Twisted vector bundles on pointed nodal curves

Transcription

1 Proc. Indian Acad. Sci. (Math. Sci.) Vol. 115, No. 2, May 2005, pp Printed in India Twisted vector bundles on pointed nodal curves IVAN KAUSZ NWF I - Mathematik, Universität Regensburg, Regensburg, Germany ivan.kausz@mathematik.uni-regensburg.de MS received 19 January 2005 Abstract. Motivated by the quest for a good compactification of the moduli space of G-bundles on a nodal curve we establish a striking relationship between Abramovich s and Vistoli s twisted bundles and Gieseker vector bundles. Keywords. Twisted bundles; nodal curves. 1. Introduction This paper is the result of an attempt to understand a recent draft of Seshadri [8] and is meant as a contribution in the quest for a good compactification of the moduli space (or stack) of G-bundles on a nodal curve. We are led by the idea that such a compactification should behave well in families and also under partial normalization of nodal curves. This statement may be reformulated by saying that we are looking for an object which has the right to be called the moduli stack of stable maps into the classifying stack BG of a reductive group G. For finite groups G the stack of stable maps into BG has been recently constructed by Abramovich, Corti and Vistoli [1,2] by means of the so-called twisted bundles. On the other hand, as shown in [6], the notion of Gieseker vector bundles leads to the construction of the stack of stable maps into BGL r. In this note we establish a connection between the straightforward generalization of the notion of twisted bundles to the case of the non-finite reductive group GL r and Gieseker vector bundles. My hope is that this relationship whose observation is entirely due to Seshadri, and which in my mind is really striking may help to find the right notion for a more general reductive groups G. 2. Twisted G-bundles Throughout this section k denotes an algebraically closed field and G a reductive group over k. A twisted G-bundle is a twisted object in the sense of 3 of [1] where the target stack M is taken to be the classifying stack BG. For convenience, we recall the necessary definitions from loc. cit. DEFINITION An n-marked curve consists of data (U S, i ) where π: U S is a nodal curve and 1,..., n U are pairwise disjoint closed subschemes whose support do not 147

2 148 Ivan Kausz intersect the singular locus U sing of π and are such that the projections i S are étale. 2. A morphism between two n-marked curves (U S,i U ) and (V S,i V ) is an S- morphism f : U V such that f(i U ) i V for each i. Such a morphism is called strict, if for each i the support of f 1 (i V ) coincides with the support of U i and if furthermore the support of f 1 (V sing ) coincides with one of the U sing. 3. The pull back of an n-marked curve (U S, i ) by a morphism S S is the n-marked curve (U S S, i S S ). 4. An n-pointed nodal curve is an n-marked curve where the projections i S are isomorphisms. 5. Let (U S, i ) be an n-marked curve. The complement (inside U) of the union of the singular locus U sing and the markings i is called the generic locus of U and is denoted by U gen. DEFINITION An action of a finite group Ɣ on an n-marked nodal curve (U S, i ) is an action of Ɣ on U as an S-scheme which leaves the i invariant. Such an action is called tame, if for each geometric point u of U the stabilizer Ɣ u Ɣ of u has order prime to the characteristic of u. 2. Let S be a k-scheme. Let (U S, i ) be an n-marked nodal curve and let η be a principal G-bundle on U.Anessential action of a finite group Ɣ on (η, U) is a pair of actions of Ɣ on η and on (U S, i ) such that (i) the actions of Ɣ on η and on U are compatible, i.e. if π: η U denotes the projection, then π γ = γ π for each γ Ɣ, (ii) if γ Ɣ is an element different from the identity and u is a geometric point of U fixed by γ, then the automorphism of the fiber η u induced by γ is not trivial. 3. An essential action of a finite group Ɣ on (η, U) is called tame, if the action of Ɣ on (U S, i ) is tame. DEFINITION2.3 Let S be a k-scheme. Let C S be an n-pointed nodal curve and let ξ be a principal G-bundle over C gen.achart (U,η,Ɣ)for ξ consists of the following data: (1) An n-marked curve U S and a strict morphism φ: U C. (2) A principal G-bundle η on U. (3) An isomorphism η U U gen ξ C U gen of G-bundles on U gen. (4) A finite group Ɣ. (5) A tame, essential action of Ɣ on (η, U). These data are required to satisfy the following conditions: (i) The action of Ɣ leaves the morphisms U C and η U U gen ξ C U gen invariant. (ii) The induced morphism U/Ɣ C is étale.

