Discriminant and Hodge classes on the space of Hitchin's covers
aa r X i v : . [ m a t h . AG ] M a r DISCRIMINANT AND HODGE CLASSES ON THE SPACE OFHITCHIN’S COVERS.
MIKHAIL BASOK Abstract.
We continue the study of the rational Picard group of the moduli spaceof Hitchin’s spectral covers started in [11]. In the first part of the paper we expand the“boundary”, “Maxwell stratum” and “caustic” divisors introduced in [11] via the setof standard generators of the rational Picard group. This generalizes the result of [11],where the expansion of the full discriminant divisor (which is a linear combination ofthe classes mentioned above) was obtained. In the second part of the paper we derivea formula that relates two Hodge classes in the rational Picard group of the modulispace of Hitchin’s spectral covers. Introduction
Hitchin’s integrable systems arise as a result of dimensional reduction of the self-dual Yang-Mils equation, see [5], [6], [2]. The Hamiltonians of a Hitchin’s system areencoded in the so-called spectral cover b Σ (see [3], [4]) which is an n -sheeted cover of a(smooth or, more generally, stable) complex projective curve Σ defined as a subvarietyof T ∗ Σ: b Σ = { ( x, v ) ∈ T ∗ Σ | P ( v, x ) = 0 } , (1.1)where P ( v, x ) = v n + q ( x ) v n − + · · · + q n ( x ) , (1.2) q j is a j -differential on Σ (i.e. a holomorphic section of K ⊗ j Σ ). In the frameworkof [3], the equation defining b Σ is given by the characteristic polynomial P ( v, x ) =det(Φ( x ) − vI ) of the so-called Higgs field Φ on Σ.We consider the moduli space P M ( n ) g of Hitchin’s spectral covers in the case ofGL( n, C ) Hitchin’s systems, when all differentials q j as assumed to be arbitrary. A pointin P M ( n ) g parametrizes a pair (Σ , [ P ]), where Σ is a genus g curve and P is a polynomialof the form (1.2) considered up to multiplication by a non-zero constant ξ given by( ξ · P )( v, x ) = ξ n P ( ξ − v, x ). As a space P M ( n ) g is a bundle over the Deligne-Mumfordcompactification of the moduli space M g of genus g curves with fibers isomorphic toa weighted projective space, see Section 3 for details. Notice that if n = 1 then P M ( n ) g is just the total space of the projectivized Hodge bundle which can be thought of asa closure of the moduli space of Abelian differentials (considered up to a constant) ongenus g smooth projective curves. A. Kokotov and D. Korotkin [7] introduced a taufunction on this moduli space called Bergman tau function. The Bergman tau function A Laboratory of Modern Algebra and Applications, Department of Mathemat-ics and Mechanics, Saint-Petersburg State University, 14th Line, 29b, 199178 Saint-Petersburg, Russia. is a generalization of the Dedeking eta function (they coincide if g = 1) and can beinterpreted as determinants of a family of Cauchy-Riemann operators in the spiritof [13]. Studying the asymptotycs of the Bergman tau function near the boundary ofthe moduli space of Abelian differentials (embedded into the total space of the Hodgebundle) D. Korotkin and P. Zograf [9] developed a new relation in the rational Picardgroup of the projectivized Hodge bundle. Thereafter the construction of the Bergrmantau function was generalized to the case of the moduli space of n-differentials (whichis closely related with P M ( n ) g when n = 2, see [10] for n = 2 and [8] for n > P M ( n ) g for any n [11]. Study of the properties of the Bergmantau function on P M ( n ) g allows to express the class of the full Discriminant locus in therational Picard group of P M ( n ) g via the set of its standard generators (see Theorem 1).The first objective of the current paper is to enhance and specify this result usingstandard methods of algebraic geometry.Let Σ be a smooth genus g curve and let P be a polynomial of the form (1.2).Then the discriminant W ( x ) = Discr( P ( · , x )) is an n ( n − W is equal to the branching divisor of the spectral cover b Σ → Σ associatedwith P . Generically, all zeros of W are simple which implies that b Σ is smooth and thecover b Σ → Σ is simply ramified. When two zeros of W coalesce, the local behavior ofthe map b Σ → Σ changes with respect to one the following three ways (we follow thenotation of [11] in this description):1) A node (normal self-crossing of b Σ) occurs at the ramification point of b Σ → Σover the double zero of W . We call the locus of such covers The “boundary”. .2) Two distinct ramification points of b Σ → Σ arise in the preimage of the doublezero of W . We call the locus of such covers The “Maxwell stratum”.
