Limit of Weierstrass Measure on Stable Curves
aa r X i v : . [ m a t h . AG ] J a n LIMIT OF WEIERSTRASS MEASURE ON STABLECURVES
Ngai-Fung Ng and Sai-Kee Yeung
Abstract
The goal of the paper is to study the limiting behavior of theWeierstrass measures on a smooth curve of genus g > as thecurve approaches a certain nodal stable curve represented by apoint in the Deligne-Mumford compactification M g of the moduli M g , including irreducible ones or those of compact type. As aconsequence, the Weierstrass measures on a stable rational curveat the boundary of M g are completely determined. In the process,the asymptotic behavior of the Bergman measure is also studied.
1. Introduction1.1
On a compact Riemann surface, an interesting geometric object tostudy is the distribution of Weierstrass points associated to the tensorpowers of an ample line bundle. It is observed by Olsen [ Ol ] that theasymptotic distribution of such Weierstrass points is dense with respectto the analytic topology. The situation is clarified by the beautiful resultof Mumford [ M1 ] and Neeman [ N ] that the asymptotic distribution isuniformly distributed with respect to the Bergman kernel of the curve.In other words, the higher Weierstrass points as defined are weaklyequidistributed with respect to the Arakelov measure. The phenomenonis interesting both from a geometric and an arithmetic point of view,such as results explained in [ B ], [ D ], [ M1 ] and [ R ].A natural problem is what happens for the corresponding distribu-tion on a singular algebraic curve, in particular for stable curves at theboundary of a moduli space in its Deligne-Mumford compactification.It has been observed by [ BG ], [ GL ], [ L1 ], [ FL ] that the asymptoticdistribution of the Weierstrass points associated to the power of an am-ple line bundle is no longer dense with respect to the complex topology Key words: moduli space, Weierstrass points, Bergman kernel
AMS 2010 Mathematics subject classification: Primary: 32G15, 14H10, 14H55,32A25
The second author was partially supported by a grant from the National ScienceFoundation. on a rational nodal curve. The main goal of this paper is to clarifythe situation and give a precise statement about the distribution of theWeierstrass points on a nodal curve sitting at the boundary of a modulispace.Let M g be the moduli space of compact Riemann surfaces of genus g >
2. Let M g be the Deligne-Mumford compactification of M g . Apoint t ∈ M g represents a Riemann surface X t of genus g . Considernow a one-parameter family of Riemann surfaces π : X → S on a curve S ⊂ M so that o ∈ M−M and a neighborhood U of o satisfies U −{ o } ⊂M . Our goal is to study the behavior of the limit of the Weierstrassmeasure on X o . Our approach is to study the relation between theBergman kernel and the Weierstrass measure for stable curves arisingfrom degeneration of a family of smooth curve. For this purpose, we haveto study the limiting behavior of the geometry of the period mappingand extract from it the geometric information needed. We refer the reader to Section 2 for various terminology used in theintroduction. For our statement here, the Bergman measure µ BX on acompact Riemann surface X is defined by µ BX = √− P gi =1 ω i ∧ ω i overan orthonormal basis { ω , . . . , ω g } of Γ( X, K X ). Our first result is theestimates on the asymptotic behavior of the Bergman measure. Theorem 1. (a) Let π : X → S be a local family of stable curvesin the sense of Deligne-Mumford so that X t = π − ( t ) is smooth for t ∈ S − { o } and X o has a single node at p ∈ X o . Let τ : b X o → X o bethe normalization of X o . Then(i) if the node p on X o is separating, the Bergman measure µ BX t → τ ∗ µ B b X o as t → o .(ii) if the node p on X o is non-separating, the Bergman measure µ BX t → τ ∗ µ B b X o + δ p as t → o , where δ p is the Dirac Delta at the node p ∈ X o .(b) Let X o be a stable curve for which p , · · · , p k are non-separatingnodes and p k +1 , · · · , p l are separating nodal points. Assume that X o has l − k + 1 irreducible components. Let b X o be the normalization of X o .Then in terms of the notations above, µ BX t → µ BX = τ ∗ µ B b X + k X i =1 δ p i as t → o In the above, we denote by τ ∗ µ B b X o the measure ( τ − ) ∗ | τ − ( X o −{ p } ) µ B b X o ,using the fact that τ − is a biholomorphism on X o − { p } .Here we remark that for a stable curve X o given by a point at theboundary of the Deligne-Mumford compactification D := ∂ M g = M g − IMIT OF WEIERSTRASS MEASURES 3 M g of M g studied in this paper, the limit µ BX t → τ ∗ µ B b X o is independentof the family of smooth curves taken. Let L be an invertible sheaf of positive degree on a stable curve X o of genus g >
2. The notion of Weierstrass points of powers of L hasbeen generalized from smooth curves to stable curves in the literature,cf. [ Wi ]. Assume X o is represented by a point o in the boundary ofthe Deligne-Mumford compactification of M g . Assume that L couldbe extended as an invertible sheaf L t to each curve X t represented bya point t in a neighborhood of U of o in M g . It is at this juncturethat we need to assume that X o is irreducible or of compact type fora general line bundle. In general it may be difficult to extend a linebundle L consistently to a nearby fiber due to the difficulty of defininglimit linear series on stable curves. This is however possible in the casethat X o is irreducible or of compact type, cf. [ AK ], [ CE ], [ CP ], [ GZ ].A typical example is given by tensor power of the (relative) dualizingsheaf of the family. The second author is grateful to Samuel Grushevskyfor pointing out the subtlety of extension of the line bundle.From the work of [ N ] and [ M1 ], if X t is a smooth curve, the discretemeasure µ WmL t associated to the set of Weierstrass points of ( L X t ) m → X t converges to 1 /g · µ BX t as m → ∞ where µ BX t is the Bergman measureof X t . Theorem 2.
Let π : X → S be a local family of stable curves whichare either irreducible or of compact type so that X t = π − ( t ) is smoothfor t ∈ S − { o } and X o has a single node at p ∈ X o . Let b X o be thenormalization of X o . Then(a) if the node p on X o is separating, the measure µ WmL o associated tothe Weierstrass points on X o satisfies µ WmL o → /g · µ B b X o as m → ∞ .(b) if the node p on X o is non-separating, the measure µ WmL o associatedto the Weierstrass points on X o satisfies µ WmL o → /g · ( µ B b X o + δ p ) as m → ∞ . The following result is a consequence of Theorems 1, 2 and induc-tion.