3 Twisted vector bundles on pointed nodal curves 149 PROPOSITION2.4 (see[1],proposition3.2.3) Let C S be an n-pointed nodal curve over a k-scheme S and let ξ be a principal G- bundle on C gen. Let (U,η,Ɣ)be a chart for ξ. Then the following holds: (1) The action of Ɣ on U gen is free. Let s be a geometric point of S and u be a closed point of the curve U s. Let Ɣ u Ɣ be the stabilizer of u. Then Ɣ u is a cyclic group. Let e be its order and let γ u be a generator of Ɣ u. Then (2) if u is a regular point, the action of γ u on the tangent space of U s at u is via multiplication by a primitive e-th root of unity, (3) if u is a singular point, Ɣ u leaves each of the two branches of U s at u invariant. The action of γ u on the tangent space of each of the branches is via multiplication with a primitive e-th root of unity. DEFINITION2.5 Let C S be an n-pointed nodal curve over a k-scheme S and let ξ be a principal G- bundle on C gen. A chart (U,η,Ɣ)for ξ is called balanced, if for each geometric fiber of U S and each singular point u on it the action of γ u on the tangent spaces of the two branches is via multiplication with primitive roots of unity which are inverse to each other. DEFINITION2.6 Let C S be an n-pointed nodal curve over a k-scheme S and let ξ be a principal G- bundle on C gen. Two charts (U 1,η 1,Ɣ 1 )and (U 2,η 2,Ɣ 2 )of ξ are called compatible,iffor each pair of u 1, u 2 of geometric points U 1, U 2 lying above the same geometric point u of C the following holds: Let C sh denote the strict henselization of C at u. Forj =1,2 let Ɣ j Ɣ j denote the stabilizer subgroup of the point u j. Let Uj sh denote the strict henselization of U j at u j, and let ηj sh := η j Uj Uj sh. Then there exists an isomorphism θ: Ɣ 1 Ɣ 2,aθequivariant isomorphism φ: U1 sh U2 sh of Csh -schemes, and a θ-equivariant isomorphism η1 sh φ η2 sh DEFINITION2.7 of G-bundles. Let g and n be two non-negative integers. An n-pointed twisted G-bundle of genus g is a triple (ξ, C S, A) where (1) S is a k-scheme, (2) C S is a proper n-pointed nodal curve of finite presentation with geometrically connected fibers of genus g, (3) ξ is a principal G-bundle on C gen, (4) A ={(U α,η α,ɣ α )}is a balanced atlas, i.e. a collection of mutually compatible balanced charts for ξ, such that the images of the U α cover C. DEFINITION2.8 Let (ξ, C S,A) be an n-pointed twisted G-bundle of genus g. A morphism of k- schemes S S induces a triple (ξ,c S,A )as follows:

4 150 Ivan Kausz The n-pointed nodal curve C S is the pull back of C S by S S. Thus we have a morphism C gen C gen, and the G-bundle ξ is the pull back of ξ by this morphism. Let {U α,η α,ɣ α }be the set of charts which make up the atlas A. Then A ={U α,η α,ɣ α}, where U α S is the pull back of the n-marked curve U α S, and η α is the pull back of η α by the morphism U α U α. Since the (U α,η α,ɣ α) are charts for ξ which are balanced and mutually compatible (see [1], Proposition 3.4.3), A is a balanced atlas. Thus the triple (ξ,c S,A ) is an n-pointed twisted G-bundle of genus g. Itis called the pull back of (ξ, C S,A) by the morphism S S. DEFINITION2.9 A morphism between two n-pointed twisted G-bundles (ξ,c S,A ) and (ξ, C S,A) consists of a Cartesian diagram C S C S and an isomorphism ξ ξ Cgen C gen such that the pull-back of the charts in A (considered as charts for ξ ) are compatible with all the charts in A. 3. Review of Gieseker vector bundles In this section I will recall some definitions from my earlier papers [4] and [5]. Let k be an algebraically closed field. Let n 1 be an integer and let R 1,...,R n be n copies of the projective line P 1. On each R i we choose two distinct points x i and y i. Let R be the nodal curve over k constructed from R 1,...,R n by identifying y i with x i+1 for i = 1,...,n 1. We call such a curve R a chain of projective lines of length n with components R 1,...,R n. On the extremal components R 1 and R n we have the two points x 1 and y n respectively, which are smooth points of R. DEFINITION3.1 A vector bundle E of rank r on R is called admissible,if (1) for each i [1,n] the restriction of E on the component R i is of the form d i O Ri (1) (r d i )O Ri for some integer d i 1 and (2) there exists no nonvanishing global section of E over R which vanishes in the two points x 1 and y n. Let C be an irreducible curve with exactly one double point p. Let C C be the normalization of C and let p 1,p 2 C be the two points lying above p. Let C 0 := C. For n 1weletC n denote the reducible nodal curve which is constructed from C and a chain R = R 1 R n of projective lines by identifying the points p 1,x 1 and p 2,y n respectively.

5 DEFINITION3.2 Twisted vector bundles on pointed nodal curves 151 A Gieseker vector bundle on C is a pair (C C, F) where C = C n for some n 0, the morphism C C is the one which contracts the chain of projective lines to the point p and F is a vector bundle on C whose restriction to the chain of projective lines is admissible in the sense of 3.1. DEFINITION3.3 A Gieseker vector bundle datum on the two-pointed curve ( C,p 1,p 2 )is a triple (C C, F, p ), where (C C, F) is a Gieseker vector bundle on C and p is a singular point on C. Let V and W be two r-dimensional k-vector spaces. In [4] I have constructed a certain compactification KGL(V, W) of the space Isom(V, W ) of linear isomorphisms from V to W which has properties similar to De Concini s and Procesi s so-called wonderful compactification of adjoint linear groups. We need the following fact about KGL(V, W ) whose proof can be found in 9 of [4]. The variety KGL(V, W) is the disjoint union of strata O I,J KGL(V, W ) indexed by pairs of subsets I,J [0,r 1] such that min(i) + min(j ) r. Let I,J be such a pair. Let us write I ={i 1,...,i n1 }and J ={j 1,...,j n2 }where i 1 < <i n1 <i n1 +1:=r and j 1 < <j n2 <j n2 +1:=r.A(k-valued) point in O I,J is given by the data where = (F (V ), F (W), ϕ 1,...,ϕ n1,ψ 1,...,ψ n2, ), (1) F (V ) denotes a flag 0 = F 0 (V )F 1 (V ) F n2 (V ) F n2 +1(V ) F n1 +n 2 +1(V ) = V with dim F ν (V ) = r j n2 +1 ν for ν [0,n 2 ] and dim F ν (V ) = i ν n2 [n 2 + 1,n 1 +n 2 +1]. (2) F (W) denotes a flag for ν 0 = F 0 (W) F 1 (W) F n1 (W ) F n1 +1(W) F n1 +n 2 +1(W ) = W, where dim F ν (W) = r i n1 +1 ν for ν [0,n 1 ] and dim F ν (W ) = i ν l for ν [n 1 + 1,n 1 +n 2 +1]. (3) The symbol ϕ ν denotes the homothety class of an isomorphism from the subquotient F n1 ν+1(w)/f n1 ν(w) of W to the subquotient F n2 +ν+1(v )/F n2 +ν(v ) of V. (4) The symbol ψ ν denotes the homothety class of an isomorphism from the subquotient F n2 ν+1(v )/F n2 ν(v ) of V to the subquotient F n1 +ν+1(w )/F n1 +ν(w ) of W. (5) The symbol denotes an isomorphism from the subquotient F n2 +1(V )/F n2 (V ) of V to the subquotient F n1 +1(W)/F n1 (W) of W. The relationship between Gieseker vector bundles and the compactification KGL(V, W ) is given by the following theorem.