3) Two ramification points of b Σ → Σ coalesce to become a ramification point oforder 3 over the double zero of W . We call the locus of such covers The “caustic”. .We use the notation of a ”Maxwell stratum” and a ”caustic” following the notationdeveloped in V. Arnold’s school in Moscow, see for example [12].The correspondence P Discr( P ) defines a map from P M ( n ) g to the modulispace of pairs (Σ , W ), where W is an n ( n − P D W denote the pullback of the divisorconsisting of those W that have at least one multiple zero (see Section 4 for details).Then the support of P D W splits into the union of three components P D ( b ) W ∪ P D ( m ) W ∪ P D ( c ) W in accordance to the three possibilities described above. We call the divisor P D W the full discriminant divisor . The class of the divisor P D W in the rationalPicard group of P M ( n ) g is called the class of the universal Hitchin’s discriminant . Thefollowing theorem was proven in [11, Theorem 3.2]: Theorem 1.
The divisor
P D W satisfies P D W = P D ( b ) W + 2 P D ( m ) W + 3 P D ( c ) W ISCRIMINANT AND HODGE CLASSES ON THE SPACE OF HITCHIN’S COVERS. 3 and the class of the universal Hitchin’s discriminant
P D W is expressed in terms of thestandard generators of Pic( P M ( n ) g ) ⊗ Q as follows: [ P D W ] = n ( n − (cid:16) ( n − n + 1)(12 λ − δ ) − g − n − n + 1) φ (cid:17) . Here δ = P [ g/ j =0 δ j is the pullback of the class of the Deligne-Mumford boundaryof M g , see Section 4.1 for details. We generalize this result by expressing the class ofeach of the three components of the full discriminant divisor via the set of generatorsof Pic( P M ( n ) g ) ⊗ Q : Theorem 2.
Let n ≥ and g ≥ . The following formulas hold in Pic( P M ( n ) g ) ⊗ Q : [ P D ( b ) W ] = n ( n − (cid:16) ( n + 1)(12 λ − δ ) − g − n + 1) φ (cid:17) [ P D ( m ) W ] = n ( n − n − n − (cid:16) λ − δ + 4( g − φ (cid:17) [ P D ( c ) W ] = n ( n − n − (cid:16) λ − δ − g − φ (cid:17) . The second objective of the paper is to relate two Hodge classes on P M ( n ) g . Notethat since the degree of the cover b Σ → Σ is equal to n and the degree of the branchingdivisor is deg div( W ) = 2 n ( n − g − b Σ is b g = g ( b Σ) = n ( g − P M ( n ) g → M g and P M ( n ) g → M b g , where the first morphismmaps (Σ , [ P ]) to the moduli of Σ and the second one maps (Σ , [ P ]) to the moduli of b Σ. Hence we can define two Hodge classes: the class λ is the pullback of the Hodgeclass from M g and the class b λ is the pullback of the Hodge class from M b g . The nexttheorem provides a formula in Pic( P M ( n ) g ) ⊗ Q which relates λ and b λ : Theorem 3.
Let n ≥ and g ≥ . The following formula holds in Pic( P M ( n ) g ) ⊗ Q : b λ = n (2 n − λ − n ( n − n + 1)( g − φ − n ( n − δ. Note that all coefficients of the right-hand side in the formula in Theorem 3 areintegers.The paper is organized as follows. In Section 2 we recall some basic facts about thegeometry of the space of monic polynomials with multiple roots and derive a technicallemma that will be used for a local analysis of the discriminant W with multiple zeros.In Section 3 we recall the construction of P M ( n ) g and some of its basic properties, andintroduce some related notation. In Section 4 we prove Theorem 2 and in Section 5 weprove Theorem 3. MIKHAIL BASOK Acknowledgments.
I would like to thank Dmitry Korotkin and Peter Zograf forhelpful discussions. The research was supported by the grant of the Government of theRussian Federation for the state support of scientific research carried out under thesupervision of leading scientists, agreement 14.W03.31.0030 dated 15.02.2018.2.
Variety of monic polynomials
Recall that a polynomial of the form P ( t ) = t n + q t n − + · · · + q n is called monic . In this section we assume that ( q , . . . , q n ) ∈ C n . The discriminant Discr( P )is a polynomial of q , . . . , q n and the equation Discr( P ) = 0 defines the affine variety D ⊂ C n of monic polynomials with multiple roots (see [12]). The variety D is notsmooth: it has a normal crossing along the subvariety D ( m ) ⊂ D that parametrizespolynomials with two multiple roots, and it has a cusp along the subvariety D ( c ) ⊂ D corresponding to polynomials that have a root of order 3 or bigger. To construct thenormalization of D let us consider the variety b D ⊂ C n × C given by b D = { ( q , . . . , q n , t ) ∈ C n × C | P ( t ) = 0 , P ′ ( t ) = 0 } , (2.1)where P ( t ) = t n + q t n − + · · · + q n . The forgetful projection C n × C → C n maps b D onto D , and the induced mapping b D → D has the degree one over an open subset of D . To see that b D is smooth observe that the functions t, P ( t ) , P ′ ( t ) , . . . , P ( n − ( t ) formanother coordinate system on C n × C . In this coordinates b D is just a linear subspacegiven by two linear equations P ( t ) = 0 , P ′ ( t ) = 0.The group C ∗ acts on the space C n of monic polynomials by the rule ( ξ · P )( t ) = ξ n P ( ξ − t ). In terms of the coefficients q , . . . , q n this action can be rewritten as ξ · ( q , q , . . . , q n ) = ( ξq , ξ q , . . . , ξ n q n ) . (2.2)Denote by P C n the projectivization under this action. The weighted projective space P C n is a smooth orbifold. The variety D is equivariant under the action of C ∗ andwe denote its projectivization by P D . The action of C ∗ on C n lifts to the action of C ∗ on C n × C given by ξ · ( P, t ) = ( ξ · P, ξt ). The variety b D is equivariant under thisaction and we denote the projectivization by P b D . Note that the map b D → D inducesthe map P b D → P D (although we do not have a map from P ( C n × C ) to P C n ) ofprojectivized varieties. Lemma 2.1.