Theorem 3. (a) Let X be a stable curve which is irreducible and p , · · · , p k are the non-separating nodes. Let b X be the normalization of X . Then µ X,L := lim m →∞ µ WmL X = 1 /g · ( k X i =1 δ p i + µ B b X ) where g is the genus of the curve from which X is obtained by pinchingcorresponding cycles. NGAI-FUNG NG AND SAI-KEE YEUNG (b) Let X be a stable curve of compact type and p , · · · , p k are the sep-arating nodes. Let b X be the normalization of X . Then µ WX,L = 1 /g · µ B b X where g is the genus of the curve from which X is obtained by pinchingcorresponding cycles. Note that the theorem shows that the measure is independent of L . As an immediate corollary, we have the following result in the caseof a rational nodal curve living on the boundary of the Deligne-Mumfordcompactification of the moduli space of curves.
Corollary 1.
Let X be a stable irreducible rational nodal curve withnodes at p , · · · , p g . Then µ X = 1 /g · ( P gi =1 δ p i ) . Related to the corollary, we remark that from the earlier work of[ BG ], [ GL ], [ L1 ], and [ FL ], it is known that µ X vanishes on X exceptpossibly on a finite number of circles and the nodes. The corollary aboveshows that the measure µ X is solely supported on the nodes. Thereom 3and Corollary 1 above complete the picture on asymptotic distributionof Weierstrass points for stable curves. In the following we outline the main steps of proof. Theorem 1follows from a careful study of the Bergman metric with respect tothe degeneration at a single node. Theorem 2 is the main result. Itfollows from the following three steps. The first is the convergence ofthe Weierstrass measure as one approaches the boundary of the moduli.The second is to prove uniform convergence of the Weierstrass measureto the Bergman measure on compacta in the complement of the nodes,which depends on the results of Neeman [ N ]. Finally we apply Theorem1 to deduce that the residual measure is supported at a node. Theorem3 follows from Theorem 2 and an induction argument. Acknowledgement
The authors are very grateful to Samuel Gru-shevsky for making many valuable comments and suggestions on thepaper, to Valery Alexeev for explaining his work on semiabelic pairs.The authors would also like to thank the referee for helpful commentson the article.
After the paper was accepted, we were kindly informed of the ear-lier articles [ Am ], [ dJ ] which are related to the study of the Bergmanmeasure in general, see Remark 16.4 of [ dJ ], and distribution of theWeierstrass point over tropical curves.
2. Preliminaries2.1
Denote by M g the moduli space of Riemann surfaces of genus g > M g be the Deligne-Mumford compactification of M g . The points on IMIT OF WEIERSTRASS MEASURES 5 the boundary M g − M g represent stable curves in the sense of Deligne-Mumford. For simplicity of notation, sometimes we just denote M g , M g by M , M when there is no danger of confusion.It is well-known that the compactifying divisor D = M g − M g has adecomposition D = D ∪ · · · D [ g/ into irreducible components, where[ x ] denotes the integral part of x . The generic point of the stratum D represents an irreducible complex curve of genus g − D i for i > j and g − j , each with a puncture and the two puncturesare identified, named as a separating node on the curve.The nodes are obtained by contracting a real 1-cycle on a smoothRiemann surface of genus g , by considering a family of curves C t ofgenus g which is smooth for t = 0 and C is the stable nodal curveconsidered.A generic point on the intersection of two components D i ∩ D j for i = j corresponds to a stable curve obtained by contracting two realcycles to two different nodes.We refer the readers to [ HM ] for basic facts about moduli space ofcurves. Let X be a compact Riemann surface of genus g >
1. The space ofholomorphic one forms Γ(
X, K X ) has dimension g . There is a natural L metric on Γ( X, K X ) defined by ( η , η ) = √− R X η ∧ η . We woulddenote by { ω , . . . , ω g } an orthonormal basis of Γ( X, K X ) on X . Definition 1.
The Bergman measure µ BX on X is defined by µ BX = √− P gi =1 ω i ∧ ω i where { ω , . . . , ω g } is any orthonormal basis of Γ( X, K X ) . It is a standard fact that the Bergman measure is independent of theorthonormal basis chosen. The Bergman measure µ BX is also given bythe pull back of the flat measure on the Jacobian of the Riemann surface X by the Abel-Jacobian map. A symplectic homology basis of X is a basis { A j , B j } j g of H ( X, Z )satisfying intersection pairings A i · A j = 0 , B i · B j = 0 , and A i · B j = δ ij for all i and j. A canonically normalized basis { ω ′ i } gi =1 of Γ( X, K X ) with respect toa symplectic homology basis { A j , B j } is a basis satisfying R A j ω ′ k = δ jk for all j and k .The period matrix of X is the g × g matrix defined by Ω ij = R B i ω ′ j .In the following we recall some standard results on the behavior ofcanonically normalized holomorphic one forms with respect to the sym-plectic bases of a deformation family. We refer the reader to [ F ], [ Y ]and [ We ] for any unexplained terminology. NGAI-FUNG NG AND SAI-KEE YEUNG