6 152 IvanKausz Theorem 3.4 ([5], Theorem 9.5). There exists a natural bijection from the set of all Gieseker vector bundle data on ( C,p 1,p 2 ) to the set of all pairs (E,), where E is a vector bundle on C and is a k-valued point in KGL(E[p 1 ], E[p 2 ]). More precisely, let (C C, F,p )be a Gieseker vector bundle datum on ( C,p 1,p 2 ). Let R = R 1 R n be the chain of projective lines in C. Let y 0 := p 1 and x n+1 := p 2. Let n 1 + n 2 = n be such that the singular point p C comes from identifying the points y n1 and x n1 +1. Let d i be the degree of F restricted to R i. Let (E,) be the pair associated to the given Gieseker vector bundle datum (C C, F,p ). Then is in fact a point in the stratum O I,J, where I ={i 1,...,i n1 },J ={j 1,...,j n2 }and the i ν, j ν are defined by n 1 i ν = r d i, i=ν j ν = r n ν+1 i=n 1 +1 d i. The special case n = 0 is included here in the sense that I = J = and O, = Isom(E[p 1 ], E[p 2 ]). 4. Twisted GL r -bundles on a fixed curve Throughout this section k denotes an algebraically closed field and r a positive integer. Let (C, p i ) be an n-pointed nodal curve over k. Let TVB r (C, p i ) be the set of isomorphism classes of n-pointed twisted GL r -bundles of the form (ξ, C Spec(k), A). The case of a one-pointed smooth curve Assume that C is smooth and that n = 1, i.e. (C, p i ) = (C, p) is a one-pointed smooth curve. Let PB r (C, p) be the set of isomorphism classes of vector bundles E of rank r on C together with a flag in the fiber at p. Theorem 4.1. There is a natural surjection TVB r (C, p) PB r (C, p). We skip the proof of Theorem 4.1, since on the one hand the result is well-known [3,7] and on the other hand there is a proof analogous to (and easier than) the proof of Theorem 4.2 which we give in detail below. The case of a nodal curve with one singularity Assume now that n = 0 and C has exactly one double point. Let GVB r (C) be the set of isomorphism classes of Gieseker vector bundles of rank r on C. Theorem 4.2. There is a natural surjection TVB r (C) GVB r (C). The rest of the paper is concerned with the proof of Theorem 4.2.

7 5. Construction Twisted vector bundles on pointed nodal curves 153 Let C be a nodal curve over Spec(k) with one singular point p. Let (ξ, C Spec(k), A) be an object of TVB r (C). Let (U,η,Ɣ)be a chart belonging to A such that there is a point q U which is mapped to p. We denote by Ô p and Ô q the completion of the local rings O C,p and O U,q respectively. Let Ɣ q Ɣ be the subgroup consisting of those elements, which leave q invariant. Ɣ q acts on Ô q, and Ô p may be identified with the set of invariants under that action. By Proposition 2.4, the group Ɣ q is cyclic of some order e (which is prime to char(k) by the tameness assumption). Let γ be a generator of Ɣ q. We choose an isomorphism Ô p k[[s, t]]/(s t). (1) It follows from Proposition 2.4(3) that there exists an isomorphism Ô q k[[u, v]]/(u v) (2) and a primitive e-th root of unity ζ such that the diagrams = Ô q k[[u, v]]/(u v) u v γ = Ô q k[[u, v]]/(u v) ζu ζ 1 v and = Ô q k[[u, v]]/(u v) u e v e = Ô p k[[s, t]]/(s t) s t are commutative. Let K p be the total quotient ring of Ô p. Then we have Spec( K p ) = Spec(Ô p ) C C gen and the isomorphism (1) induces an isomorphism K p k((s)) k((t)). We choose an isomorphism ξ Cgen Spec( K p ) GL r Spec( K p ). (3) The group Ɣ q acts on η U Spec(Ô q ) (since it acts compatibly on η, U, Spec(Ô q )). To analyse this action we need the following lemma. Lemma 5.1. Let k be an algebraically closed field. Let (R, m) be a local k-algebra with residue field R/m = k. Let Ɣ be a cyclic group of order e prime to the characteristic of k and let γ Ɣ be a generator. Assume that Ɣ acts on R such that the induced action on k is trivial. Let M be a trivial R-module of rank r on which Ɣ acts such that γ(ax)=γ(a)γ(x) for all a R, x M. Then there is a basis x 1,...,x r of M such that γ(x i ) = ζ i x i for some e-th roots of unity ζ i.

