Invariant vector bundles of rank 2 on hyperelliptic curves

Michigan Math. J. 47 (2000) Invariant Vector Bundles of Rank 2 on Hyperelliptic Curves C ha n c ha l K u m a r 1. Introduction In classical projective geometry, the Segre cubic 3-fold 6 has been extensively studied in Baker [1] and Coble [4]. It is the GIT quotient (P 1 )6//PGL(2, C) of (P 1 )6 by the diagonal action of PGL(2, C) for the natural linearization on the line bundle i=16 O P 1 (1). It has been shown in Baker [1] and Coble [4] that the Segre cubic 3-fold arises on considering the linear system of quadrics in P 3 that pass through five points in general position. The variety 6 thus embedded in P 4 as a cubic hypersurface is actually the blow-up of P 3 at these points, but with the proper transform of all lines joining any two points blown down to the ten nodes of 6. A general point ω ∈ 6 of the Segre cubic 3-fold can obviously be inter- preted as a curve C = Cω of genus g = 2 with level 2-structure. Indeed, Van der Geer [12] showed that the variety dual to 6, which is a quartic 3-fold, can be identified with the Satake compactification of the moduli space M2,2 of smooth projective curves of genus g = 2 with level 2-structure. A beautiful classical theorem (see [1; 4]) states that if ω ∈ 6 is a general point then the apparent contour—namely, the locus of points of contact of tangent to 6 from this point ω—is the Kummer surface Kum(C ) of the curve C = Cω asso- ciated to ω ∈ 6. In other words, the projection from the point ω maps 6 as a 2 : 1 covering of P 3 with Kummer surface Kum(C ) as its branch locus and the ap- parent contour as its ramification locus. The composition of the birational map P 3 99K 6 and the 2 : 1 rational map 6 99K P 3 yields a 2 : 1 rational map P 3 99K P 3, which is induced by the quadrics passing through six points in P 3 in general posi- tion. The ramification locus of this rational map is called the Weddle surface. The Weddle surface with six nodes is a birational model of the Kummer surface. A nice modern account of these results may be found in the book by Dolgachev and Ortland [8]. The aim of this paper is to generalize all this beautiful geometry to higher di- mensions. For g ≥ 2, we consider the GIT quotient (P 1 )2g+2//G of (P 1 )2g+2 by the diagonal action of G = PGL(2, C) for the natural G-linearization on the line bundle L = i=1 2g+2 O P 1 (1); we call it a generalized Segre variety or the Segre g-variety 6g . We show that the Segre g-variety 6g is obtained by the linear sys- tem  of g-forms on P 2g−1 that vanish with multiplicity g − 1 through 2g + 1 Received May 1, 2000. Revision received August 23, 2000. 575 576 C ha n c ha l K u m a r points e1, . . . , e 2g+1 in general position (cf. Theorem 4.1). In other words, the ra- tional map ι induced by  maps P 2g−1 birationally onto 6g . A general point ω ∈ 6g represents a hyperelliptic curve of genus g together with a special level-2 structure—namely, those given rise to by an ordering of the Weier- strass points (when g = 2, all level 2-structures arise in this way). If e 0 ∈ P 2g−1 such that ι (e 0 ) = ω, then we consider the partial linear system 3 of g-forms in  that vanish with multiplicity g − 1 at all the 2g + 2 points e1, . . . , e 2g+1, e 0 = e 2g+2 . The projection of 6g into |3|∗ yields a rational map of degree 2 onto its image S i , a connected component of the moduli space of semistable vector bun- dles of rank 2 with trivial determinant over C = Cω , which are invariant under the hyperelliptic involution. Also, this rational map is branched precisely along the Kummer variety Kum(C ) in S i (see