Let n ≥ . Given a monic polynomial P ( t ) = t n + q t n − + · · · + q n wedenote by P n − ( t ) = t n − + q t n − + · · · + q n − the polynomial t − ( P ( t ) − ( q n − t − q n )) .There exist polynomials R , R , S ∈ C [ q , . . . , q n ] such that the equation Discr( P ) = − q n q n − Discr( P n − ) + q n ( q n − R + q n R ) + q n − S holds for any choice of the polynomial P ( t ) = t n + q t n − + · · · + q n .Proof. Clearly, Discr( P ) lies in the ideal in C [ q , . . . , q n ] generated by q n and q n − , sowe can find S , S ∈ C [ q , . . . , q n ] such that Discr( P ) = q n S + q n − S where S isindependent of q n . Let S = q n − S + S , where S is independent on q n − . We obtainDiscr( P ) = q n S + q n − S + q n − S . (2.3) ISCRIMINANT AND HODGE CLASSES ON THE SPACE OF HITCHIN’S COVERS. 5
Consider the polynomial P z ( t ) = ( t − z )( t − z ) P n − ( t ) = t n + q ( z ) t n − + · · · + q n ( z )where z is a formal variable. Then q n ( z ) is divisible by z and q n − ( z ) is divisible by z but not by z . We haveDiscr( P z ) = q n ( z ) S ( q ( z ) , . . . , q n ( z )) + q n − ( z ) S ( q ( z ) , . . . , q n − ( z ))++ q n − ( z ) S ( q ( z ) , . . . , q n − ( z ))= q n − ( z ) S ( q , . . . , q n − ) + z Q due to (2.3), where Q ∈ C [ z, q , . . . , q n ] is some polynomial. Since Discr( P z ) is divisibleby z , we see that S must be zero, hence S = q n − S andDiscr( P ) = q n S + q n − S. (2.4)We can find R , R , R , R ∈ C [ q , . . . , q n ], where the polynomial R is indepen-dent of q n , q n − and R is independent of q n , q n − , q n − , such that S = q n R + q n − R + q n − R + R . Equation (2.4) then looks like followsDiscr( P ) = q n q n − R ( q , . . . , q n − ) + q n ( q n − R + q n R ) + q n − S + q n R ( q , . . . , q n − ) . (2.5)Put P n − ( t ) = t n − + q t n − + · · · + q n − , and consider the polynomial P z ( t ) = ( t − z )( t − z )( t + 2 z ) P n − ( t ) == ( t − zt + 36 z ) P n − ( t ) = t n + q ( z ) t n − + · · · + q n ( z )where z is a formal variable. Then q n ( z ) and q n − ( z ) are divisible by z , and q n − ( z )is divisible by z . Using these observations together with (2.5) we conclude thatDiscr( P z ) = q n ( z ) R ( q , . . . , q n − ) + z Q , where Q ∈ C [ z, q , . . . , q n ] is some polynomial. Note that the Discr( P z ) is divisibleby z (it is even divisible by z ). Since q n ( z ) is not divisible by z , the polynomial R should be zero. It follows that we can now rewrite (2.5) asDiscr( P ) = q n q n − R ( q , . . . , q n − ) + q n ( q n − R + q n R ) + q n − S. (2.6)Using this equation and the factorization properties of q n ( z ) , q n − ( z ) , q n − ( z ) again weconclude that Discr( P z ) = q n ( z ) q n − ( z ) R ( q ( z ) , . . . , q n − ( z )) + z Q , where Q ∈ C [ z, q , . . . , q n ] is some polynomial. Since Discr( P z ) is divisible by z while q n ( z ) q n − ( z ) is divisible only by z , the polynomial R ( q ( z ) , . . . , q n − ( z )) mustbe divisible by z . It can be easily shown that this is possible only if R ( q , . . . , q n − )is divisible by q n − , i.e. there exists a polynomial R ∈ C [ q , . . . , q n − ] such thatDiscr( P ) = q n q n − R ( q , . . . , q n − ) + q n ( q n − R + q n R ) + q n − S. (2.7)Now consider the polynomial P z ( t ) = ( t − z ) P n − ( t ) == t n + q t n − + ( q − z ) t n − + ( q − zq ) t n − + · · · + ( q n − − zq n − ) t − zq n − t − zq n − It follows from (2.7) thatDiscr( P z ) = − zq n − R ( q , . . . , q n − ) + z Q , (2.8) MIKHAIL BASOK where Q ∈ C [ z, q , . . . , q n ] is some polynomial. On the other hand we haveDiscr( P z ) = z · Discr( P n − ) · Res( t − z, P n − ( t )) = zq · Discr( P n − ) + z Q , where Q ∈ C [ z, q , . . . , q n ] is some polynomial. Comparing this equation with (2.8) wefind that R = − Discr( P n − ). Substituting this equality into (2.7) we get the statementof the lemma. (cid:3) Space of covers
