Mathematics 7800 Quantum Kitchen Sink Sping 2002 5. Vecto Bundles on a Smooth Cuve. We will constuct pojective moduli spaces fo semistable vecto bundles on a smooth pojective cuve C by applying GIT to a suitable Gothendieck Quot scheme. The constuction we pesent hee is due to Calos Simpson. Let E be a vecto bundle on a smooth pojective cuve C of genus g. Definition: (a) The slope µ(e) = deg(e)/k(e). (b) E is stable if µ(f ) < µ(e) fo all pope subbundles F E. (c) E is semistable if µ(f ) µ(e) fo all F E. Lemma 5.1: If 0 F E G 0 is an exact sequence of vecto bundles, then µ(f ) µ(e) (esp.>) if and only if µ(e) µ(g) (esp. >). Poof: Aithmetic! If a, b, c, d > 0, then a c > a+b c+d if and only if b d < a+b c+d. Examples: (i) Evey vecto bundle on P 1 splits as a sum of line bundles, so only the line bundles O P 1(d) ae stable, and only O P 1(d) n ae semistable. (ii) E is (semi-)stable iff the dual bundle E is (semi-)stable. (iii) E is (semi-)stable iff E L is (semi-)stable fo all line bundles L. (iv) If E is semistable of ank and degee d and: (a) d < 0, then H 0 (C, E) = 0. (b) d > (2g 2), then H 1 (C, E) = 0. (c) d > (2g 1), then E is geneated by its global sections. (Schu s) Lemma 5.2: (a) If E and F ae stable with the same slope, then any map f : E F is eithe 0 o an isomophism. (b) The only automophism of a stable bundle E is scala multiplication. (c) (Jodan decomposition) If E is semistable, thee is a filtation: 0 = E 0 E 1... E n = E such that F i := E i /E i 1 is a stable vecto bundle and each µ(f i ) = µ(e). The filtation is not canonical, in geneal, but the associated gaded bundle n i=1f i is independent of the choice of filtation. 1
Poof: If f : E F is not zeo, then both ke(f) and E/ke(f) ae bundles. If f isn t injective, then the stability of E implies µ(ke(f)) < µ(e) and by Lemma 5.1, µ(e/ke(f)) > µ(e) = µ(f ), contadictng the stability of F. So f is injective, and sujective by the stability of F. This gives (a). If α : E E is an automophism, let λ x be an eigenvalue of the estiction of α to the fibe of E ove x C. Then α λ x (id) dops ank at x, so it is not an isomophism, and must be zeo by (a) and we have (b). Finally, (c) follows fom (a) by the usual Jodan-Hölde decomposition. (Hade-Naasimhan) Lemma 5.3: If E is an any vecto bundle on C, then thee is a filtation: 0 = E 0 E 1... E n = E such that F i := E i+1 /E i ae semistable vecto bundles, with µ(f i ) > µ(f i+1 ). This filtation is uniquely detemined by the popety that if F E is any sub-bundle with µ(f ) µ(e i ), then F E i. Poof: Let S = {a a < µ(e) and a = µ(q) fo some quotient E Q}. We claim fist that S is a finite set. Indeed, let D be a diviso of lage enough degee so that E(D) is geneated by its sections. Then any Q(D) is also geneated by its sections, so deg(q(d)) 0 and µ(q) deg(d). So the elements of S ae bounded below (and above!) and since the denominatos ae bounded above by, it follows that S is finite. Finiteness of S implies that the set of slopes of sub-bundles F E is bounded fom above. Let E 1 E be the sub-bundle of maximal ank among those of maximal slope. Then E 1 is semi-stable and E/E 1 is a vecto bundle. If F E is anothe sub-bundle with µ(f ) = µ(e 1 ), then the span of F and E 1 is yet anothe sub-bundle of the same slope (since the kenel of the map fom F E 1 to the span must have the same slope). Since E 1 was of maximal ank, it follows that F E 1 which then has the desied popety. Now suppose inductively that the lemma holds fo F = E/E 1. We may use the Hade-Naasimhan filtation of F : 0 = F 0 F 1... F n 1 = F = E/E 1 to uniquely define E i+1 by the condition that E i+1 /E 1 = F i. And it follows that this filtation has the desied popety. 2