Theorem 4.2). This is the precise general- ization of the classical relationship between the Segre cubic 3-fold and curves of genus g = 2 to higher dimension. Moreover, it establishes a connection between 6g and certain moduli spaces of invariant vector bundles of rank 2 on hyperelliptic curves. A part of this generalization was carried out by Coble in his two papers [5; 7] and a survey article [6]. His aim was to find a higher-dimensional analog of the Weddle surface and study its geometry relative to the geometry of Kummer vari- ety. Coble showed that the linear system 3 is the 2θ -linear system on the Jacobian of the hyperelliptic curve C = Cω and that it induces a rational map of degree 2 onto its image, which is branched precisely along the Kummer variety; the ram- ification locus of this rational map is what Coble calls the Weddle manifold. We have given a modern account of the work of Coble and hope that this will lead to a better understanding of his work. We now give a brief overview of this paper. First we discuss certain moduli spaces of semistable vector bundles of rank 2 on a hyperelliptic curve C of genus g ≥ 2. Let K = KC and h be the canonical and hyperelliptic line bundles on C, respectively. Let W = {w1, . . . , w 2g+2 } be an ordered set of all Weierstrass points of C. Set w 0 = w 2g+2 . Then all extensions of the form 0 → O(−w 0 ) → E → K(w 0 ) → 0 are parameterized by H 1(C, K −1 ⊗ h−1) and hence there is a rational extension map ε : P = PH 1(C, K −1 ⊗ h−1) 99K SUC (2, K), where SUC (2, K) is the moduli space of semistable vector bundles of rank 2 and determinant K on the curve C. Bertram [3] showed that the rational map ε, even for C nonhyperelliptic, g−1 is induced by the linear system H 0(P, I C ⊗ O P (g)), which is canonically iso- morphic to the 2θ-linear system on the Jacobian Pic g−1(C ), where I C is the ideal sheaf of C in P and Pic g−1(C ) is the space of all line bundles of degree g − 1 on C. Since the line bundle K −1 ⊗ h−1 is invariant under the hyperelliptic involution i : C → C, there is an involution on the cohomology group H 1(C, K −1 ⊗ h−1) ' H 0(C, h2g−1)∗ . Let P + be the linear subspace of P corresponding to the positive eigenspace for this involution. Then P + is of dimension 2g − 1, that is, P + ' P 2g−1. Restricting the rational map ε to P + yields a rational map ε+ : P + 99K S inv , where S inv is the i-invariant locus in SUC (2, K). We showed that ε+ is generically 2 : 1 onto its image S i , a connected component in S inv , and it is branched along  Invariant Vector Bundles of Rank 2 on Hyperelliptic Curves 577 the Kummer variety Kum(C ) = Pic g−1(C )/± in S i (see Corollary 2.1). Then in the next section we give another proof of a result of Coble that the linear sys- tem 3 is isomorphic to the 2θ -linear system H 0(Pic g−1(C ), O(2θ )). In the last section, we established a relationship between Segre g-variety and hyperelliptic curves of genus g that generalizes the relationship between the Segre cubic 3-fold and curves of genus g = 2. Acknowledgment. The author is grateful to his thesis advisor Professor S. Ra- manan for suggesting this problem and for giving valuable help in the preparation and revision of this paper. Without his help, this paper could not have been writ- ten. We were not aware of [5] or [7] and thank the referee for drawing our attention to these works. The referee also made many useful comments that helped improve the exposition of this paper. 