Let Σ be a smooth projective curve of genus g . Denote by M ( n ) g, Σ the moduli spaceof GL( n, C ) spectral covers of Σ: M ( n ) g, Σ = n M j =1 H (Σ , K ⊗ j Σ ) (3.1)where K Σ is the canonical class of Σ anddim M ( n ) g, Σ = n ( g −
1) + 1 . (3.2)A point ( q , . . . , q n ) ∈ M ( n ) g, Σ can be considered as a polynomial P ( t, x ) = t n + q ( x ) t n − + · · · + q n ( x ). For each x ∈ Σ and v ∈ T ∗ x Σ the value P ( v, x ) is an element of ( T ∗ x Σ) ⊗ n .The spectral cover b Σ associated with P is a subvariety in T ∗ Σ defined by b Σ = { ( x, v ) ∈ Σ × T ∗ x Σ | P ( v, x ) = 0 } ;clearly, b Σ is a projective curve. Generically b Σ is smooth and all the ramification pointsof the projection b Σ → Σ are simple. If b Σ is smooth, then by the Riemann-Hurwitzformula the genus of b Σ is equal to b g = n ( g −
1) + 1. Define the action of C ∗ on M ( n ) g, Σ by ( ξ · P )( t, x ) = ξ n P ( ξ − t, x ) . (3.3)Denote by P M ( n ) g, Σ the corresponding projectivization.Let M g be the Deligne-Mumford compactification of the moduli space of genus g curves and let ν : M g, → M g be the universal curve. We define the moduli space ofGL( n, C ) Hitchin’s spectral covers by M ( n ) g = n M j =1 R ν ∗ ω ⊗ jν , (3.4)where ω ν is the relative dualizing sheaf. The forgetful projection M ( n ) g → M g is a bun-dle with fiber over Σ isomorphic to M ( n ) g, Σ (in the case when Σ is not smooth one have toreplace K Σ with the relative dualizing sheaf on Σ). The action of C ∗ on M ( n ) g, Σ definedby (3.3) extends to the action on M ( n ) g . Let P M ( n ) g denote the corresponding projec-tivization. The P M ( n ) g is a smooth orbifold (or a Deligne-Mumford stack). Denote by L → P M ( n ) g the tautological line bundle associated with the projectivization. ISCRIMINANT AND HODGE CLASSES ON THE SPACE OF HITCHIN’S COVERS. 7
Let π : C M ( n ) g → P M ( n ) g be the pullback of the universal curve M g, → M g .Denote by ω π the relative dualizing sheaf and set ψ = c ( ω π ) . (3.5)Let b π : b C M ( n ) g → P M ( n ) g be the universal family of Hitchin’s spectral curves, so thatthe fiber of b π over a point (Σ , [ P ]) ∈ P M ( n ) g is isomorphic to the curve b Σ associatedwith P . A point in b C M ( n ) g can be represented by a quadruple (Σ , x, P, v ), where P ∈ M ( n ) g, Σ , x ∈ Σ, v ∈ T ∗ x Σ and P ( v, x ) = 0. It is straightforward to check that the map p : b C M ( n ) g → C M ( n ) g that forgets v is a branched cover that coincides with b Σ → Σfiber-wise over P M ( n ) g . Let us denote by b B ⊂ b C M ( n ) g the ramification divisor of p andby B = p ( b B ) ⊂ C M ( n ) g the branching divisor. Consider the projection b B → M g, thatmaps (Σ , x, P, v ) to (Σ , x ). Let (Σ , x ) be curve with a marked point x and assume forsimplicity that Σ is smooth (otherwise one has to consider the normalization of Σ).Let ( M ( n ) g, Σ ) x ⊂ M ( n ) g, Σ be the subvariety consisting of ( q , . . . , q n ) such that q j ( x ) = 0 foreach j . The fiber of b B → M g, is (not canonically) isomorphic to P ( b D × ( M ( n ) g, Σ ) x ).Similarly, the fiber of the projection B → M g, is isomorphic to P ( D × ( M ( n ) g, Σ ) x ) (seeSection 2, where we define D and b D ). Notice that( M ( n ) g, Σ ) x ≃ C n ( g − − n +2 (3.6)(cf. (3.2)). Define the action of C ∗ on C n ( g − − n +2 via this isomorphism. The followinglemma is straightforward: Lemma 3.1. (1) The projection b B → M g, is a bundle with the fiber P ( b D× C n ( g − − n +2 ) (i.e. it can be locally represented as a projection of the form P ( b D× C n ( g − − n +2 ) × X → X ). In particular, the variety b B is smooth.(2) The projection B → M g, is a bundle with fiber P ( D × C n ( g − − n +2 ) . Inparticular, the singularities of B are normal crossings and cusps.(3) The map b B → B is a bundle morphism that is given fiber-wise by the map b D → D .(4) The ramification of the map p : b C M ( n ) g → C M ( n ) g is simple at a generic point. Components of the universal discriminant locus