8 154 IvanKausz Proof. Let e 1,...,e r be an arbitrary basis of M. Let a = (a i,j ) GL r (R) be defined by γ(e j )= r i=1 a i,j e i. Since γ is of order e, it follows that e 1 γ j (a) = 1. j=0 We have to show that there is a matrix b GL r (R) such that a γ(b)=b z for some diagonal matrix z GL r (k) with z e = 1. Representation theory of finite groups tells us that there is a matrix c GL r (k) and a diagonal matrix z GL r (k) with z e = 1 such that a c c z modulo m. Let a := c 1 a c and let b be the matrix ( ) b e 1 i 1 := γ j (a ) z i. i=0 j=0 Since b e 1 modulo m, it follows that b GL r (R). Using the fact that e 1 i=0 γ i (a ) = 1 a simple calculation shows that γ(b )=(a ) 1 b z. Therefore, if we set b := c b, we get the desired equality. COROLLARY5.2 There exists an isomorphism η U Spec(Ô q ) GL r Spec(Ô q ) (4) of principal GL r -bundles on Spec(Ô q ), and elements α 1,...,α r Z/eZ such that the following diagram commutes: η U Spec(Ô q ) γ η U Spec(Ô q ) = GL r Spec(Ô q ) diag(ζ α 1,...,ζ αr ) γ = GLr Spec(Ô q ) where the morphism diag(ζ α 1,...,ζ α r):gl r GL r is a multiplication from the left with the matrix whose only non-zero entries are the values ζ α 1,...,ζ α r on the diagonal. Proof. This is immediate from Lemma 5.1. Let K q be the total quotient ring of Ô q. The Ɣ-equivariant isomorphism η U U gen ξ Cgen U gen, which is part of the data of the chart (U,η,Ɣ), induces a Ɣ q - equivariant isomorphism η U Spec( K q ) ξ Cgen Spec( K q ) (5) of principal GL r -bundles over Spec( K q ).

9 Twisted vector bundles on pointed nodal curves 155 Via the isomorphisms (3) and (4) such an isomorphism is given by a matrix F GL r ( K q ) such that γ(f)=f diag(ζ α 1,...,ζ α r ). The isomorphism (2) induces an isomorphism GL r ( K q ) GL r (k((u))) GL r (k((v))) and we denote by (F 1 (u), F 2 (v)) the image of F under this isomorphism. The above condition on F translates into the conditions Fi,j 1 (ζ u) = ζ α j Fi,j 1 (u), (6) Fi,j 2 (ζ 1 v) = ζ α j Fi,j 2 (v), (7) for the entries Fi,j 1 (u) k((u)) and F 2 i,j (v) k((v)) of the matrices F 1 (u) and F 2 (v). After possibly changing the isomorphism (4) by a permutation matrix, we can choose integers a 1,...,a r with 0 a 1 a 2 a r <e and a i α i mod ez. (8) Conditions (6), (7) imply that there are matrices H 1 (s) and H 2 (t) with entries Hi,j 1 (s) k((s)) and Hi,j 2 (t) k((t)) such that F 1 i,j (u) = ua j H 1 i,j (ue ), (9) F 2 i,j (v) = v a j H 2 i,j (ve ). (10) We will now use the GL r -bundle ξ over C gen, the isomorphisms (1) and (3), the numbers a 1,...,a r and the matrices H 1 (s) and H 2 (t), to construct a Gieseker vector bundle of rank r on the curve C. Let p 1 and p 2 denote the closed point of Spec(k[[s]]) and Spec(k[[t]]) respectively. Let V be the trivial vector bundle O [1,r] on the disjoint union Spec(k[[s]]) Spec(k[[t]]) (the normalization of Spec(k[[s, t]]/(s t))), and let V and W be its fiber at p 1 and p 2 respectively. Of course, both V and W are naturally identified with k [1,r]. The numbers a 1,...,a r define a partition [1,r]=D 1 D 2 D m characterized by the following properties: (1) D 1 is the (possibly empty) set of all indices i such that a i = 0. (2) For ν 2 the set D ν is non-empty. (3) If 1 ν<ν m,i D ν and j D ν then a i <a j. (4) For all ν [1,m] and i, j D ν we have a i = a j. We define filtrations 0 = F 0 (V ) F 1 (V ) F 2 (V ) F m 1 (V ) F m (V ) = V, 0 = F 0 (W) F 1 (W) F 2 (W) F m 1 (W ) F m (W ) = W,