2. Invariant Vector Bundles of Rank 2 Let E be an invariant vector bundle of rank 2 on a hyperelliptic curve C of genus g ≥ 2. Let j : E → E be a lift of i-action to E. Then (E, j ) is called a vector bundle pair. Two vector bundle pairs (E, j ) and (E 0 , j 0 ) are said to be equiva- lent if there is a vector bundle isomorphism f : E → E 0 such that j 0 B f = f B j. We say that the vector bundle pair (E, j ) is semistable (resp., stable) if, for every j -invariant line subbundle F of E, deg(E ) deg(F ) = µ(F ) ≤ µ(E ) = (resp., µ(F ) < µ(E )). 2 Let W = {w1, . . . , w 2g+2 } be the ordered set of all Weierstrass points of C. Consider a vector bundle pair (E, j ). Then, for every w ∈ W, jw : Ew → Ew is an involution on the fiber Ew . Let S i0 be the moduli space of semistable vector bun- dle pairs (E, j ) of rank 2 on the hyperelliptic curve C with det(E ) = K and trace Tr(jw ) = 0 for all w ∈ W. The existence of the moduli space S i0 follows from the work of Seshadri [11] on π-vector bundles. Let p : S i0 → S inv be the map given by p((E, j )) = E and let S i be the image of p. Then we show that S i0 is a ramified double cover of S i . Theorem 2.1. The map p : S i0 → S i given by p((E, j )) = E is generically 2 : 1 with the Kummer variety Kum(C ) in S i as its branch locus. Proof. If (E, j ) and (E, j 0 ) are two vector bundle pairs over E ∈ S i , then j 0 = Aj for some A ∈ Aut(E ). If E is stable, then Aut(E ) ' C ∗ . Thus j 0 = ±j and so, for every stable bundle E ∈ S i , there are two nonequivalent vector bundle pairs (E, j ), (E, −j ) over E. This shows that p is generically 2 : 1. Now the Kummer variety Kum(C ) of the curve C is embedded in S i by the map α 7 → α ⊕ i ∗ α, and it corresponds to strictly semistable (i.e., semistable but not stable) bundles in S i . If E = α ⊕ i ∗ α for some α ∈ Pic g−1(C ), then any two lifts of E in S i0 are equivalent. 578 C ha n c ha l K u m a r We claim that the rational extension map ε+ : P + 99K S i lifts to the rational map ε̄ : P + 99K S i0 . For v ∈ P + , the two extensions 0 → O(−w 0 ) → Ev → K(w 0 ) → 0 and 0 → O(−w 0 ) → i ∗(Ev ) → K(w 0 ) → 0 are isomorphic, so Ev comes with a lift jv of i-action. Thus (Ev , jv ) is a vector bundle pair. Also the trace Tr((jv )w ) = 0 for each w ∈ W. Since a generic extension is semistable, (Ev , jv ) ∈ S i0 for a generic v ∈ P + . Thus we define a rational map ε̄ : P + 99K S i0 by ε̄(v) = (Ev , jv ). Theorem 2.2. The rational map ε̄ : P + 99K S i0 is birational. Proof. It suffices to prove that, for a generic (E, j ) ∈ S i0 , there exists a unique v ∈ P + such that ε̄(v) = (E, j ). Let 20i be the generalized theta divisor on S i0 ; that is, Supp(20i ) = {(E, j ) ∈ S i0 : H 0(C, E ) 6= 0}. If (E,  j) ∈ / 20i then, from the short exact sequence 0 → E → E(w 0 ) → E(w 0 ) w 0 → 0, we have dim(H 0(C, E(w 0 ))) ≤ 2. Since the Euler characteristic χ(E(w 0 )) = 2, we have dim(H 0(C, E(w 0 ))) = 2. Then involution j on E induces an involution j¯ on H 0(C, E(w 0 )). Now, by the Atiyah–Bott fixed point theorem (see [2]), the trace Tr( j¯) = 0. Thus dim(H 0(C, E(w 0 ))+ ) = dim(H 0(C, E(w 0 ))− ) = 1 and / 20i , there exists a unique extension 0 → O(−w 0 ) → E → so, for each (E, j ) ∈ K(w 0 ) → 0, where the inclusion O(w 0 ) → E is induced by the unique invariant nonzero section of E(w 0 ). Clearly, E and i ∗E are the same as extensions. Hence there is a unique v ∈ P + such that ε̄(v) = (E, j ). Corollary 2.1. The rational map ε+ : P + 99K S i is generically 2 : 1 with the Kummer variety Kum(C ) in S i as its branch locus. Proof. Since ε + = p B ε̄, the proof follows from Theorems 2.1 and 2.2. 3. 