Let P = t n + q t n − + · · · + q n represent an element in M ( n ) g, Σ . Consider thediscriminant W ( x ) = Discr( P ( · , x )). Recall that W is an N -differential, where N = n ( n − W is equal to the branching divisor of the spectral cover b Σ → Σ associated with P . Generically all zeros of W are simple, thus b Σ is smooth andthe ramification of b Σ → Σ is simple. If x is a zero of order 2 of W then there are threepossibilities that describe the local behaviour of the cover b Σ → Σ; we will follow thenotation of [11]:1) There is one simple ramification point of b Σ → Σ over x and b Σ has a node(normal crossing) at this point. We call the locus of such covers the “boundary” . MIKHAIL BASOK
2) The cover b Σ → Σ has two simple ramification points of order 2 over x and b Σis smooth at these points. We call the locus of such covers the “Maxwell stratum” .3) The cover b Σ → Σ has a ramification point of order 3 over x and is b Σ is smoothat this point. We call the locus of such covers the “caustic” .Let Q Ng be the moduli space of pairs (Σ , W ), where Σ is a curve of genus g and W is an N -differential on it (or a section of ω ⊗ N Σ if Σ is not smooth). The map P Discr( P ) gives rise to a map Discr : M ( n ) g → Q Ng . Let D d eg be the divisor in Q Ng parametrizing pairs (Σ , W ), where W has multiple zeros. The locus Discr − ( D d eg )has three components D ( b ) W , D ( m ) W and D ( c ) W in accordance with the three possibilitiesdescribed above. Put D W = Discr ∗ D d eg . A local analysis (see [11]) yields D W = D ( b ) W + 2 D ( m ) W + 3 D ( c ) W (alternatively, one can use Lemma 3.1 to show this). We callthe divisor D W the universal Hitchin’s discriminant . Note that D W , D ( b ) W , D ( m ) W and D ( c ) W are equivariant under the action of C ∗ on M ( n ) g . Therefore, we can define theirprojectivizations P D W , P D ( b ) W , P D ( m ) W and P D ( c ) W that are divisors in P M ( n ) g . Ourgoal is to represent the classes of P D ( b ) W , P D ( m ) W and P D ( c ) W as linear combinations ofstandard generators of Pic( P M ( n ) g ) ⊗ Q .4.1. Generators of
Pic( M ( n ) g ) ⊗ Q . Put φ = c ( L ) , (4.1)where L → P M ( n ) g is the tautological line bundle associated with the action of C ∗ on M ( n ) g . By construction, the space P M ( n ) g is a bundle over M g whose fibers are weightedprojective spaces. Therefore Pic( P M ( n ) g ) ⊗ Q is generated by the class φ and thepullbacks of the generators of Pic( M g ) ⊗ Q . Classically, the standard set of generatorsof Pic( M g ) ⊗ Q consists of the Hodge class λ and the classes of boundary divisors δ , . . . , δ [ g/ (see [1]). We will keep the same notation for the pullbacks of these classesto P M ( n ) g . Let δ = P [ g/ j =0 δ j denote the full boundary class. Let ν : M g, → M g be theuniversal curve and let ω ν denote the relative dualizing sheaf. Pulling the Mumford’sformula for ν ∗ c ( ω ν ) to P M ( n ) g we get π ∗ ψ = 12 λ − δ (4.2)where π and ψ where defined in Section 3.4.2. Expansion of the classes of components of the universal discriminantlocus.
In this section we will prove Theorem 2. The following lemma is straightfor-ward:
Lemma 4.1.