10 156 IvanKausz by setting F i (V ) := k D 1 D i and F i (W ) := k D m i+1 D m for i = 0,...,m.Fori=1,...,m 1, let ϕ i : F m i (W)/F m i 1 (W) = k D i+1 k D i+1 = F i+1 (V )/F i (V ) be the identity morphism on k D i+1 and let ϕ i be the homothety class of ϕ i. Finally let : F 1 (V )/F 0 (V ) = k D 1 k D 1 = F m (W )/F m 1 (W ) be the identity morphism on k D 1. By [4], 9.3 the data ((F (V ), F (W)), ϕ 1,...,ϕ m 1, ) defines a k-valued point of KGL(V, W), i.e. a generalized isomorphism from V to W. Let C C be the normalization of the curve C. By a slight abuse of notation we denote also by p 1, p 2 the two points of C which lie above the singular point p of C. Let E ξ be the rank r vector bundle on C gen = C\{p 1,p 2 }associated to the principal GL r -bundle ξ. We use the isomorphism (k((s)) k((t))) [1,r] (H 1,H 2 ) (k((s)) k((t))) [1,r] = (1), (3) V k[[s]] k[[t]] (k((s)) k((t))) E ξ O C K p as a glueing datum to define a vector bundle E on C, whose fibers at the points p 1 and p 2 are naturally identified with V and W respectively. By Theorem 3.4 the pair (E,) induces a Gieseker vector bundle datum (C C, F,p ) on ( C,p 1,p 2 ) which in turn induces a Gieseker vector bundle (C C, F) on C. For the convenience of the reader I will now describe the Gieseker vector bundle (C C, F) explicitly. Let R 0 := Spec(k[[s]]), R m := Spec(k[[t]]). Ifm=1, we set R = Spec(k[[s, t]]/(s t)), which is nothing else but the nodal curve which arises from R 0 R m by identifying the points p 1 and p 2.Ifm 2, let R 1,...,R m 1 be m 1 copies of the projective line P 1 and let x i,y i be two distinct points in R i. Let R be the nodal curve which arises from the union R 0 R 1 R m 1 R m by identifying p 1 R 0 and p 2 R m with x 1 R 1 and y m 1 R m 1 respectively and by identifying y i R i with x i+1 R i+1 for i [1,m 2]: R 0 R 1 R 2 R 3 y 1 = x 2 y 2 = x 3 p 1 = x 1 R m R m 2 R m 1. y m 2 = x y m 1 = p 2 m 1

11 Twisted vector bundles on pointed nodal curves 157 Let O Ri (1) be the defining bundle on R i = P 1 together with isomorphisms O Ri (1)[x i ] k and O Ri (1)[y i ] k. (11) We define the rank r vector bundles E i := O D 1 D i R i on R i together with the isomorphisms O Ri (1) D i+1 O D i+2 D m R i E i [x i ] k [1,r] and E i [y i ] k [1,r] (12) induced by (11). The maximal ideal sk[[s]] of k[[s]] is a free module of rank one and as such defines a line bundle O R0 ( 1) on R 0 = Spec(k[[s]]). We consider this line bundle together with the isomorphism O R0 ( 1)[p 2 ] k (13) given by sk[[s]]/s 2 k[[s]] k, s 1. The generic fiber of O R0 ( 1) is identified with k((s)) via the inclusion sk[[s]] k[[s]]. Then we have the rank r vector bundles E 0 := O D 1 R 0 O R0 ( 1) D 2 D m and E m := O [1,r] R m on R 0 and R m together with isomorphisms E 0 [p 1 ] k [1,r] and E m [p 2 ] k [1,r] (14) (the first one being induced by (13)) and isomorphisms E 0 OR0 k((s)) k((s)) [1,r] and E m ORm k((t)) k((t)) [1,r]. (15) The vector bundles E 0,...,E m glue together via the isomorphisms (12) and (14) to form a rank r vector bundle E on R. Let C C be the modification of C obtained by glueing together R and C gen along the isomorphism Spec(k((s))) Spec(k((t))) R (1) Spec( K p ) and let F be the rank r vector bundle on C obtained by glueing together E and E gen via the isomorphism C gen (k((s)) k((t))) [1,r] (H 1,H 2) (k((s)) k((t))) [1,r] = (15) = (1), (3) E OR (k((s)) k((t))) E ξ O C K p It is easy to check that (C C, F) is indeed a Gieseker vector bundle on C. It remains to be shown that the association (ξ, C Spec(k), A) (C C, F) is surjective and does not depend on the choices (1), (2), (3) and (4) which we made during the construction. This will be done in the following sections.

12 158 IvanKausz 6. Independence of the isomorphisms (1) and (2) Let Ô p k[[s, t]]/(s t) (1 ) be another isomorphism and let Ô q k[[u, v]]/(u v) (2 ) be an isomorphism with the required property with respect to (1 ). For the moment we make the following assumption: The images of the two minimal ideals of Ô p under (1) and (1 ) are the same. (*) Then there are units σ (s), π(s) k[[s]] and τ(t), ω(t) k[[t]] such that the following diagrams commute: s sσ (s) k[[s, t]]/(s t) k[[s, t]]/(s t) t tτ(t) (1) (1 ), Ô p u uπ(u e ) k[[u, v]]/(u v) k[[u, v]]/(u v) v vω(v e ) (2) (2 ) Ô p Furthermore we have π e = σ and ω e = τ. It should be noticed that the e-th root of unity ζ is independent of whether we choose (1) or (1 ), since it is the eigenvalue of γ operating on the tangent space of one of the branches of Spec(Ô q ) and by assumption ( ) both the isomorphisms (1) and (1 ) map that branch Spec(Ô q ) to the same branch {v = 0} of Spec(k[[u, v]]/(u v)). Therefore the elements α 1,...,α r Z/eZ and the numbers a 1,...,a r are independent of whether we choose (1) or (1 ). Let ( F 1 (u), F 2 (v)) be the image of F under the isomorphism GL r ( K q ) GL r (k[[u]]) GL r (k[[v]]) induced by (2 ). Then we have F 1 (u) = F 1 (uπ(u e )) and F 2 (v) = F 2 (vω(v e )) and it follows that where F i,j 1 (u) = uaj H i,j 1 (ue ), F i,j 2 (v) = v aj H i,j 2 (ve ), H i,j 1 (s) = π a j Hi,j 1 (sσ ), H 2 i,j (t) = ω a j H 2 i,j (tτ).