2θ -Linear System In this section, we identify the 2θ -linear system on the Jacobian Pic g−1(C ) of a hyperelliptic curve C with the linear system 3C = 3 on P + ' P 2g−1. From the g−1 canonical isomorphism H 0(Pic g−1(C ), O(2θ )) ' H 0(P, I C ⊗ O(g)), we ob- tain a linear map g−1 res : H 0(P, I C ⊗ O(g)) → H 0(P + , O(g)) g−1 by restricting the sections of H 0(P, I C ⊗ O(g)) to P + . We recall that the lin- ear system 3C = 3 consists of all the g-forms on P + ' P 2g−1 that vanish with multiplicity g − 1 at the Weierstrass points w1, . . . , w 2g+2 in P + . We will prove that the mapping res induces an isomorphism between the 2θ -linear system g−1 H 0(P, I C ⊗ O(g)) and the linear system 3. But first we prove the following results. Lemma 3.1. Let Q ∈ H 0(P N , O(n)), and let A and B be any two distinct points on P N . Suppose the n-form Q vanishes with multiplicity l and m at A and B, re- spectively. Then Q vanishes along the line AB with multiplicity at least l + m − n. If l + m − n ≤ 0, then the conclusion is vacuous.  Invariant Vector Bundles of Rank 2 on Hyperelliptic Curves 579 Proof. Let r = l + m − n. We need only consider the case 0 < r ≤ l, m. Let ∂ |r−1|Q be a partial derivative of Q of order r − 1. Then deg(∂ |r−1|Q) = n − r + 1 and ∂ |r−1|Q vanishes with multiplicity l − r + 1 and m − r + 1 at A and B, respec- tively. Since (l − r +1) + (m − r +1) = n − r + 2 > n − r +1 = deg(∂ |r−1|Q), the line AB intersects ∂ |r−1|Q = 0 in a divisor greater than its degree deg(∂ |r−1|Q). Hence ∂ |r−1|Q vanishes identically on AB. Corollary 3.1. Let Q ∈ H 0(P N , O(n)). Let {ui : i ∈ 1} be a collection of finitely many points in P N in general  position such that Q vanishes with multi- plicity n − 1 at the ui . Then Q P(I ) = 0, where P(I ) = hui : i ∈ I i ⊂ P N is the linear subspace spanned by ui with i ∈ I ⊂ 1 and #(I ) ≤ n − 1. Proof. Let #(I ) = r ≤ n − 1. Then we claim that the n-form Q vanishes with multiplicity n − r on P(I ). Using Lemma 3.1, this claim can be proved by induc- tion on r.  Remark. With notation  as in Corollary 3.1, if Q P(J ) = 0 for every J ⊂ 1 with #(J ) = n, then Q P(1) = 0. By induction, one proves that Q P(H ) = 0 for  H ⊂ 1 with #(H ) ≥ n. For instance, if #(H ) = n + 1, then by assumption Q P(J ) = 0 for every J ⊂ H with #(J ) = n. Thus Q P(H ) is a product of n + 1 hyperplanes in P(H ). Since Q is a n-form, it is absurd unless Q P(H ) = 0. g−1 Lemma 3.2. The linear map res: H 0(P, I C ⊗ O(g)) → H 0(P + , O(g)) is in- jective, and its image is contained in 3.  Proof. Let Q ∈ H 0(P, I C ⊗ O(g)) be such that res(Q) = Q P + = 0. Let g−1 z1, . . . , zg be any general g points on the hyperelliptic curve C in P. Consider the g-secant P g−1 = hz1, . . . , zg i spanned by the z i . Since the g-form Q vanishes on the curve with multiplicity g − 1, by Corollary 3.1 it follows that the g-form Q P g−1 is (up to a constant factor) a product of g hyperplanes of the form P g−2 = hz1, . . . , ẑi , . . . , zg i in P g−1. But P g−1 ∩ P + 6 = ∅ and, for a general g-secant P g−1, we may assume that P + does not meet any of these hyperplanes P g−2 in P g−1. Since Q P + = 0 and P + meets  P g−1 in the complement of the hyperplanes just  described, we must have Q P g = 0. Thus, the g-form Q vanishes on a general g-secant to the hyperelliptic curve C in P. Since C is nondegenerate in P, by the remark to Corollary 3.1 we have that Q is identically zero. This proves that the mapping res is injective. Also C ∩ P + = W, the set of all Weierstrass points of C in P. Thus res(Q) ∈ 3. g−1 Remarks. (i) Since dim(H 0(P, I C ⊗ O(g))) = 2g and res is injective, we have dim(3) ≥ 2g . Thus, to show that res is an isomorphism onto 3, it is enough to prove that dim(3) ≤ 2g . (ii) Every Q ∈ 3 vanishes with multiplicity g − 2 on the rational normal curve S in P + besides vanishing with multiplicity g − 1 at the Weierstrass points (see [7, Thm. 1.4]). 