Let X be a complex orbifold, let h : L → X be a line bundle and n be aninteger. Consider the factor space L ( n ) = { ( v, α ) ∈ L × C } / ∼ ISCRIMINANT AND HODGE CLASSES ON THE SPACE OF HITCHIN’S COVERS. 9 modulo the relation ( ξv, α ) ∼ ( v, ξ n α ) that holds for any ξ ∈ C ∗ . Then L ( n ) is a complexorbifold, the projection L ( n ) → X given by ( v, α ) h ( v ) is a line bundle on X and themap ( v, α ) α · v ⊗ n is an isomorphism between L ( n ) and L ⊗ n . As above we denote by b π : b C M ( n ) g → P M ( n ) g the universal Hitchin’s spectralcurve and by π : C M ( n ) g → P M ( n ) g is the universal curve over P M ( n ) g . The branchedcover p : b C M ( n ) g → C M ( n ) g is given fiber-wise by the projection b Σ → Σ. The divisor b B ⊂ b C M ( n ) g denotes the ramification divisor of p and the divisor B = p ( b B ) ⊂ C M ( n ) g is the branching divisor of p . The class c ( ω π ) is denoted by ψ as before. Lemma 4.2.
The following relation holds in
Pic( P M ( n ) g ) ⊗ Q : b π ∗ ( p ∗ ψ · h b B i ) = n ( n − λ − δ − g − φ ) . Proof.
It follows from Lemma 3.1 and the construction of b D that the projection b B → B is of degree one. Therefore, b π ∗ h p ∗ ψ · b B i = π ∗ p ∗ h p ∗ ψ · b B i = π ∗ h ψ · p ∗ b B i = π ∗ h ψ · B i (4.3)Define the map h : π ∗ L ( n ( n − → ω ⊗ n ( n − π (cf. Lemma 4.1) as follows. Let (Σ , x, P ) ∈C M ( n ) g , so that P ∈ L| (Σ , [ P ]) . Set h ( P, ξ ) = ξ · Discr( P ) | x . Lemma 4.1 implies that h ( P, ξ ) is well-defined and depends linearly of (
P, ξ ) ∈ L ( n ( n − . Moreover, we havediv h = B . Therefore π ∗ h ψ · B i = π ∗ h ψ · ( n ( n − ψ − π ∗ φ )) i = n ( n − λ − δ − g − φ ) (4.4)where we used (4.2) in the last equation. Combining (4.3) and (4.4) we get the result. (cid:3) Proof of Theorem 2.
Let (Σ , x, P, v ) be a point in b B , so that ( x, v ) ∈ b Σ is a ram-ification point of b Σ → Σ. Let P b D W ⊂ b B denote the closure of the locus in b B that parametrizes those (Σ , x, P, v ) for which x is a double zero of Discr( P ). Then b π ( P b D W ) = supp( P D W ) (for a divisor D = a D + · · · + a k D k we denote the supportby supp( D ) = D ∪ · · · ∪ D k ). The divisor P b D W splits into three components P b D ( b ) W , P b D ( m ) W and P b D ( c ) W in accordance with the three possibilities described in the beginningof Section 4. We have b π ∗ P b D ( b ) W = P D ( b ) W , b π ∗ P b D ( m ) W = 2 P D ( m ) W and b π ∗ P b D ( c ) W = P D ( c ) W .Now let (Σ , x , P, v ) be a generic point in b B . Without loss of generality we canassume that Σ is smooth at x . Let U ⊂ Σ be a small neighborhood of x and v be aholomorphic 1-differential on U such that v ( x ) = v . Consider the polynomial P ( t + v ( x ) , x ) = t n + q ( x ) t n − + · · · + q n ( x ) where x ∈ U . Note that we have an equality Discr( P ( t + v ( x ) , x )) = Discr( P ( t, x ))for discriminates with respect to t , because the discriminant is invariant under anargument shift. Put P n − ( t, x ) = t n − + q ( x ) t n − + · · · + q n − ( x )as in Lemma 2.1. Let z be a local coordinate on Σ at x such that z ( x ) = 0. Since t = 0 is a zero of the second order of P ( t + v , x ) we have q n − ( x ) = O ( z ( x )) and q n ( x ) = O ( z ( x )). Using the decomposition of the discriminant obtained in Lemma 2.1we getDiscr( P ( t + v ( x ) , x )) = − q n ( x )( q n − ( x )) Discr( P n − ( t, x )) + O ( z ( x ) ) (4.5)as x → x . Note that the first summand on the right-hand side of (4.5) has a zeroof order 1 at x if the point (Σ , x , P, v ) belongs to an non-empty open subset of b B . On the other hand, if the point (Σ , x , P, v ) belongs to P b D W , then we havethe relation Discr( P ( t + v ( x ) , x )) = O ( z ( x ) ) as x → x , which is equivalent to thecondition q n ( x )( q n − ( x )) Discr( P n − ( t, x )) = O ( z ( x ) ) on the first summand. Since q n ( x )( q n − ( x )) Discr( P n − ( t, x )) is a product of three differentials we get the followingthree possibilities for this condition to hold:
1) The formula for [ P D ( b ) W ] . Assume that q n ( x ) = O ( z ( x ) ) as x → x . ThenDiscr( P ( t + v ( x ) , x )) = O ( z ( x ) ) as x → x due to (4.5). Notice that in this case v is aroot of order 2 of P ( t, x ) and Discr( P n − (0 , x )) = 0 in general, thus (Σ , x , P, v ) doesnot belong to P b D ( m ) W or P b D ( c ) W . Therefore, (Σ , x , P, v ) ∈ P b D ( b ) W . Vice versa, assumethat (Σ , x , P, v ) ∈ P b D ( b ) W . Write P ( v, x ) = F ( v, x ) dz n , where F is a holomorphicfunction defined in a neighborhood T ∗ U of ( x , v ) ∈ T ∗ Σ. By the definition b Σ ∩ T ∗ U is given by the equation F = 0 in T ∗ U . Since b Σ is not smooth at ( x , v )we must have dF ( v , x ) = 0. Because v is a root of second order of P , we have dF ( v , x ) = ∂ F ( v , x ), where ∂ denotes the partial derivative with respect to thesecond argument. Now assume that q n ( x ) = f n ( x ) dz n . Then dF ( v , x ) = ∂ F ( v , x ) = df n ( x ) . (4.6)Therefore, the equality dF ( v , x ) = 0 is equivalent to the fact that q n ( x ) = O ( z ( x ) )as x → x . We conclude that the equality q n ( x ) = O ( z ( x ) ) is equivalent to the factthat (Σ , x , P, v ) ∈ P b D ( b ) W . Introduce the notationΦ(Σ , x , P, v ) = dF ( v , x ) dz n ( x ) ∈ ( T ∗ x Σ) ⊗ ( n +1) . (4.7)Since F ( v , x ) = 0 this is well-defined (i.e. does not depend on the choice of a localcoordinate). Note that if ξ ∈ C ∗ then Φ(Σ , x , ξ · P, ξv ) = ξ n Φ(Σ , x , P, v ), wherethe action of C ∗ is defined by (3.3). It follows from Lemma 4.1 that Φ extends to ahomomorphism b Φ : b π ∗ L ( n +1) | b B → p ∗ ω ⊗ nπ | b B . (4.8)defined by ( P, ξ ) ξ Φ(Σ , x , P, v ). Computations made above shows that the vanish-ing locus of b Φ coincides with P b D ( b ) W . Moreover, it is straightforward that div b Φ = P b D ( b ) W so that we have P b D ( b ) W ≡ ( np ∗ ψ − ( n + 1) b π ∗ φ ) · b B (4.9) ISCRIMINANT AND HODGE CLASSES ON THE SPACE OF HITCHIN’S COVERS. 11 in the Chow ring of b C M ( n ) g , where we used that L ( n +1) ≃ L ⊗ ( n +1) . Applying Lemma 4.2we conclude from (4.9) that[ P D ( b ) W ] = [ b π ∗ P b D ( b ) W ] = n ( n − (cid:16) ( n + 1)(12 λ − δ ) − g − n + 1) φ (cid:17) .
2) The formula for [ P D ( m ) W ] . Assume that the equality Discr( P n − ( t, x )) = 0holds. This is equivalent to the fact that (Σ , x , P, v ) ∈ P b D ( m ) W by definition of b D ( m ) W .Introduce the notationΦ(Σ , x , P, v ) = Discr( P n − ( t, x )) ∈ ( T ∗ x Σ) ⊗ ( n − n − . Note that Φ(Σ , x , P, v ) does not depend on the choice of the differential v (althoughwe used v to define P n − ) and we have Φ(Σ , x , ξ · P, ξv ) = ξ ( n − n − Φ(Σ , x , P, v ).It follows that Φ can be extended to a homomorphism b Φ : b π ∗ L (( n − n − | b B → p ∗ ω ⊗ ( n − n − π | b B (4.10)defined by ( P, ξ ) ξ Φ(Σ , x , P, v ). We have div b Φ = P b D ( m ) W . From here we get that P b D ( m ) W ≡ ( n − n − (cid:16) p ∗ ψ − b π ∗ φ (cid:17) · b B (4.11)in the Chow ring of b C M ( n ) g , where by Lemma 4.1 L (( n − n − ≃ L ⊗ ( n − n − . Lemma 4.2together with the eq. (4.11) imply that2[ P D ( m ) W ] = [ b π ∗ P b D ( m ) W ] = n ( n − n − n − (cid:16) λ − δ + 4( g − φ (cid:17) .