Consider a one-parameter family of Riemann surfaces π : X → S with S − { o } ⊂ M and o ∈ M − M . The stable nodal curve X o is a singularcurve with nodal points as the only singularities, which are also calledpunctures of X o . X o can be considered as a union of finitely manycompact Riemann surfaces b X o with some particular points identifiedcorresponding to the nodal points where b X o is the normalization of X o . The local defining equation for a neighborhood of a node can bedescribed by zw = 0 in C .We will assume that X o has only one node. The cases of more thanone node will follow from induction.We recall the limit of canonically normalized holomorphic one formsas t → o . There are two cases according to whether a node is separatingor non-separating. In the case of separating node p , X t degenerates into X ∪ X as t → o , where X and X are Riemann surfaces of genus g = g ( X ) > g = g ( X ) >
0, and p is represented by x ∈ X and x ∈ X .Here g = g + g . Analytic structure of the degeneration is understoodfrom the following model. For i = 1 ,
2, let x i ∈ X i representing p and U i be a neighborhood of x i in X i with coordinates z i : U i → ∆ centeredat x i . Let S = { ( x, y, t ) : xy = t, x, y, t ∈ ∆ } . Denote the fiber at t ∈ ∆ by S t . Here ∆ r denotes the disk of radius r in C . For | t | < X − z − (∆ | t | ) and X − z − (∆ | t | ) according to the recipe z ( z , tz , t ) , z ( tz , z , t ) . The resulting surfaces gives rise to an analytic family
X → ∆ withsmooth fibers X t for t = 0 centered around X = X o . For z ∈ X i − { p } and | t | sufficiently small, there is a natural section z ( t ) of X → ∆ with z (0) = z . In the notation of [ We ], we say that z ∈ X i ∩ X t if z ( t ) ∈ X i − z − i (∆ | t | / ) for all small t .Let { ω (1) ′ i } , { ω (2) ′ j } be normalized bases with respect to some sym-plectic homology bases on X and X respectively. Proposition 1. ( [ We ] page 433, [ F ] page 38, [ Y ] page 129) We canfind a normalized basis of Γ( X t , K X t ) for t sufficiently close to o suchthat ω ′ i ( x, t ) = ( ω (1) ′ i ( x ) + O ( t ) , for x ∈ X − U , − tω (1) ′ i ( x ) ω (2) ′ ( x, p ) + O ( t ) , for x ∈ X − U ,ω ′ j ( x, t ) = ( ω (2) ′ j ( x ) + O ( t ) , for x ∈ X − U , − tω (2) ′ j ( x ) ω (1) ′ ( x, p ) + O ( t ) , for x ∈ X − U , where i g , g + 1 j g + g = g , and ω (1) ′ ( x, p ) and ω (2) ′ ( x, p ) are canonical differentials of second kind on X and X respectively. IMIT OF WEIERSTRASS MEASURES 7
We refer the readers to [ F ] for standard terminology of canonicaldifferentials of second kind and just remark for example that ω (1) ′ ( x, p )is evaluated at p with respect to the local coordinate U . In the case of non-separating node p , X t degenerates into a sta-ble curve X o with node at p , which can be considered as a connectedRiemann surface b X o with two points a, b ∈ b X o identified. Again, thereexists small coordinate neighborhoods U a , U b of a and b respectively,and b X o is the normalization of X o . We may regard U a and U b as disksof fixed radius δ in some local coordinates around a and b respectively.The analytic structure can be given as in . Let 0 < ρ <
1. Denoteby ρU a and ρU b disks of radius ρδ . Proposition 2. ( [ We ] page 437, [ F ] page 51, [ Y ] page 135) We canfind a normalized basis of Γ( X t , K X t ) for t sufficiently close to o suchthat for x ∈ b X o − ρU a − ρU b ,ω ′ i ( x, t ) = ω ′ i ( x ) − t [ ω ′ i ( b ) ω ′ ( x, a ) + ω ′ i ( a ) ω ′ ( x, b )] + O ( t ) , (1 i g − ω ′ g ( x, t ) = ω ′ b − a ( x ) − t [ γ ω ′ ( x, b ) + γ ω ′ ( x, a )] + O ( t ) , where γ i ’s are some constants. Moreover, the expression lim t → O ( t ) /t is a meromorphic form with poles only at a and b , and the coefficientshas uniform convergence on b X o − ρU a − ρU b . In the above, ω ′ b − a ( z ) = πi ∂ z log E ( z,b ) E ( z,a ) and E ( z, a ) is the prime formof b X o . Since E ( z − a ) in local coordinates is given by z − a , we concludethat ω ′ b − a ( z ) = πi ( z − b − z − a ) in local coordinates. We recall the definition of generalized Weierstrass points on a pro-jective algebraic curve as given in [ L ] and [ Og ]. A point p ∈ X is calleda Weierstrass point of the holomorphic line bundle L (represented by z as above) if there is an s ∈ Γ( X, L ) whose vanishing order at p is at least h ( L ) := dim C Γ( X, L ). As in the case of the usual Weierstrass points,the Weierstrass points of a line bundle can also be defined in terms ofthe Wronskian of a basis of sections of L in [ L ] and [ Og ].Consider the case that X is a smooth curve of genus g >
2. Denoteby J d the Picard variety of degree d . Denote by Θ the theta divisorof X in J g − . There is a mapping f n : X × Θ → J g − n defined by f n ( x, θ ) = nx + θ . Then it is well-known that x is a Weierstrass point of the line bundle z if and only if(1) z = f n ( x, θ )for some θ ∈ Θ, which was taken as definition in [ N ].Let p be a Weierstrass point of L over X . Then the weight of p ,denoted w L ( p ), is defined as follows: Let s , ..., s m be a basis of Γ( X, L )with distinct vanishing orders α < · · · < α m at p , then w L ( p ) := α + · · · + α m − − − − − · · · − ( h ( L ) − NGAI-FUNG NG AND SAI-KEE YEUNG
Notice that non-Weierstrass points have weight 0. Denote by W ( L ) theset of all Weierstrass points of L over X .Let h : X → R be a continuous function on X . Let L be any holo-morphic line bundle of degree g − m over X ( m > g − µ WX,L := P p ∈ W ( L ) w L ( p ) · δ p P p ∈ W ( L ) w L ( p )where δ p is the Dirac Delta at p . In case that there is no danger ofconfusion, we would simply denote µ WX,L by µ WL . Then Z X h · µ WL = P p ∈ W ( L ) h ( p ) · w L ( p ) P p ∈ W ( L ) w L ( p ) = 1 gm · X p ∈ X h ( p ) w L ( p ) . Recall the following result of [ N ], see also [ M1 ]. Proposition 3.