13 Twisted vector bundles on pointed nodal curves 159 Therefore the following diagram commutes: (k((s)) k((t))) [1,r] (H 1,H 2 ) (k((s)) k((t))) [1,r] = (1), (3) E ξ s sσ t tτ O C K p (1 ), (3) = (k((s)) k((t))) [1,r] ( H 1, H 2 ) (k((s)) k((t))) [1,r] where the left vertical arrow maps an element (x(s), y(t)) to the element (diag(π a 1,...,π a r )x(sσ ), diag(ω a 1,...,ω a r )y(tτ)). Let Ẽ be the vector bundle on C obtained by glueing datum ( H 1, H 2 ). Then the above diagram shows that there is an isomorphism E Ẽ which induces the isomorphisms E[p 1 ] = k [1,r] diag(π(0) a 1,...,π(0) ar ) E[p 2 ] = k [1,r] diag(ω(0) a 1,...,ω(0) ar ) k [1,r] = Ẽ[p 1 ], k [1,r] = Ẽ[p 2 ], between the fibers at p 1 and p 2 respectively. Thus it maps the generalized isomorphism KGL(k [1,r],k [1,r] ) = KGL(E[p 1 ], E[p 2 ]) to the generalized isomorphism KGL(k [1,r],k [1,r] ) = KGL(Ẽ[p 1 ], Ẽ[p 2 ]). This shows that the pairs (E,) and (Ẽ,) are isomorphic. Consequently this is also true for the associated Gieseker vector bundles. To get rid of the assumption ( ) we now investigate what happens if we change the isomorphisms (1), (2) by composing them with the automorphisms k[[s, t]]/(s t) k[[u, v]]/(u v) s t t s k[[s, t]]/(s t), u v v u k[[u, v]]/(u v), respectively. This means that ζ 1 takes the role of ζ and consequently the set {α 1,...,α r } Z/eZ from 5.2 is replaced by the set { α 1,..., α r }. It follows that in (8) we would choose integers ã 1,...,ã r instead of a 1,...,a r, where { a i, for i [1,i 1 ]=D 1, ã i = e a r+i1 +1 i, for i [i 1 + 1,r]. Then the matrix F is replaced by the matrix F = F, where I i1 0 = 1 0 1

14 160 IvanKausz is the permutation matrix belonging to the permutation λ S r, where { i for i [1,i1 ] λ(i) = r +i 1 +1 i, for i [i 1 + 1,r] and the matrices H 1 (s) and H 2 (t) are replaced by the matrices H 1 (s) and H 2 (t) respectively, where [ ] H 1 (s) = H 2 Ii1 0 (s) 0 s 1 I r i1 and [ ] H 2 (t) = H 1 Ii1 0 (t). 0 ti r i1 The numbers ã 1,...,ã r define the partition [1,r]= D 1 D 2 D m, where D 1 = D 1 and D i = λ(d m+2 i ) for i [2,m]. Let ( R = R 0 R m,ẽ) be the nodal curve associated to this partition, together with isomorphisms Ẽ O R k((s)) k((s)) [1,r] and Ẽ O k((t)) k((t))[1,r] R as in (15). Now one checks easily that there is an isomorphism ρ: (R, E) ( R, Ẽ) which sends the component R i to R m i (i = 0,...,m), such that the following diagram commutes: E OR (k((s)) k((t))) = (k((s)) k((t))) [1,r] ρ ρ, Ẽ O (k((s)) k((t))) = R (k((s)) k((t))) [1,r] where the morphism ρ is given by ( [ ] [ ] ) Ii1 0 Ii1 0 (x(s), y(t)) 0 s 1 y(s), x(t). I r i1 0 ti r i1 From the commutativity of the diagram (k((s)) k((t))) [1,r] ρ (k((s)) k((t))) [1,r] (H 1 (s),h 2 (t)) ( H 1 (s), (k((s)) k((t))) [1,r] t s s t H 2 (t)) (k((s)) k((t))) [1,r] (1),(3) (1 ),(3) E ξ O C E ξ O C K p K p, it finally follows that the Gieseker vector bundle ( C C, F) constructed from the data ξ, (1 ), (3), (ã 1,...,ã r ), H 1 (s) and H 2 (t) is isomorphic to the Gieseker vector bundle (C C, F) constructed from the data ξ, (1), (3), (a 1,...,a r ),H 1 (s) and H 2 (t).