Thus evey vecto bundle on C is an extension of stable vecto bundles, which ae the indecomposable objects. They also have the smallest possible goup of automophisms, namely C, though thee ae vecto bundles with this automophism goup (called simple vecto bundles) which ae not stable. The following theoem is due to Naasimhan and Seshadi: Theoem 5.4: Fo each pai (, d) of copime positive integes, the functo: Obj : schemes S with an equivalence class of vecto bundles E on C S with the popety that each E s is stable, of ank and degee d (and E F E = F π Λ fo some line bundle Λ on S) Mo : mophisms φ : S S such that (φ, id) E E is epesented by a pojective scheme M C (, d) which is ieducible and smooth, of dimension 2 (g 1) + 1. Poof: We need two key lemmas, the fist solving a GIT poblem, and the second having to do with the boundedness of families of sheaves on C. (GIT) Lemma 5.5: If V and W ae vecto spaces and M is an intege, let G(V W, M) be the Gassmannian of M-dimensional quotients of V W. Then a point ψ G(V W, M) is semistable (esp. stable) with espect to the natual line bundle and lineaization of SL(V ) if and only if dim(h) dim(v ) dim(ψ(h W )) M (esp. <) fo evey pope subspace H V. Poof: Let N = dim(v ) and R = dim(w ) with a fixed basis w 1,..., w R. An point ψ G(V W, M) lifts to ψ = M ψ M (V W ) in the natual lineaization. Given any basis e 1,..., e N of V and dual basis x 1,..., x N, we ll call e i1 w j1... e im w jm the induced basis of Plücke vectos. Thus the coodinates of ψ ae the values: M ψ(e i1 w j1... e im w jm ) = ψ(e i1 w j1 )... ψ(e im w jm ) C which ae zeo if and only if the ψ(e ik w jl ) ae not linealy independent. 3
If λ = diag{t 1,..., t N } is a 1-PS of SL(V ) and x 1,..., x N is the associated (dual) basis, we ll say the weight of the Plücke vecto above is M j=1 ij. Then ψ is λ-unstable fo this λ if and only if M ψ vanishes on evey Plücke vecto of nonpositive weight. Suppose H V has dimension n, dim(ψ(h W )) = m and n > m. N M Let e 1,..., e n be a basis of H, extended to a basis e 1,..., e N of V and let λ = diag{t n N,..., t n N, t n,..., t n } fo the dual basis. Fo each Plücke vecto, if M ψ(e i1 w j1... e im w jm ) 0, then ψ(e i1 w 1 ),..., ψ(e im w M ) must be linealy independent, so the e ij must involve at most m of the e 1,..., e n vectos, thus its weight must be at least m(n N) + (M m)n. But Mn mn > 0 by assumption, so M ψ is λ-unstable fo this λ. Convesely, let λ be any 1-PS, diagonalized as λ = diag{t 1,..., t N } fo a basis x 1,..., x N. If ψ is λ- unstable, let H n be the span of e 1,..., e n, and let m n = dim(ψ(h n W )). Then λ-instability tells us: ( ) 1 m 1 + 2 (m 2 m 1 ) +... + N (M m N 1 ) > 0 because it is the minimal weight of a Plücke vecto on which M ψ is nonzeo. I claim that fo some n, the aveaged weights also satisfy: 1 n ( 1 +... + n )m n + 1 N n ( n+1 +... + N )(M m n ) > 0 It then follows that n > mn holds fo H = H N M n. To see the claim, notice fist that if m i+1 m i m i m i 1, then we may combine i and i+1, eplacing them with thei aveage i+ i+1 without deceasing ( ). The aveaged weights 2 ae the same as the oiginal, so we may assume the sequence of diffeences is inceasing: 1 := m 1 < 2 := m 2 m 1 <... < N := m N m N 1. Now conside the linea function: L(t 1, t 2,..., t N ) = 1 t 1 +... + N t N which by assumption satisfies L( 1,..., N ) > 0, and conside its values at the points: ni=1 ni=1 i Ni=n+1 i Ni=n+1 i p n := (,...,, n n N n,..., i N n ) RN These points ae linealy independent and L(p N ) = 0 (the i sum to zeo). Thus they span the hypeplane { t i = i } R N and in paticula, ( 1,..., N ) = N 1 i=1 y i p i y N p N fo positive values y i, so some L(p i ) > 0. 4