580 C ha n c ha l K u m a r Lemma 3.3. Let {ui : i ∈ 1} be a finite collection of points in P 2g−1 in general position. Let Q be a n-form on P 2g−1 for n ≤ g. Suppose Q vanishes at ui with multiplicity n − 1 for i ∈ 1. Let P(I ) = hui : i ∈ I i for I ⊂ 1. Let P 2g−n be a linear subspace   that P of P 2g−1 such 2g−n ∩ P(I ) = ∅ for I ⊂ 1 with #(I ) =   n − 1. If Q P 2g−n = 0, then Q P(1) = 0. Proof. From Corollary 3.1, we may assume that #(1) ≥ n. Also, in view of the remark to Corollary 3.1, it is enough to prove that Q P(J ) = 0 for J ⊂ 1 with #(J ) = n. But again by Corollary3.1, Q vanishes on hyperplanes P(I ) in P(J ), I ⊂ J, with #(I ) = n−1. Thus Q P(J ) is a product of n hyperplanes. Since  P 2g−n intersects P(J ) in the   complement of the hyperplanes P(I ) and since Q P 2g−n = 0, we must have Q P(J ) = 0. Lemma 3.4. Let Q ∈ 3. Suppose {w i : i ∈ 1} is a subset of W and P g+r is a lin- ear subspace of P + ' P 2g−1 such that P  g+r ∩P(I ) = ∅ for I ⊂ {w i : i ∈ 1} with   #(I ) = g−r −1. If Q P g+r = 0 then Q S r(1) = 0, where S r(1) = Sec r(S )∗P(1) is the join of rth-order secant variety to the rational normal curve S in P + and the linear space P(1). For r = 0, S 0(1) = P(1). Proof. We proceed by an induction on r. For r = 0, it follows  from Lemma 3.3 that Q P(1) = 0. Thus, by induction we assume that Q S r−1(1) = 0. Now con- sider r general points z1, . . . , z r on S. Let P(z1, . . . , z r , 1) = z1 ∗ · · · ∗ z r ∗ P(1). Then, by induction assumption, Q P(z1,. . . ,zr ,1) is a product of r hyperplanes of the form P(z1, . . . , ẑ i , . . . , z r , 1) and a (g − r)-form Q 0 in P(z1, . . . , z r , 1). Since every Q ∈ 3 vanishes with multiplicity g − 2 along the rational normal curve S (see [7, Thm. 1.4]), the (g − r)-form Q 0 vanishes with multiplicity g − r − 1 at z1, . . . , z r and w i (i ∈ 1).  Because z1, . . . , z r are general points  of S, it follows from Lemma 3.3 that Q 0  P(z1, ...,z r ,1) = 0. This implies that Q S r(1) = 0. We now proceed to show that the dimension of the linear system 3C = 3 is 2g . Let I n = {1, . . . , n} for n ≤ 2g and let P(I n ) = hw i ∈ W : i ∈ I n i ⊂ P + . Then P(I n ) ' P n−1 and we have a complete flag P(I1 ) ⊂ P(I 2 ) ⊂ · · · ⊂ P(I 2g ) ' P + +  P ' P . We define ª a decreasing filtration on 3 as 2g−1 for the projective space  follows. Let Fk 3 = Q ∈ 3 : Q P(Ig+k−1) = 0 for 0 ≤ k ≤ g + 1. Since Q ∈ 3 vanishes with multiplicity g−1 at w ∈ W, we have Q P(Ig−1) = 0. Thus, F0 3 = 3; also, Fk 3 ⊃ Fk+13 and Fg+13 = 0. Hence we have a finite decreasing filtration 3 = F0 3 ⊃ F13 ⊃ · · · ⊃ Fg 3 ⊃ Fg+13 = 0 of the 3. The associated graded linear space for this L glinear systemL Pfiltration is given g g by k=0 Grk 3  = k=0 (Fk 3/F k+1 ª 3). Therefore, dim(3) = k=0 dim(Gr k 3).  Let 3k = Q P(Ig+k ) : Q ∈ Fk 3 . Then we have a short exact sequence 0 → Fk+13 → Fk 3 → 3k → 0, where Fk 3 → 3k is the natural restriction map. Thus dim(Gr k 3) = dim(3k ). ¡ g
Lemma 3.5. dim(Gr k 3) ≤ g−k .  Invariant Vector Bundles of Rank 2 on Hyperelliptic Curves 581 ¡ g
Proof. Since dim(Gr k 3) = dim(3k ), we show that dim(3k ) ≤ g−k . For I ⊂ Ig = {1, . . . , g} with #(I ) = g − k, we define linear subspaces P(I ; g − k) of P(Ig+k ) by P(I ; g − k) = span of {w i : i ∈ I } and {wg+1, . . . , wg+k }. Then each P(I ; g −k) is ¡ isomorphic to a P g−1, and  the number of such ª P(I ; g −k)-subspaces g