Let X be a Riemann surface of genus g > . Let h : X → R be a continuous function on X . Let L be any line bundle ofdegree g − m over X ( m > g − ). Then P p ∈ W ( L ) h ( p ) · w L ( p ) P p ∈ W ( L ) w L ( p ) = Z X h · µ WL converges to the constant R X h · ( ω ∧ ω + · · · + ω g ∧ ω g ) R X ( ω ∧ ω + · · · + ω g ∧ ω g ) = 1 g · Z X h · µ BX as m → ∞ .In the above, { ω , ..., ω g } is an orthonormal basis of Γ( X, K X ) and µ BX is the Bergman measure on X . Let P g,d be the variety consisting of pairs [ C, L ], where C ∈ M g and L is a line bundle on C of degree d . In general, it is a subtle problem tohave a natural canonical compactification of P g,d sitting above M g . Thedifficulty is shown by the non-uniqueness of extension of line bundle inthe following example. Consider a one-parameter family of stable curves π : C → ∆ with ∆ ∗ = ∆ − { } ⊂ M g , where fibers C t , t ∈ ∆ is smoothand C is nodal consisting of two components C and C meeting at apoint p . Let L be a line bundle on π − (∆ ∗ ) so that L t = L| C t is a linebundle on C t for t ∈ ∆ ∗ . Then the extension of L over ∆ is not unique,since L + O C C would give another possible extension apart from agiven extension L over ∆. Here ( L + O C C ) | C = ( L + ( p )) | C hasdegree deg( L| C ) + 1 and ( L + O C C ) | C = ( L − ( p )) | C has degreedeg( L| C ) − IMIT OF WEIERSTRASS MEASURES 9
For the case of irreducible stable C , the problem of compactificationof P g.d is resolved by considering torsion-free coherent sheaves of rankone as given by [ DS ], and the above difficulty of uniqueness in exten-sion does not occur since there is only one irreducible component. Inparticular, a line bundle with a fixed degree on X extends to a linebundle on C t for t ∈ U , a neighborhood of 0 in M g .In the example above with two irreducible components meeting at apoint, the problem of compactification was resolved in [ C ], in which theextension is unique by considering line bundles of appropriate bidegreein Pic ( d ,d ) ( C ) = Pic d ( C ) × Pic d ( C ), where the choice of bidegreeis finite. Recall that a nodal curve is of compact type if every node isseparating. In such a case, once we fix a multi-degree corresponding toa choice in [ C ], the extension is unique. In particular, a line bundle witha fixed multi-degree on X extends to a line bundle on C t for t ∈ U , aneighborhood of 0 in M g .In this article we study which are either irreducible or of compacttype and consider line bundles which extends to a neighborhood U of 0in M g .
3. Convergence of Bergman measure on a family of curves3.1 Proof of Theorem 1
Let us first give a short outline of proof of Theorem 1a. It is well-known that in the setting of Theorem 1(a), a holomorphic one-form on X t gives rise to a one form with at most a log pole at the node p . Re-call that the total residue of a meromorphic one-form on a connectedRiemann surface is trivial. If p is separable so that X o consists of twoirreducible components X and X of genus a and g − a respectively,the residue argument as above applied to the normalization b X i of eachcomponent X i , i = 1 ,
2, implies that a meromorphic one form cannothave pole at a single point and hence the form is actually holomorphic.In this case, the sum of the Bergman kernels on X and X is pre-cisely the limit of the Bergman kernel on X t . If p is non-separable, thiscorresponds to a meromorphic one form with a single pole at p , p ofopposite residues, where { p , p } = τ − ( p ). In such case, there is a g − X o coming from the convergenceof the space of holomorphic one-forms from X t . One expects that theBergman kernel of X t approaches the Bergman kernel of X o as t → o .It is however a bit tedious to describe the convergence of the Bergmankernel since orthonormality is imposed in the definition of Bergman ker-nel as given in and a log pole is not L -integrable. We provide somedetails below. Theorem 1(a)(i)—
This is already observed in Lemma 6.9 of [We].For completeness of presentation, we explain the reason here parallel toour argument for (ii) in . We are given a family of curves π : X → S with o ∈ S representing X o = X ∪ X . Let { ω (1) ′ ( x, , . . . , ω (1) ′ g ( x, } and { ω (2) ′ g +1 ( x, , . . . , ω (2) ′ g + g ( x, } be canonically normalized bases withrespect to symplectic homology bases on X and X respectively.On X , let { ω (1)1 ( x, , . . . , ω (1) g ( x, } be an orthonormal basis of Γ( X , K X )with respect to the natural L norm as defined in . Similarly for { ω (2) g +1 ( x, , . . . , ω (2) g + g ( x, } . Let H (0) , H (0) be the transformationsso that ω (1) i ( x,
0) = H (0) ij · ω (1) ′ j ( x,
0) for 1 i, j g ω (2) j ( x,
0) = H (0) ij · ω (1) ′ j ( x,
0) for g + 1 i, j g + g Let H ( t ) be the transformation (for | t | sufficiently small) such that ω (1) i ( x, t ) = H ( t ) ij · ω (1) ′ j ( x, t ) for 1 i, j g ω (2) i ( x, t ) = H ( t ) ij · ω (2) ′ j ( x, t ) for g + 1 i, j g + g (3)and H ( t ) − exists. Indeed,(4) H ( t ) = (cid:18) H ( t ) 00 H ( t ) (cid:19) where H and H are square matrices of size g and g respectively.Since µ BX t = g X i =1 ω X t ,i ∧ ω X t ,i = X i,j,k H ( t ) ij H ( t ) ik ω ′ j ( x, t ) ω ′ k ( x, t ) , taking limit on both sides yields the result. Theorem 1(a)(ii)—