Thus, we ve shown that n > m if and only if ψ is λ-unstable fo some λ. N M By the numeical citeion, this poves the semi-stable pat of the lemma, and the stable pat is poved by eplacing each > by a. Lemma 5.6: Let p C, and O C (1) := O C (p). If n > 2g 1 d then: (a) If E is semistable of ank and degee d, then H 1 (C, E(n)) = 0, E(n) is geneated by global sections and fo all subbundles F E: h 0 (C, F (n)) ank(f ) h0 (C, E(n)) ank(e) with equality if and only if F is semistable, h 1 (C, F (n)) = 0 and fo all m: χ(c, F (m)) ank(f ) = χ(c, E(m)) ank(e) (b) If E is any coheent sheaf of the same Hilbet polynomial χ(c, E(n)) = P (m) = m + d (g 1) as a vecto bundle of ank and degee d, and if evey vecto bundle quotient E G satisfies: h 0 (C, G(n)) ank(g) P (n), then E is itself a semistable vecto bundle of ank and degee d. Poof: The key point is the following. If E is a semistable bundle of ank and if h 1 (C, E) 0, then h 0 (C, E) g independent of the degee of E. This is well-known fo line bundles, since evey L with h 1 (C, L) 0 is a subsheaf of the canonical line bundle and h 0 (C, ω C ) = g. But hee s a poof that genealizes. If h 1 (C, L) 0, then deg(l) 2g 2. If h 0 (C, L) g, then thee is a section s H 0 (C, L) vanishing at any p 1,..., p g 1 C. If the p i ae geneal, then h 0 (C, O C ( p i )) = 1 and fom: 0 O C ( p i ) L τ 0 and deg(τ) g 1 it follows that h 0 (C, L) 1 + h 0 (C, τ) g. If E is semistable of ank and h 1 (C, E) 0, then by Example (iv) we have deg(e) (2g 2) and deg(f ) (2g 2) fo any subbundle F E of ank. If h 0 (C, E) g, then thee is a section s H 0 (C, E) vanishing at any g 1 points, and then we get O C ( p i ) E spanning a line bundle L E of degee 2g 2 satisfying h 0 (C, L) g as above. 5
If h 0 (C, E) g, then h 0 (C, E/L) ( 1)g and we can find a section s H 0 (C, E/L) which again can be chosen to vanish at g 1 geneal points. The two sections s, s will span a sub-bundle F E of degee 2(2g 2) which then has at most 2 + 2(g 1) = 2g sections fom the exact sequence: 0 O C ( p i ) O C ( p i) F τ 0 and then one consides sections of E/F, etc. We aleady saw that the fist pat of (a) is satisfied in Example (iv). Notice that: χ(c, E(n)) = P (n) = n + d (g 1) Thus any semistable bundle F of ank and slope µ d must satisfy h 0 (C, F (n)) = χ(c, F (n)) P (n), o else h0 (C, F (n)) g < P (n) by the key point above (and the lowe bound n > 2g 1 d). If F E and E is semistable, then evey F i in the Hade-Naasimhan filtation of F has slope at most d, so each subquotient F i satisfies h 0 (C, F i (n)) k(f i ) P (n) and by Lemma 5.1, we have the same inequality fo F. If equality holds, then it must hold fo evey F i, and we conclude that evey F i has slope exactly d, so F is semistable, and fo all m. This poves (a). χ(c,f (m)) k(f ) = P (m) k(e) If E is the sheaf in (b), let T E be the tosion subsheaf, and let G be the (semistable) quotient of smallest ank in the Hade-Naasimhan filtation of E/T. Since µ(g) µ(e/t ) d, it follows as above that: h 0 (C, G(n)) k(g) P (n) with equality if and only if T = 0 and µ(g) = µ(e). But this means E = G! We ae eady fo the poof of Theoem 5.4 now. Let P (m) = m + d (g 1) be the Hilbet polynomial of a bundle of ank and degee d as in Lemma 5.6, and fo fixed n > 2g 1 d, conside the Quot scheme Quot(V O C ( n), P (m)) whee V is a vecto space of ank P (n) (with SL(V ) action). 6