15 Twisted vector bundles on pointed nodal curves Independence of the isomorphisms (3) and (4) Independence of (3) is immediate, since if we change it by an automorphism of GL r Spec( K p ) (which can be written as an element in GL r (k((s))) GL r (k((t)))), then (H 1 (s), H 2 (t)) is changed by that same matrix. Two isomorphisms (4) differ by a matrix A = (A i,j ) GL r (Ô q ) such that A = diag(ζ α 1,...,ζ α r ) γ (A) diag(ζ α 1,...,ζ α r ). (16) After identifying Ô q with the ring k[[u, v]]/(u v) via the isomorphism (1), we can write A = A 0 + u A 1 (u) + v A 2 (v) with uniquely determined matrices A 0 GL r (k), A 1 (u) M(r r, k[[u]]) and A 2 (v) M(r r, k[[v]]). Condition (16) implies that A 0 is a block matrix of the form A 0 = A0 1 0, (17) 0 A 0 m where A 0 i are blocks of size D i for i = 1,...,m. Condition (16) implies furthermore that there are matrices B 1 (s) = (B 1 i,j (s)) GL r(k[[s]]) and B 2 (t) = (B 2 i,j (t)) GL r(k[[t]]) such that and such that A 1 (u) = u 1 diag(u a 1,...,u a r ) B 1 (u e ) diag(u a 1,...,u a r ), A 2 (v) = v 1 diag(v a 1,...,v a r ) B 2 (v e ) diag(v a 1,...,v a r ) B 1 i,j (0) = 0 for a i a j 0, B 2 i,j (0) = 0 for a j a i 0. (18) The change of (4) by the matrix A means that we have to replace F by the matrix F = F A and that consequently we have to replace the matrices H 1 (s) and H 2 (t) by the matrices H 1 (s) = H 1 (s) (A 0 + B 1 (s)) and H 2 (t) = H 2 (t) (A 0 + B 2 (t)) respectively. The pair of matrices (A 0 + B 1 (s), A 0 + B 2 (t)) defines an automorphism of V which induces the automorphisms A 0 + B 1 (0) and A 0 + B 2 (0) on the special fibers V and W respectively. From (17) and (18) it follows that the induced automorphism of KGL(V, W ) maps the generalized isomorphism to itself. It follows that the pair (Ẽ, ) obtained by the glueing datum ( H 1, H 2 ) is isomorphic to the pair (E,)obtained by the glueing datum (H 1,H 2 ). Therefore the induced Gieseker vector bundles are also isomorphic.

16 162 IvanKausz 8. Surjectivity Let (C C, F) be a Gieseker vector bundle on C. By definition, C is either isomorphic to C, or it is the union of the normalization C of C and a chain R of projective lines which intersects C in the two points p 1 and p 2 lying above the singularity p C. In the first case we let p := p, and in the second case we let p = p 2. Then the triple ( C C, F,p ) is a Gieseker vector bundle datum in the sense of Definition 3.3. By Proposition 3.4 such a datum induces a vector bundle E on the curve C together with a generalized isomorphism from V := E[p 1 ]tow :=E[p 2 ]. More precisely, is a k-valued point of KGL(V, W ) which lies in the stratum O I,J for some I [0,r 1] and J =. As we have recalled in 3, such a point is given by a tupel where m := I +1, ((F (V ), F (W)), ϕ 1,...,ϕ m 1, ), 0 = F 0 (V ) F 1 (V ) F 2 (V ) F m 1 (V ) F m (V ) = V, 0 = F 0 (W) F 1 (W) F 2 (W) F m 1 (W ) F m (W ) = W are flags in V and W respectively, ϕ i is the homothety class of an isomorphism ϕ ν : F m ν (W)/F m ν 1 (W) F ν+1 (V )/F ν (V ) and denotes an isomorphism F 1 (V )/F 0 (V ) F m (W )/F m 1 (W ). There is a basis v 1,...,v r of V and w 1,...,w r of W and a partition with the property that [1,r]=D 1 D 2 D m (1) i D ν, j D ν and i<jimplies ν ν. (2) F ν (V ) is generated by {v i i D 1 D ν }and F ν (W ) is generated by {w i i D m ν+1 D m }. (3) For i D ν+1 the isomorphism ϕ ν sends the residue class of w i mod F m ν 1 (W ) to the residue class of v i mod F ν (V ). (4) For i D 1 the isomorphism sends v i to the residue class of w i mod F m 1 (W ). For i = 1, 2 we choose an isomorphism E O C Ô pi Ô r p i (19) which induces the isomorphism V k r, v i e i and W k r, w i e i from the fibres at p 1 and p 2 respectively, where e 1,...,e r is the canonical basis of k r. Let ξ be the prinicpal GL r -bundle on C gen of local frames of the restriction of the vector bundle E to C gen = C\{p 1,p 2 }. Then the isomorphism (19) induces the isomorphism ξ Cgen Spec( K p ) GL r Spec( K p ). (20)

17 Twisted vector bundles on pointed nodal curves 163 Lemma 8.1. There is a morphism f : U C, an integer e m prime to the characteristic of k and an operation of Ɣ := Z/eZ on U such that (1) Ɣ leaves f invariant and the induced morphism U/Ɣ C is étale, (2) U has exactly one singular point q and f 1 (p) ={q}, (3) the action of Ɣ on f 1 (C gen ) is free. Proof. Since p is an ordinary double point of C, there exists a diagram of pointed schemes and étale morphisms as follows: (C, p) étale (U 0,q 0 ) étale (V 0,y 0 ):=((Spec(k[s, t]/(s t)), (s, t)). After removing from U 0 the points q 0 inthe fiber of U 0 Cwe may assume that q 0 is the only point lying above p. Choose e Z prime to char(k) with e m. Let (V, y) := (Spec(k[u, v]/(u v)), (u, v)) and let (V, y) (V 0,y 0 ) be defined by s u e, t v e. Let γ be a generator of Ɣ = Z/eZ and let ζ k be a primitive e-th root of unity. We define an action of Ɣ on (V, y) by letting γ (u) = ζuand γ(v)=ζ 1 v. Now we set U := U 0 V0 V and let f : U C be the composition U U 0 C. From V the scheme U inherits an action of the group Ɣ. Since V/Ɣ = V 0 and U 0 V 0 is flat we have U/Ɣ = U 0 which by construction is étale over C. The only point in the fiber of f over p is the point q = (q 0,y) U. Since U 0 V 0 is étale and V V 0 is smooth outside the point y, it follows that the fiber product U = U 0 V0 V is regular outside q. Furthermore, since the action of Ɣ on V \{y} is free the same holds for the action of Ɣ on U\{q}. In what follows we will construct a chart (U,η,Ɣ) for ξ where U C and Ɣ are chosen as in the lemma and the GL r -bundle η with Ɣ-operation is glued together from an object η gen over U gen and an object η q over the completion of U at the singular point q. To fix the notation, let Ô q be the completion of the local ring O U,q and let γ be a generator of Ɣ. There exists an isomorphism Ô q k[[u, v]]/(u v) (21) and a primitive e-th root of unity ζ such that the automorphism γ : Ô q Ô q translates into the automorphism u ζu,v ζ 1 v of k[[u, v]]/(u v) (see [2], 2.1.2). Let a i [0,e 1] (i [1,r]) be chosen such that a i = 0 for i D 1, a i <a j for i D ν,j D ν,ν<ν, a i =a j for i, j D ν,ν [1,m]. Let η q := GL r Spec Ô q together with the Ɣ-operation be defined by diag(ζ a 1,...,ζ a r ) γ:gl r Spec Ô q GL r Spec Ô q.