We consider degeneration of the Weierstrasspoints for a stable nodal curve with a node p and degeneration asgiven in Proposition 2. Hence we have a family of curves π : X → S with o ∈ S representing X o . b X o is the normalization of X o . Let { ω ′ ( x, , . . . , ω ′ g − ( x, } be a canonically normalized basis with respectto a symplectic homology basis on b X o .On b X o , we let { ω ( x, , . . . , ω g − ( x, } be an orthonormal basis ofΓ( b X o , K b X o ) with respect to the natural L norm as defined in . Let J (0) be the transformation so that(5) ω i ( x,
0) = J (0) ij ω ′ j ( x,
0) for 1 i, j g − . Let { ω ′ i ( x, t ) } gi =1 be the set of one forms on X t given by Proposition 2.There exists an invertible transformation J ( t ) (for | t | sufficiently small)satisfying ω i ( x, t ) = J ( t ) ij ω ′ j ( x, t ) for 1 i, j g − IMIT OF WEIERSTRASS MEASURES 11 and { ω i } g − i =1 being an orthonormal basis of span( ω ′ , . . . , ω ′ g − ) ⊂ Γ( X t , K X t ).Adding one more one form ω g ( x, t ) so that { ω i } i =1 ,...,g gives an orthonor-mal basis of Γ( X t , K X t ), it follows that we can find a transformation H containing J as a submatrix so that(6) ω i ( x, t ) = H ( t ) ij ω ′ j ( x, t ) for 1 i, j g. Indeed,(7) H ( t ) = (cid:18) J ( t ) 0 a T ( t ) b ( t ) (cid:19) with a T ( t ) = ( a ( t ) , · · · , a g − ( t )). It follows that the inverse of H isgiven by(8) H − = (cid:18) J − ( t ) 0 − b a T · J − ( t ) 1 /b ( t ) (cid:19) From (6), we know that(9) ω g ( x, t ) = g − X i =1 a i ( t ) ω ′ i ( x, t ) + b ( t ) ω ′ g ( x, t ) . By construction, a i ( t ) is smooth in t for 1 i g −
1. Moreover, ω ′ i ( x, t ) is uniformly bounded on X o when t = 0 for 1 i g −
1, andthe expression varies smoothly with respect to t . Hence the expression P g − i =1 a i ( t ) ω ′ i ( x, t ) above is uniformly bounded for small t . It remainsto estimate the term b ( t ) ω ′ g ( x, t ).Now, since ω g ⊥ span { ω ′ , ..., ω ′ g − } , we have0 = Z X t ω g ( x, t ) ∧ g − X i =1 a i ( t ) ω ′ i ( x, t ) , plugging (9) into the above gives(10)0 = Z X t ( g − X i =1 a i ( t ) ω ′ i ( x, t )) ∧ g − X i =1 a i ( t ) ω ′ i ( x, t )+ g − X i =1 b ( t ) a i ( t ) Z X t ω ′ g ( x, t ) ∧ ω ′ i ( x, t ) . We claim that for 1 i g −
1, there is the estimate Z X t ω ′ g ( x, t ) ∧ ω ′ i ( x, t ) = o ( t ) . This follows from smoothness of π and(11) Z X o ω ′ g ( x, ∧ ω ′ i ( x,
0) = 0where ω ′ g ( x,
0) = ω ′ b − a ( x ). The above identity is true because from ourassumption, ω ′ i ( x,
0) for 1 i g − { A i } i =1 ,...,g − . Hence R A i ω ′ g ( x,
0) = 0 for 1 i g − and so the claim is valid.It follows from the claim and (10) that(12) Z X t ( g − X i =1 a i ( t ) ω ′ i ( x, t )) ∧ g − X i =1 a i ( t ) ω ′ i ( x, t ) = o ( t )Recall that Proposition 2 gives rise to(13) ω ′ g ( x, t ) = ω ′ b − a ( x ) − t [ γ ω ′ ( x, b ) + γ ω ′ ( x, a )] + O ( t ) , for x ∈ b X o − ρU a − ρU b and the estimates in t [ γ ω ′ ( x, b ) + γ ω ′ ( x, a )]and O ( t ) are uniform. Hence for fixed ρ >
0, given any small ǫ > | ω ′ g ( x, t ) − ω ′ b − a ( x ) | < ǫ if t is sufficiently small and 0 < t < ρ . Now k ω ′ b − a k b X o − ρU − ρU := Z b X o − ρU − ρU ω ′ b − a ∧ ω ′ b − a > Z ( U − ρU ) ∪ ( U − ρU ) ω ′ b − a ∧ ω ′ b − a > c | log ρ | (15)for some constant c > k ω g ( · , t ) k X t = 1, and P g − i =1 a i ( t ) ω ′ i ( x, t ) is uniformly boundedfor small t , it follows from identity (9) and the estimate (15) that b ( t ) >c | log ρ | for some constant c > t < ρ . Hence the one form ω g ( x, t )converges to 0 on compacta on X o − { p } ∼ = b X o − { a, b } as t → x ∈ b X o − { p } , Proposition 2 implies that ω X t ,i ( x ) → ω b X o ,i ( x ) for 1 i g − t →
0. Since µ BX t = P gi =1 ω X t ,i ∧ ω X t ,i ,it follows from the last paragraph that the limit of ω X t ,g ∧ ω X t ,g wouldconcentrate at the node p as t →
0. Here we note that the total measure Z X t µ BX t = g X i =1 k ω X t ,i k = g and P g − i =1 k ω b X o ,i k = g − . The discrepancy is precisely given by thedelta measure at the point p , since the mass cannot be concentratedanywhere else according to the discussions above. Theorem 1(b)—
This follows from and induction. Suppose k = 2. Suppose X o is a stable curve with two nodes obtained aftercontracting two nodes from families of smooth curves, correspondingto a point at the boundary of the Deligne-Mumford compactification D i ∩ D j ⊂ D = M g − M g for some i = j . We may consider a local twodimensional holomorphic family of curves X ( s, t ) for ( s, t ) ∈ ∆ × ∆, so IMIT OF WEIERSTRASS MEASURES 13 that X ( s, t ) is smooth for s = 0 and t = 0, X (0 , t ) ∈ D i and X ( s, ∈ D j , and X (0 ,
0) = X o .We consider first a point X (0 , t ) at ∂ M g obtained by contracting 1real cycle giving rise to a node p ( t ) , which may be assumed to be afixed node p with respect to a local trivialization of the family. This isobtained by letting s → X ( s, t ). Let b X (0 , t ) be the normalizationof X (0 , t ). Theorem 1 implies thatlim s → µ BX ( s,t ) = µ BX (0 ,t ) = µ B b X (0 ,t ) + δ p .X o is obtained by contracting a real 1-cycle on X (0 , t ) to a node p ,which corresponds to contracting a real cycle on b X (0 , t ) to a node ˆ p ,where ˆ p corresponds exactly to the node p on X o . Now the normal-ization of b X (0 ,