If E is any semistable bundle of ank and degee d, then as we have aleady emaked, E(n) is geneated by global sections and H 1 (C, E(n)) = 0, so h 0 (C, E(n)) = P (n) and the global section map V = H 0 (C, E(n)) E(n) twists to give a point V O C ( n) E of the Quot scheme. Recall that the Quot scheme embeds in Gassmannians: ι m : Quot(V O C ( n), P (m)) G(V W, M) fo each M = P (m) and sufficiently lage m, and W = H 0 (C, O C (m n)). We will conside the GIT quotient of the Quot scheme fo the action of SL(V ) induced fom the Gassmannian (and lineaized as in Lemma 5.5). Fo lage enough m, the two notions of vecto bundle (semi-)stability and GIT (semi-)stability will coincide. When n, d ae copime, semi-stability equals stability, and the GIT quotient will epesent the functo. Defomation theoy will then show that the quotient is smooth, of the indicated dimension. If x Quot(V O C ( n), P (m)), let q x : V O C ( n) E x be the coesponding quotient. Such a quotient induces a map V H 0 (C, E x (n)) and fo each (lage enough) m, let ψ x : V W H 0 (C, E x (m)) be the image point in the Gassmannian. Let X U (m), X SS (m) and X S (m) be the loci of unstable, semistable and stable points fo this embedding. Step 1: Fo lage enough m (independent of x), if (i) E x is a semistable vecto bundle and (ii) V H 0 (C, E x (n)) is an isomophism, then x X SS (m). Poof: If x X U (m), then by Lemma 5.5, thee is an H V so that: ( ) dim(h) P (n) > dim(ψ x(h W )) P (m) and we need to show that the existence of such an H violates (i) o (ii). Fo each H V, let F x,h E x be the subsheaf geneated by H O C ( n). Assuming (ii), we see that H = H 0 (C, F x,h (n)). Conside: 0 K x,h H O C ( n) F x,h 0 and choose m 0 so that m m 0 implies that H 1 (C, K x,h (m)) = 0 and H 1 (C, F x,h (m)) = 0, fo all H V and all x in the Quot scheme. Then ψ x (H W ) = H 0 (C, F x,h (m)) is of dimension χ(c, F x,h (m)). 7
Thus if ( ) holds, then: dim(h 0 (C, F x,h (n))) P (n) > χ(c, F x,h(m)). P (m) On the othe hand, if we assume (i), then Lemma 5.6 (a) gives us: dim(h 0 (C, F x,h (n)) P (n) < ank(f x,h) (equality would foce equality in the pevious fomula). But χ(c, F x,h (m)) = m + d (g 1) fo = k(f x,h ) and d = deg(f x,h ), so we ae getting: > dim(h0 (C, F x,h (n)) P (n) > (m + d (g 1)) (m + d (g 1)) Thee ae only finitely many d and, so since the ight side appoaches the left as m, we obtain a contadiction when m is sufficiently lage. Step 2: Afte possibly inceasing m again, if x X SS (m) then: (a) The map V H 0 (C, E x (n)) is an isomophism and (b) The quotient E x is a semistable vecto bundle. Poof of Step 2: By Lemma 5.5, if x X SS (m) (fo any m), then V H 0 (C, E x ) must be injective, because any kenel would yield an H such that ψ x (H W ) = 0. Similaly, fo all H V, we must have: ( ) dim(h) dim(ψ x (H W )) P (n) P (m) Suppose E x wee not a bundle o not semistable. Then by Lemma 5.6(b), we could find a quotient bundle E x G so that h0 (C,G(n)) < P (n). Let H be k(g) the kenel of the map V H 0 (C, G(n)) fo such a quotient, and let F x,h be the image of H in E x. If F x,h is tosion, then thee is a univesal bound on its length, say K, and we can choose m so that P (n) < 1 violating ( ). P (m) K Othewise, by the aithmetic of Lemma 4.0, we have: ( ) dim(h) ank(f x,h ) > P (n) 8