18 164 IvanKausz Let η gen := ξ Cgen U gen together with the Ɣ-operation be given by id γ : ξ Cgen U gen ξ Cgen U gen. Now we glue together η q and η gen along Spec K q = k((u)) k((v)) via the isomorphism η q Ôq Spec( K q ) (20) GLr Spec( K q ) F 1 F 2 GLr Spec( K q ) = η gen Ugen Spec( K q ), where F 1 = diag(u a 1,...,u a r ) and F 2 = diag(v a 1,...,v a r ). This gives a principal GL r -bundle η on U. From the commutativity of the diagram η q Ôq Spec( K q ) diag(ζ a 1,...,ζ ar ) γ η q Ôq Spec( K q ) = η gen Ugen Spec( K q ) id γ, = ηgen Spec( Ugen K q ) it follows that the Ɣ-operation on η q and η gen induces a Ɣ-operation on η. It is clear from the construction that the triple (U,η,Ɣ)forms a chart for ξ. There is a chart (U 1,η 1,Ɣ 1 ) for ξ, where U 1 := C gen, η := ξ, Ɣ := (1). This chart together with the chart (U,η,Ɣ)make up a balanced atlas A for ξ. It is clear by construction that the twisted G-bundle (ξ, C Spec(k), A) is mapped to the Gieseker vector bundle (C C, F). 9. Further directions The relationship between twisted GL r -bundles and Gieseker vector bundles should be further investigated since it might lead to a clue as to what the right notion of stable maps to the classifying stack of a reductive group are. The next step would be to try to extend the mapping given in Theorem 4.2 so that it works for families. For example let A := C[[t]], S := Spec A and C S be a stable curve over S. Let ( ) C C, F S be a Gieseker vector bundle of rank r on C. Assume in particular that the generic fiber of C S is smooth and that its special fiber is irreducible with one double point p. Then it can be shown that there is a twisted GL r - bundle (ξ, C S,A) such that if we apply the mappings from Theorems 4.1 and 4.2 to the isomorphism class of the generic and special fiber of (ξ, C S,A), then we obtain the generic and special fiber of the Gieseker vector bundle (C C, F) respectively.

19 Twisted vector bundles on pointed nodal curves 165 Indeed, in the neighbourhood of p one may chose a chart (U,η,Ɣ)for ξ, where U C étale locally looks like Spec A[u, v]/(uv t) Spec A[x,y]/(xy t e ), u e x, v e y. On the other hand, assume C = C 0 S, where C 0 is an irreducible curve with one ordinary double point p, and assume that C C induces an isomorphism of the generic fibers and the morphism C 1 C 0 on the special fibers. For simplicity let us assume furthermore that the rank r of the Gieseker bundle is one. In this situation it would be interesting to know, whether there is a twisted GL 1 -bundle (ξ, C S,A) such that the map of Theorem 4.2 maps the generic and the special fiber of (ξ, C S,A) to the generic and special fiber of (C C, F) respectively. Acknowledgements The author would like to thank Seshadri for generously imparting his ideas. This paper owes very much to long discussions which the author had with Nagaraj in November and December The author would also like to thank the Institute of Mathematical Sciences in Chennai, whose hospitality made these discussions possible. This work was partially supported by the DFG. References [1] Abramovich D and Vistoli A, Compactifying the space of stable maps, J. Am. Math. Soc. 15(1) (2002) [2] Abramovich D, Corti A and Vistoli A, Twisted bundles and admissible covers. Special issue in honor of Steven L Kleiman, Comm. Algebra 31(8) (2003) [3] Biswas Indranil, Parabolic bundles as orbifold bundles, Duke Math. J. 88(2) (1997) [4] Kausz I, A modular compactification of the general linear group, Documenta Math. 5 (2000) [5] Kausz I, A Gieseker type degeneration of moduli stacks of vector bundles on curves, math.ag/ (to appear in Trans. AMS) [6] Kausz I, Stable maps into the classifying space of the general linear group (Preliminary draft) [7] Mehta V and Seshadri C S, Moduli of vector bundles with parabolic structures, Math. Ann. 248 (1980) [8] Seshadri C S, Moduli spaces of torsion free sheaves and G-spaces on nodal curves, (Preliminary draft) November 2002