3) The formula for [ P D ( c ) W ] . Finally, we consider the case q n − ( x ) = 0 that isequivalent to the fact that (Σ , x , P, v ) ∈ P b D ( c ) W . SetΦ(Σ , x , P, v ) = q n − ( x ) ∈ ( T ∗ x Σ) ⊗ ( n − . The value q n − ( x ) is independent of the choice of v and we have Φ(Σ , x , ξ · P, ξv ) = ξ n − Φ(Σ , x , P, v ). Hence Φ can be extended to a homomorphism b Φ : b π ∗ L ( n − | b B → p ∗ ω ⊗ ( n − π | b B (4.12)defined by ( P, ξ ) ξ Φ(Σ , x , P, v ) and we have div b Φ = P b D ( c ) W . It follows that P b D ( c ) W ≡ ( n − (cid:16) p ∗ ψ − b π ∗ φ (cid:17) · b B (4.13)in the Chow ring of b C M ( n ) g , where we use that L ( n − ≃ L ⊗ ( n − by Lemma 4.1.Lemma 4.2 and (4.13) imply that[ P D ( c ) W ] = [ b π ∗ P b D ( c ) W ] = n ( n − n − (cid:16) λ − δ + 4( g − φ (cid:17) . (cid:3) The Hodge classes on P M ( n ) g . In this section we will prove Theorem 3. We proceed using the notation introducedin two previous sections.
Lemma 5.1.
The following formula holds in
Pic( P M ( n ) g ) ⊗ Q : [ b π ∗ ( b B · b B )] = − n ( n − λ − δ − g − φ ) + 12 ([ P D ( b ) W ] + [ P D ( c ) W ]) . Proof.
Recall that b B is smooth due to Lemma 3.1. The projection b B → P M ( n ) g is abranched cover of degree 2 n ( n − g −
1) (equal to the number of zeros of Discr( P )counted with multiplicities). From the discussion in Section 4 it follows that the ram-ification divisor of this branched cover is P b D ( b ) W + P b D ( c ) W . These observations yield thefollowing expression for the canonical class of b B : c ( K b B ) = b π ∗ c ( K P M ( n ) g ) · b B + P b D ( b ) W + P b D ( c ) W . (5.1)Another expression for the canonical class of b B comes from the adjunction formula: c ( K b B ) = ( c ( K b C M ( n ) g ) + b B ) · b B . (5.2)Using these two expressions we get b B · b B = ( b π ∗ c ( K P M ( n ) g ) − c ( K b C M ( n ) g )) · b B + P b D ( b ) W + P b D ( c ) W = − c ( ω b π ) · b B + P b D ( b ) W + P b D ( c ) W (5.3)where ω b π is the relative dualizing sheaf. Recall that the map p : b C M ( n ) g → C M ( n ) g is abranched cover with a simple ramification along b B due to Lemma 3.1. Therefore, c ( ω b π ) = p ∗ ψ + b B . (5.4)Substituting this expression for c ( ω b π ) into (5.3) we find that2 b B · b B = − p ∗ ψ · b B + P b D ( b ) W + P b D ( c ) W . (5.5)Applying b π ∗ to this equation and using Lemma 4.2 we get the statement of the lemma. (cid:3) Lemma 5.2.
The following formula holds in
Pic( P M ( n ) g ) ⊗ Q : b π ∗ c ( ω b π ) = 6 n (3 n − λ − n ( n − g − φ − n (3 n − δ + 12 (cid:16) [ P D ( b ) W ] + [ P D ( c ) W ] (cid:17) . (5.6) Proof.
Using (5.4) we can write b π ∗ c ( ω b π ) = b π ∗ (cid:16) p ∗ ψ + 2 p ∗ ψ · b B + b B · b B (cid:17) = nπ ∗ ψ + b π ∗ (cid:16) p ∗ ψ · b B + b B · b B (cid:17) . Combining this formula with the eq. (4.2), Lemma 4.2 and Lemma 5.1 we get thedesired eq. (5.6). (cid:3)
ISCRIMINANT AND HODGE CLASSES ON THE SPACE OF HITCHIN’S COVERS. 13
Proof of Theorem 3.
Let V nodal ⊂ b C M ( n ) g denote the locus of nodal points of fibers of b π , i.e. V nodal = { (Σ , x, P, v ) | ( x, v ) is a node of b Σ } . (5.7)Note that P b D ( b ) W is a component of V nodal , and we have b π ∗ V nodal = nδ + [ P D ( b ) W ] . (5.8)Now let us apply the Grothendieck-Riemann-Roch formula to the structure sheaf O b C M ( n ) g and the morphism b π : b C M ( n ) g → P M ( n ) g . Taking the degree one components ofboth sides of the formula we get the following relation in Pic( P M ( n ) g ) ⊗ Q :12 b λ = b π ∗ ( c ( ω b π ) + V nodal ) . (5.9)Using Lemma 5.2 and (5.8) we conclude that12 b λ = 6 n (3 n − λ − n ( n − g − φ − n (3 n − δ + 32 [ P D ( b ) W ] + 12 [ P D ( c ) W ] (5.10)The formulas of Theorem 2[ P D ( b ) W ] = n ( n − (cid:16) ( n + 1)(12 λ − δ ) − g − n + 1) φ (cid:17) [ P D ( c ) W ] = n ( n − n − (cid:16) λ − δ − g − φ (cid:17) . together with the eq. (5.10) give the desired formula for b λ . (cid:3) References [1] E. Arbarello and M. Cornalba. The picard groups of the moduli spaces of curves.
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