whee the ank of F x,h is the geneic ank, which is the coefficient of m in χ(c, F x,h (m)). Since χ(c, F x,h (m)) = dim(ψ x (H W )) (see Step 1) we get a contadiction to ( ), pehaps afte boosting m again, fom the fact that thee is a unifom uppe bound on the constant tems of the Hilbet polynomials of the F x,h. So E x is semistable. Finally, since E x is semistable, the map V H 0 (C, E x (n)), which we aleady saw was injective, must be an isomophism by Lemma 5.6(a). Step 3: Fo sufficiently lage m (a) x X S (m) x X SS (m) and E x is stable. (b) Fo any x X SS (m), the closed obit O( x ) O( x) coesponds to an E x that is isomophic to the associated gaded of E x. Poof of Step 3: (a) is the same agument as Steps 1 and 2. Fo (b), if x X SS (m) X S (m), let F E x be a pope subbundle of the same slope, and let H V be the kenel of the map V H 0 (C, G(n)), whee G = E x /F. Conside the induced extension: ( ) : 0 F E x G 0 of vecto bundles of the same slope. If we take e 1,..., e n spanning H, extend to a basis of V, and conside the 1-PS subgoup λ = diag{t n N,..., t n N, t n,..., t n }, then λ acts on the extension class of in H 1 (C, G F ) by multiplication by t N, taking it to the split extension in the limit as t 0. We can epeat the pocess until we get to the associated gaded of E x. Since the associated gaded is uniquely detemined by Schu s Lemma, and thee must be some closed obit in the closue of the obit of E x, this must be the one! We have poved that fo lage m (and abitay (, d)!), the GIT quotient: G(V W, M) Quot(V O C ( n), P (m)) f > M C (, d) has the following popeties: (i) M C (, d) is a pojective scheme (ii) The points of M C (, d) coespond to associated gadeds of semistable vecto bundles of ank and degee d. The M C (, d) ae independent of the choice of (lage enough) m because they ae all categoical quotients of the same open subscheme X SS (m)! 9
Now take any vecto bundle E on S C and conside the sheaf π S (E(n)), whee E(n) = E π CO C (n). If E is a family of semi-stable bundles of ank and degee d (and n > 2g 1 d ), then π S (E(n)) is locally fee of ank P (n) and the natual map: π Sπ S (E(n)) π CO C ( n) E is a sujective map of vecto bundles. Locally (on S) we may tivialize π S (E(n)) each tivialization detemines U i Quot(V O C ( n), P (m)) which do not patch as maps to the Quot scheme, but do patch to: φ : S M C (, d) So to pove that M C (, d) epesents the functo, we need only to find a univesal vecto bundle U on C M C (, d) with the popety that any vecto bundle E as above satisfies E (φ, 1) U. We will use the following: (Descent) Lemma 5.7: Given a lineaized G-action on (X, L) and a vecto bundle F on X SS (L) with a G-action, then F descends to the GIT quotient: f : X SS (L) X G if and only if fo each closed obit O( x), the stabilize G x G also stabilizes the fibe F x of F at x X SS (L). Poof (Kempf): If F descends, then by definition, F = f (F ) is the pull-back of the descended bundle F, so G x acts tivially on the fibes F x. To pove the convese, it suffices to find, fo each x X SS (L), an affine neighbohood V of y := f(x ) X G and a tivialization of F 1 f (V by G- ) invaiant sections. Given x f 1 (y) with closed O( x), thee ae = k(f ) G-invaiant sections of the estiction F O(x) which tivialize F along the obit. Indeed, since we assumed that G x acts tivially on F x, we can tanslate a basis e 1,..., e F x by G to obtain the desied sections Ge 1,..., Ge. Let y V = D(h) G fo a homogeneous, invaiant h in the homogeneous coodinate ing of X. Then f 1 (V ) = D(h) X SS (L) is also affine and, as in pojective GIT, the map D(h) D(h) G is the affine GIT quotient. I claim that thee is a Reynolds opeato E : H 0 (D(h), F ) H 0 (D(h), F ) G. To see this, it suffices to show that G acts ationally on H 0 (D(h), F )). 10