 is precisely g−k . Let 3P(I ;g−k)L = Q P(I ;g−k) : Q ∈ 3 and consider the natu- ral restriction map r : 3k → #(I )=g−k (3P(I ;g−k) ), where the direct sum is taken over all I ⊂ Ig with #(I ) = g − k. We  claim that r is injective.  Let Q ∈ 3k be such that r(Q) = 0. Then Q ∈ 3, Q P(Ig+k−1) = 0, and Q P(I ;g−k) = 0 for every I ⊂ Ig with #(I ) = g − k. We need to show that Q P(Ig+k ) = 0. For k ≤ 1 this is trivial, so assume that k ≥  2. Since P(Ig+k−1) ' P g+k−2 and Q P g+k−2 = 0, we deduce from Lemma 3.3 that Q k−1 0 = 0, where W 0 = {w i ∈ W : i ∈ S (W ) / Ig+k−1}. Now consider P g = span of {wj ; j ∈ J } and  {wg+1, . . . , wg+k }, where J ⊂ Ig with #(J ) = g −  k + 1. By as- sumption, Q  = 0 for I ⊂ J with #(I ) = g − k, and Q  =0  P(I ;g−k)  P(J ;g−k+1)   because Q P(Ig+k−1) = 0. This shows that Q P g is a product of g − k + 2 hyper- planes and a (k − 2)-form Q 0 on P g . Also, Q 0 vanishes with multiplicity k − 2 at the points wg+k , wj (j ∈ J ) whereas it vanishes with multiplicity k − 3 at the remaining points wg+1, . . . , wg+k−1. This implies that Q 0 must be a cone over a (k − 2)-form Q 00 on P k−2 = hwg+1, . . . , wg+k−1i. Now, for a general k − 2 points z1, . . . , z k−2 ∈ S we have P g ∩ P(z1, . . . , z k+2 , W 00 ) 6 = ∅, where W 00 = W 0 − {wg+k } and P(z1, . . . , z k−2 , W 00 ) ' P g−1. Since Q S k−1(W 00 ) = 0, it follows  that QP g = 0 contains a (k − 2)-dimensional subvariety of P g . The same is true for Q 0  P g = 0 and hence also for Q 00 = 0 in P k−2 , since Q 0 is a cone over Q 00 . Thus we must have Q 00 ≡ 0, and so Q P g ≡ 0. On similar lines, we can deduce that Q P g+i ≡ 0, where P g+i = span of {wj : j ∈ J } and {wg+1, . . . ,wg+k }, and that J ⊂ Ig with #(J ) = g − k +1+ i. Thus, for i = k − 1, we have Q P(Ig+k ) = 0 and hence r is injective. Now, in view of Corol- ¡L
¡lary 3.1, dim(3P(I ;g−k) ) ≤ 1 and so dim(3k ) ≤ dim
#I =g−k (3P(I ;g−k) ) ≤ g g−k . Theorem 3.1 (Coble). The linear system 3C = 3 on P + is isomorphic to the 2θ-linear system on the Jacobian Pic g−1(C ) of the hyperelliptic curve C. g−1 Proof. Since dim(H 0(P, I C ⊗ O(g))) = 2g and the linear map res : H 0(P, g−1 g I C ⊗O(g)) → 3 is Pinjective, it follows that ¡ g
≥ 2 g. But from Lemma 3.5g P gdim(3) g we have dim(3) = k=0 dim(Gr k 3) ≤ k=0 g−k = 2 . Thus dim(3) = 2 g−1 and res induces an isomorphism of the 2θ -linear system H 0(P, I C ⊗ O(g)) ' g−1 H (Pic (C ), O(2θ)) with 3. 0 ¡ g
Remark. Since dim(3) = 2g , we have dim(Gr k 3) = g−k . Theorem 3.2. The rational map ι3 : P + 99K |3|∗ induced by the linear system 3C = 3 is generically 2 : 1 onto S i , and its branch locus is the Kummer variety Kum(C ) in S i . Proof. From Theorem 3.1, the pull-back of the linear system H 0(S i , 2 i ), which is isomorphic to the 2θ-linear system H 0(Pic g−1(C ), O(2θ )) under the rational 582 C ha n c ha l K u m a r map ε+ : P + 99K S i , is isomorphic to the linear system 3, where 2 i is the gen- eralized theta divisor on S i . Since S i is embedded in the linear system |2 i |∗, the rational map ε + : P + 99K S i is induced by the linear system 3. Now the theorem follows from Corollary 2.1. 4. Higher-Dimensional Segre Varieties In this section we discuss a higher-dimensional analog of the Segre cubic 3- fold. As in Section 1, we consider the GIT quotient (P 1 )2g+2//G of (P 1 )2g+2 by the diagonal action of G = PGL(2, C) for the natural G-linearization on L = i=1 2g+2 O P 1 (1) and call it the Segre g-variety 6g . Using the theory of associated point sets [8], we have a duality isomorphism (P 1 )2g+2//G ' (P 2g−1)2g+2//G 0 , where G 0 = PGL(2g, C) acts diagonally on (P 2g−1)2g+2 for the natural G 0 - linearization on the line bundle M = i=1 2g+2 O P 2g−1 (g). Moreover, we have 0 H 0(L)G ' H 