0) is precisely b X o . Hence Theorem 1a again implies thatlim t → µ B b X (0 ,t ) = µ B b X (0 , + δ p = µ B b X o + δ p . Combining the above two identities, we see thatlim t → lim s → µ BX ( s,t ) = µ B b X o + δ p + δ p . Note that the arguments of [ F ], [ Y ] and [ We ] concerning behaviorof period matrices corresponding to contraction of a real 1-cycle appliesequally well to a family of degenerating curves obtained by contract-ing two different non-intersecting real 1 cycles as well. The end resultdepends only on X o and is independent of the paths of degenerationtaken.Hence Theorem 1(b) is proved for k = 2. The same proof clearlyworks for k >
4. Convergence of Weierstrass measure on a family of curves4.1
Suppose C is a stable curve with nodal singularities at z i , i =1 , . . . , n . Let π : e C → C be the normalization of C so that π − ( z i ) = { a i , b i } . Let U be a small coordinate neighborhood of z i . Then the du-alising sheaf ω C is generated by holomorphic 1-forms in a neighborhoodof a regular point on C or e C , and by meromorphic 1-forms η with atworst simple poles at a i , b i over U , satisfyingRes a i ( η ) + Res b i ( η ) = 0 . Now let L be an ample line bundle on a stable curve C . Let ψ, τ bethe generators of L and ω C over U respectively. Let n = h ( C, L ) and φ , . . . , φ n be a basis of H ( C, L ). Define F i,j ∈ Γ( U, O C ) inductivelyby F ,j ψ := φ j | U j = 1 , . . . , n,F i,j τ := dF i − ,j i = 2 , . . . , n, j = 1 , . . . , n Define also ρ = det( F i,j ) ψ n τ ( n − n/ . It follows easily by checking compatibility on different charts that ρ defines a section in H ( C, L n ⊗ ω ( n − n/ C ). Then as in [ LW ], we define p ∈ C to be a Weierstrass point of L n ⊗ ω ( n − n/ C if and only if(16) ord p ρ > . It follows from [ LW ] that the number of Weierstrass points countedwith multiplicity is given by n deg( L )+( n − n ( g − n > g − L ) = g − n and hence thenumber of Weierstrass points counted with multiplicity is then given by n g . Let us consider first in details the situation that b X o has genus 0, acase that partly motivates the present paper. Lemma 1.
Let X o be an irreducible rational nodal curve. For each m ∈ N , lim t → µ WX t ,mK t = µ WX o ,mK o . Proof
We assume that X o is a rational curve with g double points.Hence X o is formed by identifying g pairs of distinct points b i and c i , i = 1 , . . . , g on P C . In this case, the discussions in could be realizedconcretely as follows (for details, see [ M2 ],[ L1 ]).The dualizing sheaf of X o is spanned by ω i = dzz − b i − dzz − c i i = 1 , , ..., g Thus the period lattice Λ is generated by the g vectors { (2 π √− , , ..., , (0 , π √− , , ..., , ..., (0 , ..., , π √− } Hence the generalized Jacobian is C g / Λ ∼ = ( C ∗ ) g . Let X so be the setof points on X o with all b i , c i removed, i.e., the set of smooth points.And we further assume x o = ∞ ∈ X so , this can be done by choosingappropriate coordinate. Choosing x o as basepoint, we define the Abelmapping ϕ : X so → J ( X o ) ∼ = ( C ∗ ) g by ϕ ( x ) = (cid:18) exp (cid:18)Z xx o ω (cid:19) , · · · , exp (cid:18)Z xx o ω g (cid:19)(cid:19) = (cid:18) x − b x − c , · · · , x − b g x − c g (cid:19) This induces a map ϕ : ( X so ) m → ( C ∗ ) g given by ϕ ( X k n k x k ) = Y k (cid:18) x k − b x k − c (cid:19) n k , · · · , Y k (cid:18) x k − b g x k − c g (cid:19) n k ! IMIT OF WEIERSTRASS MEASURES 15
Let λ i = exp ( z i ) be coordinates on ( C ∗ ) g . Define τ on J by τ ( λ , . . . , λ g ) = det − λ · · · − λ g b − c λ · · · b g − c g λ g ... ... b g − − c g − λ · · · b g − g − c g − g λ g Let X t be a family of smooth curves of genus g degenerating to X o . Itis a standard fact ([ F ]) that the period matrices Ω ij ( t ) of X t satisfy Im (Ω ii ( t )) → ∞ as t → ij ( t ) are continuous for | t | < ǫ Let Ω ii ( t ) be the diagonal of Ω( t ), we have, upon direct computation,that, as t → θ (cid:18) z −
12 Ω ii ( t ) , Ω( t ) (cid:19) converges to τ up to a constant multiple. ([ M2 ], page 3.253.)Hence τ defines the Jacobian divisor θ o on X o .To sum up, consider a family of stable curves X t with smooth X t when t = 0 and X o is a stable curve with double points in the sense of Deligne-Mumford. In the case that X o is just a rational curve with doublepoints, we know that there is a convergence of θ t to θ o . Hence fromthe definition of Weierstrass points earlier, there is a convergence of theWeierstrass divisors as claimed in the statement of the lemma. q.e.d. In this subsection, we generalize the argument in the previous sub-section to the case of arbitrary genus.Consider now the family of stable algebraic curves of genus g , π : X → S as before so that fibers X t are smooth except for X o whichhas nodal singularity with a single node at p . We define µ WmL t as thedistribution associated to Weierstrass points on X t as in (2). Lemma 2.