But if we choose an open affine U D(h) on which F tivializes, then an s H 0 (U, F ) gives ise to a egula function φ : G U C defined by (g, x) gs(g 1 x). Then if G = Spec(A) and U = Spec(B), we have φ = a i b i, whee a i A and b i : U C, and as befoe, Gs U is contained in the span, W, of the b i. Since the estiction of sections fom D(h) to U is injective, we pove ationality by intesecting W with H 0 (D(h), F ). Now take the sections Ge 1,..., Ge spanning F O(x) and extend them to sections s 1,..., s of F D(h), which is possible since D(h) is affine. Apply the Reynolds opeato to get invaiant sections E(s 1 ),..., E(s ), which still estict to Ge 1,..., Ge on O(x) (by popety (i) of the Reynolds opeato). Finally, conside the closed invaiant subset Z D(h) whee E(s 1 ),..., E(s ) fail to span F. The image φ(z) D(h) G is closed and does not contain f(x ), so we can shink V = D(h) to a smalle open neighbohood x V fo which E(s 1 ),..., E(s ) do span, finishing the poof. Now suppose (, d) ae copime and conside the univesal quotient: V O C XSS ( n) E on C X SS C Quot(V O C ( n), P (m)). It is a consequence of flatness that E is a vecto bundle of elative ank and degee d ove X SS, and E is an SL(V )-bundle by vitue of the fact that the Quot scheme epesents the functo. That is, the action on E is obtained by pulling back the univesal quotient unde the action of SL(V ) on C X SS. Since (, d) ae copime, each of the bundles E x is stable (thee is no smalle with d = d ) and since Aut(E x ) = C, it follows that each stabilize SL(V ) x = SL(V) C = µp (n) is the goup of P (n)-th oots of unity. Again, since (, d) ae copime, it follows that P (n) = n + d (g 1) and ae copime, so we can solve(!) 1 + a = bp (n) and it follows that the action of the stabilizes SL(V ) x on E ( E) a is tivial, and this bundle, at least, descends. In the ank = 1 case, take b = 1 and a = P (n) 1, and this gives us a bundle L n on C Pic d (C) fo the Picad scheme Pic d (C) = M C (1, d), which has the popety that (L n ) x is the P (n)th tenso powe of the line bundle associated to x M C (1, d). But we may descend fo two consecutive values of n, which gives consecutive values of P (n) = n + d (g 1) and we get L := L n+1 L n which then has the desied univesal popety. 11
In the abitay ank case, we have the deteminant mophism: X SS M C (, d) det M C (1, d) coming fom the family of line bundles E on C X SS which factos though M C (, d) because it is a categoical quotient! Thus we may take the vecto bundle E n on C M C (, d) descended fom E ( E) a and tenso back by ((1, det) L) a to obtain U which has the desied univesal popety. Claim: Each M C (, d) is ieducible. Poof: Fist of all, notice that thee ae isomophisms: O C(1) M C (, d) O C(1) M C (, d + ) O C(1) which, in case (, d) ae copime ae obtained by taking the tenso poduct U π CO C (1) (and in the non-elatively pime case ae obtained by consideing E π CO C (1) on the Quot scheme and using the categoical quotient). Thus we may assume that d > (2g 1) so all bundles ae geneated by sections. When = 1, this gives us a sujective map (in fact a pojective bundle) u d : Sym d (C) M C (1, d); D O C (D) defined igoously by using the univesal Catie diviso D C Sym d (C) and using it to constuct the family O C Sym d (C)(D) of line bundles. Evidently, Sym d (C) is ieducible, as it is the quotient of C d by the pemutation goup. In ank, a choice of + 1 geneal sections of a bundle E of ank and degee d gives a sujection OC +1 E with kenel ( E). Dually, this means we can exhibit E as the kenel of a map: 0 E O +1 C L 0 whee L = ( E) (and then dualize to get E). It tuns out that a geneal choice of + 1 sections of a line bundle L of degee d has a semi-stable E as its kenel, and this gives a sujective mophism: G( + 1, π L) U M C (, d) fom an open subset U of the Gassmann bundle ove M C (1, d) to M C (, d). Since M C (1, d) is ieducible, it then follows that M C (, d) is ieducible too. 12