0(M)G . Now let e1, . . . , e 2g+1 be any 2g + 1 points in general position in P 2g−1. Without loss of generality, we may assume that ej = [0 : · · · : 1 : · · · : 0] for j = 1, . . . , 2g and e 2g+1 = [1 : · · · : 1]. Then we define an inclusion f : P 2g−1 → (P 2g−1)2g+2 by e 7→ (e1, . . . , e 2g+1, e). On composing f with the GIT quotient map and using the preceding duality isomorphism, we derive a ratio- nal map f¯ : P 2g−1 99K 6g . Any two general points t = (t1, . . . , t 2g+2 ) ∈ (P 1 )2g+2 and z = (z1, . . . , z 2g+2 ) ∈ (P 2g−1)2g+2 are associated to each other under the above duality isomorphism if and only if there is a rational normal curve γ : P 1 → P 2g−1 such that γ (tj ) = zj for 1 ≤ j ≤ 2g + 2 (see [8]). Any 2g +1 points in general po- sitions in P 2g−1 can be mapped to e1, . . . , e 2g+1 by an automorphism T of P 2g−1, so if T (γ (t 2g+2 )) = e then f¯(e) is the image of t = (t1, . . . , t 2g+2 ) under the GIT quotient map. For a general point e ∈ P 2g−1, there is a unique rational normal curve through e1, . . . , e 2g+1, e. This shows that the rational map f¯ : P 2g−1 99K 6g is birational. Let  be the linear system of g-forms on P 2g−1 that vanish with multiplicity g − 1 at 2g + 1 points e1, . . . , e 2g+1 in P 2g−1. We then show that the rational map f¯ is induced by the linear system . Theorem 4.1. The linear system  on P 2g−1 is isomorphic to H 0(L)G , and the rational map ι : P 2g−1 99K | ∗ | induced by the linear system  is birational onto 6g . Proof. We consider the birational map f¯ : P 2g−1 99K 6g induced by the above du- ality isomorphism. By the Hilbert–Mumford numerical criterion for semistability (see [10]), we check that the indeterminacy locus of f¯ consists of all the (g − 1)- planes hej1 , . . . , ejg i spanned by ej (j = 1, . . . , 2g + 1). The Segre g-variety 6g embeds in P(H 0(L)G )∗ and so, for a section s ∈ H 0(6g , O6g (1)) ' H 0(L)G , the pull-back section f¯∗(s) ∈ H 0(P 2g−1, O P 2g−1 (g)) is a g-form that vanishes on the indeterminacy locus of f¯. In other words, the g-form f¯∗(s) vanishes on all the  Invariant Vector Bundles of Rank 2 on Hyperelliptic Curves 583 (g − 1)-planes spanned by the ej . But these conditions are equivalent to the con- dition that f¯∗(s) vanish with multiplicity g − 1 at the 2g + 1 points e1, . . . , e 2g+1. Thus the pull-back f¯∗ yields a linear map ρ : H 0(L)G → . Since f¯ is bira- tional, ρ is nontrivial. Hence, to complete this proof we need only show that ρ is an isomorphism. We now compute the dimension of . Let Nk = {1, . . . , k}, R = {I ⊂ N2g : #(I ) = g}, and C = {J ⊂ N2g : #(J ) = g − 2}, and let xI denote the monomial x i1 . . . x ig if I = {i1, . . . , ig }. The g-form Q vanishes with P multiplicity g − 1 at the points e1, . . . , e 2g if and only if it is expressed as Q = I ∈R aI xI with aI ∈ C. If Q also vanishesP with multiplicity g − 1 at e 2g+1 then we have the condition that, for each J ∈ C, J⊂I ∈R aI = 0. Therefore X X  = aI xI : aI = 0 ∀J ∈ C . I ∈R J⊂I ∈R The incidence matrix (λIJ )I ∈R,J∈C , given by λIJ = 1 if J ⊂ I and λIJ = 0 if J 6⊂ I, is of maximal rank, so all conditions among the generators {xI :¡I ∈
R}¡of the 2g
