = n .(ii) Denote by s lt and s rt the generating functions obtained by shifting s t by one to theleft and to right, respectively, i.e. s lt = t + s t + · · · + s k − t k + · · · ,s rt = s + s t + · · · + s k +1 t k + · · · . Then the coefficient s n − ( L [ k ] ) ∨ ) s n +1 ( L [ k ] ) ∨ ) of t n in s lt ∗ s rt is X n ,n ,... ≥ n ! n ! · · · (1! ch ) n (2! ch ) n · · · n n · · · X l ,l ,... ≥ l ! l ! · · · (1! ch ) l (2! ch ) l · · · l l · · · (22) where n + 2 n + · · · = n − and l + 2 l + · · · = n + 1 Proof. (i) This follows by standard results on symmetric functions. In particular, one hasthat s t ∗ s t = k Y i,j =1 − α i α j t , X n ≥ t n n k X i,j =1 ( α i α j ) n = ln( s t ∗ s t ) . Using the fact that ch n = p n n ! we get P ki,j =1 ( α i α j ) n = p n = ( n !) ch n , where p n denotesas before the n -th power sum symmetric polynomial in the Chern roots of L [ k ] . Theexpression of the coefficient of t n follows immediately.(ii) This follows by a standard combinatorial computation and recognising that s lt = ts t and s rt = t ( s t − Z C [2 k ] s k ( L [ k ] ) − s k − ( L [ k ] ) s k +1 ( L [ k ] ) , it is enough to make sense of the integrals of the expressions (21) and (22). To do so weuse Corollary 3.3 to find separate expressions for Z C [2 k ] s k ( L [ k ] ) (23)and Z C [2 k ] s k − ( L [ k ] ) s k +1 ( L [ k ] ) . (24)To obtain (23) we must therefore consider the integral Z C [2 k ] X n ,n ,... ≥ C n ,n ,... ch n ( L [ k ] ) ch n ( L [ k ] ) · · · where n + 2 n + · · · = k and C n ,n ,... denote the constants from (21). Now, for eachchoice of n i with n + 2 n + · · · = k , Corollary 3.3 yields Z C [2 k ] ch n ( L [ k ] ) ch n ( L [ k ] ) · · · = L + Y i (cid:16)Z C [ i ] ch i ( L [ i ] ) (cid:17) n i ++ X ≤ i ≤ n ≤ i ≤ n ... L i ...i j (cid:16)Z C ch ( L ) (cid:17) i (cid:16)Z C [2] ch ( L [2] ) (cid:17) i · · · , where L and the L i ...i j are linear functions as in the context of Corollary 3.3. Note thatfor simplicity we omit the scalar multiples before each term in the sum. Finally, eachterm between parantheses in the above equality can be found by either reading off thecoefficients of the generating function F g,d from Step 1 of the proof of Proposition 4.3, ordirectly by a recursive argument as done in the case of the coefficients b k,g,d from loc.cit.With either method one obtains: Z C [ i ] ch i ( L [ i ] ) = ( − i − i ! ( d + (1 − i )(1 − g )) . This then yields the desired structure result for (23). A similar expression can then beobtained for (24). 20 eferences [1] E. Arbarello et al.
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Albert-Ludwigs-Universität Freiburg, Mathematisches Institut, Abteilung Reine Mathematik,Ernst-Zermelo-Str 1, 79104 Freiburg
E-mail address: [email protected]@math.uni-freiburg.de