Fo smoothness and the dimension count, we will use defomation theoy. Given a stable bundle E of ank and degee d, the Zaiski tangent space is the space of (equivalence classes of) vecto bundles E ɛ on C Spec(k[ɛ]) with the popety that E ɛ C = E. We may tivialize E on an open cove Ui = C with intesections U ij = U i U j, and then E is detemined by tansition functions: G ij GL(O C (U ij )) satisfying the cocycle condition: G jk G ij = G ik on tiple intesections U ijk. An extension of E is given by an extension of the tansition functions: G ij + ɛh ij GL(O C (U ij )[ɛ]) (the invetibility puts no constaint on the matix H ij ) satisfying: (G jk + ɛh jk )(G ij + ɛh ij ) = G ik + ɛh ik o the oiginal cocycle condition togethe with: H jk G ij + G jk H ij = H ik on tiple intesections U ijk. But if we egad the H ij as sections of the (tivialized!) bundle End(E), then this is pecisely the cocycle condition to define an element of: H 1 (C, End(E)) (as the G ij tansition the H ij to allow us to compae them on U ik ). And coboundaies ae cocycles that give tivial defomations of E, so this is indeed the tangent space. Similaly, one checks that the obstuction space is: H 2 (C, End(E)) = 0 Thus on a cuve C, thee is no obstuction space, so M C (, d) is smooth, and: dim(m C (, d)) = dim(h 1 (C, End(E))) = χ(c, End(E)) + 1 = 2 (g 1) + 1 by Riemann-Roch (and Schu: h 0 (C, End(E)) = 1 since E is stable). 13
Finally, I want to use descent to descibe: The Deteminantal Line Bundle: Going back to the semi-stable points of the Quot scheme X SS Quot(V O C ( n), P (m)) let s assume that, in fact, fa fom being elatively pime, we actually have: d = (g 1) Poposition 4.5: Thee is a scheme stuctue on the subset Θ := {E H 0 (C, E) 0} M,(g 1) (C) making it an ample Catie diviso. Poof: Let n be chosen as in Lemma 4.4, let X SS Quot P (V O C ( D)/C) be the semistable locus, whee P (m) = m, and D = n i=1 p i is a diviso on C consisting of distinct points. If U is the univesal quotient on C X SS, then pushing down the exact sequence: yields the sequence: 0 π X SS 0 U U(D) n i=1u(d) pi 0 U π X SSU(D) f n i=1u(d) pi R 1 π X SS U 0 whee the middle two sheaves ae both locally fee of ank N = n. Moeove, since thee exist semistable bundles E of degee (g 1) with H 1 (C, E) = 0, (e.g. E = L whee H 1 (C, L) = 0), the fist sheaf vanishes! Finally, the map f is G-invaiant, so f descends, and N (f), a (nonzeo) section of the line bundle L := Hom( N π X SS U(D), n i=1 U(D) pi ) descends to a section s which vanishes pecisely on Θ. If m > M is fixed, then O X (1) := m π U(m) is the lineaization used in Theoem 4VB to define X SS. In paticula, some powe of O(1) descends to an ample line bundle on M,d (C). We claim that thee ae integes a and b such that L a and O(b) diffe by the pullback of a line bundle fom Pic d (C). This implies that Θ is ample. But N π X SS diffeence between c π X SS U(D) is tivial, natually isomophic to N V O, and the U(c) and (c+1) π X SSU(c + 1) is a tanslate of the bundle U p by the pullback of a line bundle fom Pic d (C) (p C is an abitay point). The esult is theefoe immediate, since up to tanslation, L and O(1) ae powes of the same line bundle. 14