linear system  are independent. Thus dim() = #(R) − #(C ) = 2g g − g−2 . Now let Wk be the symmetric group on k symbols. We recall that H 0(L)G is an irreducible W 2g+2 -module corresponding to the Young tableau consisting of 2-rows and (g + 1)-columns; by the Hook length formula, dim(H 0(L)G ) = (2g+2)! (g+2)! (g+1)! (see [8]). Forgetting the last symbol, H 0(L)G is also an irreducible W 2g+1-module. For every σ ∈ W 2g+1, there is a unique automorphism Tσ of P 2g−1 such that Tσ (ej ) = eσ(j ) for j = 1, . . . , 2g +1. Now the maps Q 7 → Tσ∗(Q) for σ ∈ W 2g+1 define an action of W 2g+1 on , and it can be checked that ρ is equivariant for these W 2g+1-actions. Since H 0(L)G is an irreducible W 2g+1-module, ρ must be injective. Also, since dim(H 0(L)G ) = dim(), ρ must be an isomorphism. A general point on the Segre g-variety ω ∈ 6g represents a hyperelliptic curve C = Cω with a special level 2-structure as mentioned in Section 1. If e 0 ∈ P 2g−1 such that ι (e 0 ) = ω, then we consider the linear system 3 of g-forms on P 2g−1 that pass with multiplicity g − 1 through 2g + 2 points e1, . . . , e 2g+1, e 0 = e 2g+2 . Then 3 is a partial linear system of . We can identify P 2g−1 with P + by a unique projective transformation taking ei to w i for i = 1, . . . , 2g + 2. In view of Theorem 3.2, we now have our main theorem. Theorem 4.2. The Segre g-variety 6g embeds in the projective space ||∗ . Projecting 6g into the linear system |3|∗ yields a rational map of degree 2 onto its image S i , and it is branched precisely along the Kummer variety Kum(C ) in S i . Proof. By Theorem 4.1, the Segre g-variety 6g embeds into ||∗ . Also, the linear system 3 corresponds to the linear system 3C under the foregoing identification of P 2g−1 with P + . The result then follows from Theorem 3.2. As an application of Theorem 4.2, we give an alternative proof of a result of Narasimhan and Ramanan [9]. 584 C ha n c ha l K u m a r Theorem 4.3 (Narasimhan–Ramanan). The moduli space SUC (2, K) is isomor- phic to P 3 for a smooth projective curve C of genus g = 2. Proof. For g = 2, i ∗E = E for all E ∈ SUC (2, K); thus S i = SUC (2, K). The Segre cubic 3-fold 6 is a cubic in ||∗ ' P 4 , and projecting away from a general point ω ∈ 6 yields a rational map of degree 2 from 6 onto |3|∗ ' P 3 . Thus, from Theorem 4.2, we derive that S i ' P 3 . References [1] H. F. Baker, Principles of geometry, vol. IV, Unger, New York, 1925 (reprinted 1963). [2] N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, Grund- lehren Math. Wiss., 298, Springer-Verlag, Berlin, 1992. [3] A. Bertram, Moduli of rank-2 vector bundles, theta divisors and the geometry of curves in projective spaces, J. Differential Geom. 35 (1992), 429–469. [4] A. B. Coble, Algebraic geometry and theta functions, Amer. Math. Soc. Colloq. Publ., 10, Providence, RI, 1929 (3rd ed. published 1969). [5] , A generalization of the Weddle surface, of its Cremona group, and of its parametric expression in terms of hyperelliptic theta functions, Amer. J. Math. 52 (1930), 439–500. [6] , The geometry of the Weddle manifold Wp , Bull. Amer. Math. Soc. 41 (1935), 209–222. [7] A. B. Coble and J. H. Chanler, The geometry of the Weddle manifold Wp , Amer. J. Math. 57 (1935), 183–218. [8] I. Dolgachev and D. Ortland, Point sets in projective spaces and theta functions, Astérisque 165 (1988). [9] M. S. Narasimhan and S. Ramanan, Moduli of vector bundles on a compact Riemann surface, Ann. of Math. (2) 89 (1969), 19–51. [10] P. E. Newstead, Introduction to moduli problems and orbit spaces, TIFR Lecture Notes Math. Phys., 51, Narosa Publishing, New Delhi, 1978. [11] C. S. Seshadri, Moduli of π -vector bundles over an algebraic curve, Questions on algebraic varieties (CIME, III Ciclo, 1969), pp. 141–260, Edizioni Crenonese, Rome, 1970. [12] G. Van der Geer, On the geometry of a Siegal modular threefold, Math. Ann. 260 (1982), 317–350. Department of Mathematics University of Jammu, New Campus Jammu 180 006 India Chennai Mathematical Institute 92 G.N. Chetty Road, T. Nagar Chennai 600 017 India [email protected]