Assume that X o is stable. Then for each m ∈ N , lim t → µ WX t ,mL t = µ WX o ,mL o . Proof
Since there is only one node, X o is either irreducible or of compacttype. A Weierstrass point on a stable curve is defined by (16). For thecase of irreducible stable curve, the lemma follows from [ L2 ] Theorem1. In the case that X t is stable and of compact type, it follows from[ EN ] Theorem 8.4, [ ES ] Theorem 6. In either case, this follows from theconvergence of the Wronskian in the definition of Weierstrass points. An alternative approach closer to the description in (4.2) for irre-ducible stable rational curve can be given as follows, making use of thealternative definition of Weierstrass points as given in (1). Denote byPic d = J d the Picard variety of degree d on X , parametrizing line bun-dles of degree d . Pic d may not be projective if X is not smooth. Insuch case, we may consider compactified Jacobian and theta divisor asdefined in [ A1 ], [ A2 ]. If X is a smooth curve, the theta divisor can bedefined intrinsically as the locus of L ∈ Pic g − ( X ) with h ( X, L ) = 0.This is used as definition for stable curve as well in the following way.According to [ A1 ], there is a complete moduli of semiabelic pairs de-fined in [ A1 ], for which the compactified Jacobian and its theta divisoris such a pair. From the work of Simpson in 1.21 of [ S ], see also theexplanation in 1.2-1.3 of [ A2 ], there is a family π : J → S of com-pactified Jacobians of degree g − S in which eachfiber is the compactified Jacobian of X t , for which it follows from [ A2 ]that we may choose an arbitrary polarization. Let L be the ample linebundle which gives the polarization. From [ A2 ], π ∗ L is invertible by co-homology and base change. Choosing a trivialization of π ∗ L , this givesa section s ∈ H ( S, π ∗ L ) whose restriction s t to the fiber over t is theunique section of L t . This gives a family of theta divisors Θ t , ∀ t ∈ S .The setting above implies that the theta divisor on J t for t = o converges to J o as t → o , which is a restatement of Mumford in highergenus case at the central fiber.Then from the definition of Weierstrass points and weights, it followsthat lim t → µ WX t ,mL t = µ WX o ,mL o . q.e.d.We remark that the assumption that X t is stable irreducible or ofcompact type is used in the second approach above. It is well knownthat there is isomorphism between Picard varieties of different degreesfor smooth curves, after translation by a based line bundle with degreethe difference of the two. This could also be done for stable irreduciblecurves or curves of compact type. The problem is in general subtle forarbitrary stable curves. We refer the readers to [ GZ ] for some resultsin this direction.
5. Weierstrass measure on stable curves5.1
Recall that p is the node on X o , corresponding to a and b on b X o .Let U be a small neighborhood of p corresponding to the union of twodisks U a and U b around a and b respectively. Let ∆ be a sufficientlysmall neighborhood of o in S . We may assume that U can be extendedto U on π − (∆) ∩ X so that for U t = U ∩ X t , X t − U t is diffeomorphic to IMIT OF WEIERSTRASS MEASURES 17 X o − U o for all t ∈ ∆ − { o } . The following lemma follows immediatelyfrom the steps of proof in [ N ]. Lemma 3. (17) lim m →∞ lim t → o µ WmL t ( x ) = lim t → o lim m →∞ µ WmL t ( x ) uniformly for x ∈ ( π − (∆) − U ) . Proof
We are going to follow the proof of Neeman in Chapter 2 of [ N ]on x ∈ ( π − (∆) − U ). The reason that the argument goes through isthat we have nice convergence of the Bergman metric and geometry on( π − (∆) − U ) as t → o .As in [ N ], we consider f t,n : X t × Θ t → J t,g − n given by f t,n ( x, θ ) = nx + θ . Let F t ⊂ X t × Θ t be the union of the singular set of C t × Θ t and the ramification locus of f t,n . By abuse of language, we denote by J t,g − n ∩ (( π − (∆) − U )) the subset of J t,g − n given by the image ofthe Jacobian image of X t ∩ (( π − (∆) − U )). We are actually consideringonly the restriction of f t,n to f − ( J t,g − n ∩ ( π − (∆) − U )), namely f t,n : C t × Θ t | f − ( J t,g − n ∩ (( π − (∆) −U )) → J t,g − n ∩ (( π − (∆) − U )) . Now for a translational invariant vector field V t on J t,g − n ∩ (( π − (∆) −U )), we have f − t,n ( V t ) = V nt, ⊕ V nt, . As in Lemma 2.3 of [ N ], we have(18) V nt, = 1 n V t, = 1 n V t, , V nt, = V t, = V t, . As in Lemma 2.3 of [ N ], the operators f − t,n ( V t ) · · · f − t,n ( V t ) on com-pact space D t ⊂ ( X t × Θ t − F t ) | f − ( J t,g − n ∩ (( π − (∆) −U )) are uniformlybounded as operators C r + mD t → C rD t as n varies and is uniform in t ∈ ∆.Let h t : X t × Θ t | f − ( J t,g − n ∩ (( π − (∆) −U )) → R be a smooth functionwith compact support. Define Av n h t : J t,g − n → R by(Av n h t )( z ) = 1 gn X x ∈ W t ( z ) h t ( x, z ) , where W t ( z ) denotes the set of Weierstrass points of z with multiplici-ties. Again, the argument of Lemma 2.6 of [ N ] implies that there exists M ∈ R such that for all n > g − t ∈ ∆ that k V t · · · V mt ( Av n h t ) k ∞ M. Similarly, it follows as Lemma 2.7 of [ N ] that the k -th Fourier coefficientof Av n h t for k = ( k , . . . , k g ) = (0 , , ..., n h t )ˆ( k ),satisfies (Av n h t )ˆ( k ) M P gi =1 | k i | g +1 . Then Lemma 2.8 of [ N ] implies that the 0-th Fourier coefficient ofAv n h t , denoted by (Av n h t )ˆ(0), satisfies(Av n h t )ˆ(0) = Z C t × Θ t | f − Jt,g − n ∩ (( π − −U )) h t · dB t . The argument of Lemma 2.9 of [ N ] then implies thatAv n h t − Z C t × Θ t | f − Jt,g − n ∩ (( π − −U )) h t · dB t converges uniformly to 0 as n → ∞ and uniformly in t → o . Now wemay use the argument of Lemma 2.9 of [ N ] to show that the aboveconvergence actually holds for arbitrary continuous function h t on C t × Θ t | f − ( J t,g − n ∩ (( π − (∆) −U )) , uniformly in n → ∞ and in t → o . In par-ticular, we may interchange the order of limits as given in our statement.q.e.d. Let V be a small neighborhood of a node p ∈ X o as mentioned at the beginning of . We extend V smoothlyto a neighborhood V in the total family and use the same notationto denote V ∩ X t for t sufficiently small. Note that all of this can beperformed in a local coordinate as discussed in . It follows fromLemma 2, Lemma 3 and Theorem 1 that for any small neighborhood U of the node and any x ∈ U ,lim m →∞ µ mL o = lim m →∞ lim t → o µ WmL t (19) = lim t → o lim m →∞ µ WmL t = 1 g · lim t → o µ BX t . Hence by shrinking V to the node, we conclude that(20) lim m →∞ µ mL o | X o −{ p } = 1 /g · lim t → o µ BX t | X o −{ p } = 1 /g · µ BX o | X o −{ p } . Consider first the case (a) that the nodal point p is separating. Inthis case, the sum of genera of the two components of X o is precisely g .Hence(21) 1 g Z X o µ BX o = 1 g Z X o −{ p } µ BX o = 1 . On the other hand,(22) lim m →∞ µ mL o | X o −{ p } = lim m →∞ lim t → o µ mL t | X o −{ p } IMIT OF WEIERSTRASS MEASURES 19 and each lim m →∞ µ mL t is given by g · µ BX t and hence(23) Z X o lim m →∞ µ mL o = 1 . It follows from equations (20), (21) and (23) that there is no masstrapped in p and hence(24) lim m →∞ µ mL o | X o = 1 /g · lim t → o µ BX t | X o = 1 /g · τ ∗ µ B ˆ X o . This corresponds to (a).Consider now (b) for which X o is obtained from contracting a realcycle on X t to a non-separating node on X o . The Genus of X t is g foreach t = 0 and the genus of b X o is g −
1. It follows thatlim m →∞ Z X t µ mL t = g for t = 0;lim m →∞ Z X o µ mL o = g − . It follows that the difference in the measure is supported at the nodeas t →
0. Hencelim m →∞ µ mL o | X o = 1 /g · (lim t → o µ BX t | X o + δ p ) = 1 /g · ( τ ∗ µ B ˆ X o + δ p ) . q.e.d. Let X o be a stable curve represented by a point on M g − M g . Wemay regard X o = X as the degeneration of a family of smooth curves X t , t ∈ ∆ ∗ by contracting a finite number of real 1-cycles γ i , i = 1 , . . . , l .Denote by p i the nodes on X o .Consider first the case (b) that X o is of compact type. In such case,the normalization ν : b X o = ∪ l +1 i =1 Y i → X o has l + 1 disconnected com-ponents and there exist q ∈ Y , q i , q i, ∈ Y i for 2 i l −
1, and q l, ∈ Y l so that ν ( q i +1 , ) = ν ( q i, ) = p i . It the known that the sumof the genera satisfies P li =1 g ( Y i ) = g . Let V be a neighborhood of thenodes. V consists of several components if l >
1. In such case, identity(19) still holds and for x ∈ X o − V in the notation of proof of Theorem2,(25) lim m →∞ µ WmL o = 1 g · lim t → o µ BX t . From Theorem 1, we conclude that lim t → o µ BX t = µ B ( b X ) = P li =1 µ B ( Y i ),since all the nodes are separating. Hence we conclude that(26) lim m →∞ µ WmL o = 1 g ( l X i =1 µ B ( Y i )) on X o − V. After shrinking V to the points p i , we conclude that theidentity (25) holds everywhere on X o − { p , . . . , p l } . Since P li =1 g ( Y i ) = g , we know that Z X o −∪ li =1 { p i } µ WmL o = 1 g Z X o −∪ li =1 { p i } l X i =1 µ B ( Y i ) = 1 , where we used the fact that Y i are smooth for 1 i l and hence µ B ( Y i ) are smooth measures as well. It follows that no mass is trappedin p i , i = 1 , . . . , l . Hence the identity (26) holds as a measure everywhereon X o .Consider now the case (a). In this case, X o is of irreducible with k non-separable nodes p i , i = 1 , . . . , k. The normalization ν : b X o → X o is smooth and irreducible. There are points q ij , j = 1 , , i = 1 , . . . , k on b X o such that ν ( q ij ) = p i . The identity (25) still holds in this case.Hence as in (b), g lim m →∞ µ WmL o = ν ∗ µ B b X o + µ o , where µ o is supported on the nodes { p , . . . , p l } . From our normaliza-tion, R X o µ WmL o = 1 and R X o µ B b X o = g − l , we conclude that R X o µ o = l. Hence we may assume that µ o = P li =1 a i δ p i with 0 a i and P li =1 a i = l .In other words,(27) g lim m →∞ µ WmL o = ν ∗ µ B b X o + l X i =1 a i δ p i . We claim that a i i l . For simplicity of explanation, weconsider first the case that X o has exactly two nodal points p , p ∈ X o in M g . In our setting, the line bundle L o at X o extends to a neigh-borhood U of o in M g . Consider a deformation family of stable curves X t centered at X o so that each X t has precisely one nodal point p t for t = 0 and X o corresponds to t = 0. In other words, p on X o is theresult of contracting a real cycle C on X t . Here we may assume that t ∈ ∆, a small disk centered at o and that coordinates described as in are used.Since b X t has just a single node at p t , we know that gµ WX t ,L t = g lim m →∞ µ WX t ,mL t = ( ν X t ,p t ) ∗ µ BX t + a t δ p t with a t = 1 from Theo-rem 2, where ν b X t ,p t : b X t → X t is the normalization X t at p t on X t .This holds for all t ∈ ∆ ∗ . In particular, in taking t →
0, and applyingFatou’s Lemma, we conclude that a = a lim inf t → a t . Similarly, a IMIT OF WEIERSTRASS MEASURES 21
In the general situation of k > , we choose a local family of irre-ducible stable curves X t centered at X o such that X t has only a nodeat p t and is smooth elsewhere for t ∈ ∆ ∗ . The same argument as aboveshows that a . Applying the same argument to p ti for 1 i k ,we conclude that a i i and hence the claim is proved.Since P li =1 a i = l , it follows from the claim that actually a i = 1 for1 i k . q.e.d. Let X be a Riemann surface of genus g . After contracting a non-separating real one cycle which is homologically non-trivial, we obtaina stable curve X with a node. X lies in the boundary of the Deligne-Mumford compactification M g − M g . The normalization of b X is acurve of genus g −
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