ON SINGULARITIES OF MIP 3 (c1 , c2 ) Vincenzo Ancona and Giorgio Ottaviani * arXiv:alg-geom/9502008v1 13 Feb 1995 alg-geom/9502008 Summary. Let MIP 3 (c1 , c2 ) be the moduli space of stable rank-2 vector bundles on IP with Chern classes c1 , c2 . We prove the following results. 1) Let 0 ≤ β < γ be two integers, (γ ≥ 2), such that 2γ − 3β > 0; then MIP 3 (0, 2γ 2 − 3β 2 ) is singular (the case β = 0 was previously proved by M. Maggesi). 2) Let 0 ≤ β < γ be two odd integers (γ ≥ 5), such that 2γ − 3β + 1 > 0; then MIP 3 (−1, 2(γ/2)2 − 3(β/2)2 + 1/4) is singular. In particular MIP 3 (0, 5), MIP 3 (−1, 6) are singular. 3 The first examples of singular moduli spaces of stable vector bundles on a projective space were found by the authors in [AO1], where it is shown that the symplectic special instanton bundles on IP 5 with second Chern class c2 = 3, 4 correspond to singular points of their moduli space. Later R. M. Miró-Roig [MR] detected an example in the case of rank-3 vector bundle on IP 3 . Recently M. Maggesi [M] has pointed out that pulling back some particular instanton bundles with second Chern class c2 = 2 by a finite morphism IP 3 → IP 3 one always obtains singular points in the corresponding moduli spaces, thus giving examples in the rank-2 case. His result is that for any integer d ≥ 2 the moduli space MIP 3 (0, 2d2 ) is singular; d = 2 gives MIP 3 (0, 8). The aim of this paper is to exhibit some more examples. The idea is to replace the usual pull-back by the more general construction of ”pulling back over C I 4 \0 ”, introduced in [Ho] and developed in [AO2]. The result is the following: let 0 ≤ β < γ be two integers, (γ ≥ 2), such that 2γ −3β > 0; then the moduli space MIP 3 (0, 2γ 2 −3β 2 ) is singular (Main theorem I). In particular taking β = 1, γ = 2 we obtain that MIP 3 (0, 5) is singular; in this case the singular points we have detected fall in the closure of the open set consisting of instanton bundles [Ra]. We prove a similar result in the case c1 = −1 (Main Theorem II). In particular we find that MIP 3 (−1, 6) is singular. §1 * Both authors were supported by MURST and by GNSAGA of CNR 1 Let U be a 2-dimensional complex vector space, V = U ⊕ U , and IP 3 = IP (U ⊕ U ) the projective space of hyperplanes in V . We denote by MIP 3 (c1 , c2 ) the moduli space of stable rank-2 vector bundles on IP 3 with Chern classes c1 , c2 . Let us choose homogeneous coordinates (a, b, c, d) in IP 3 so that (a, c) and (b, d) are coordinates in U ⊕ 0 and 0 ⊕ U respectively. It is well known ([H],[LP],[NT]) that any stable rank-2 vector bundle E on IP 3 belonging to MIP 3 (0, 2) can be described by a monad A B I 2 ⊗ OIP 3 (1) → 0 0→ C I 2 ⊗ OIP 3 (−1) → U 3 ⊗ OIP 3 → C where A=     −d B=   b  a 0 −d −c b a b 0 c a b 0 d 0 c d −d −c −c  b a a (1.1)    α3    α2   α1  α0  α4 α3   α2 α1 and α0 . . . α4 are constant coefficients subject to the condition  α0 det α1 α2 α1 α2 α3  α2 α3  6= 0 α4 There is a natural action of SL(2) ∼ = SL(U ) on the matrices A and B through the transformations         a a b b →g , →g (g ∈ SL(2)) c c d d ∼ It follows that SL(2)  = SL(U  ) acts on the monad (1.1), hence on its cohomology E. x y let More precisely, for g = z w  x    Qg =  y  z x z x w y w y 2    z    w It is easy to check that g ∗ A = AQg , g ∗ B = Q−1 g B, which implies that the monads (A, B) and (g ∗ A, g ∗ B) are equivalent. Moreover, the monad (1.1) is SL(U )-invariant (SL(U ) acts trivially on C I 2 and diagonally on U 3 ). Lemma 1. The bundle E defined by the monad (1.1) admits the following SL(U )invariant minimal resolution: 0 → U ⊗ OIP 3 (−4) → (S 2 U )2 ⊗ OIP 3 (−3) → [S 3 U ⊗ OIP 3 (−2)] ⊕ [ C I 2 ⊗ OIP 3 (−1)] → E → 0 (1.2) Proof: The minimal resolution of E is well known and is clearly SL(U )-invariant; (1.2) follows by inspecting in it the cohomology groups of E as SL(U )-representations. Lemma 2. H 0 E(1) = C I 2 , H 0 E(2) = S 3 (U ) ⊕ U 2 as SL(U )-representations; moreover, for t ≥ 3 H 0 E(t) is a direct sum of symmetric powers S m (U ) with 0 ≤ m ≤ t + 1. Lemma 3. H 1 End E(−1) = U 4 , H 1 End E = C I 4 ⊕(S 2 U )3 , H 1 End E(1) = U 2 ⊕(S 3 U )2 , H 1 End E(2) = S 4 U as SL(U )-representations. Moreover, H 1 End E(t) = 0 for t ≤ −3 and t ≥ 3, and H 1 End E(−2) is equal to C I if I = α0 α4 −4α1 α3 +3α22 = 0 , to 0 otherwise. Proof. The statement about H 1 End E(−2) can be found in [LP],[NT]. The remaining equalities can be easily obtained by tensoring the sequence (1.2) by E(t) and inspecting the corresponding cohomology sequences. §2 Let 0 ≤ β < γ be two integers, (γ ≥ 2); let f1 , . . . , f4 be homogeneous polynomials in the variables a, b, c, d without common zeroes of degree γ −β, γ −β, γ +β, γ +β respectively. Let us take into account the diagram ω C I 4 \0 → η↓ IP 3 C I 4 \0 η↓ IP 3 where ω is defined by f1 , . . . , f4 . According to [Ho], [AO2], from any bundle E defined by the monad (1.1) we construct a rank-2 bundle Eβ,γ such that η ∗ Eβ,γ = ω ∗ η ∗ E . The bundle Eβ,γ is the cohomology of the monad Aβ,γ Bβ,γ 0→ C I 2 ⊗ OIP 3 (−γ) → U 3 → C I 2 ⊗ OIP 3 (γ) → 0 (2.1) where U = OIP 3 (−β) ⊕ OIP 3 (β), and Aβ,γ , Bβ,γ are obtained from the matrices A, B in (1.1) replacing a, b, c, d by f1 , f2 , f3 , f4 respectively. In particular the Chern classes of Eβ,γ are c1 = 0, c2 = 2γ 2 − 3β 2 . 3 Of course Eβ,γ depends on f1 , . . . , f4 but for simplicity we omit this fact in the notations. The cohomology groups of Eβ,γ (t) can be computed by [AO2 §2 ]; in particular, the theorem 2 of [AO2] can be rephrased as follows. Theorem 4. Let H i E(t) = T t (U ) where T t is a representation of SL(U ). Then hi Eβ,γ (t) = 4 XX (−1)j h0 [ h∈Z j=0 j ^
U 2 (−γ) ⊗ T h (U) ⊗ OIP 3 (t − hγ)] In practice the groups hi Eβ,γ (t) can be computed as follows. Let sp the dimension of the the degree p summand of the artinian algebra S = C[a, I b, c, d]/(f1, f2 , f3 , f4 ); for h ∈ Z we write M (2.2) T h (U)(hγ) = OIP 3 (µs,h ) s Let bq (E) = ♯ {(s, h) : µs,h = q} Then X hi Eβ,γ (t) = sp bq (E) (2.3) p+q=t Moreover, sp = 4 X j=0 j 0 (−1) h [ j ^ U 2 ⊗ OIP 3 (p − jγ)] (2.4) The formulae (2.3), (2.4) are easy adaptations of the results of [AO2 §2]. The above formulae still hold if we replace Eβ,γ (resp E) by End Eβ,γ (resp End E). Proposition 5. Let E be defined by the monad (1.1). Then (i) Eβ,γ is stable if and only if 2γ − 3β > 0. (ii) h1 End Eβ,γ = kβ,γ + s2γ h1 End E(−2) where kβ,γ is a constant not depending on E. Proof. (i) We need to show that h0 Eβ,γ = 0 iff 2γ − 3β > 0 . We compute h0 Eβ,γ substituting t = 0 in the formula (2.3) . Since h0 E(h) = 0 for h ≤ 0 and sp = 0 for p < 0, a contribution to the right-hand side of (2.3) can occur only for q ≤ 0. By lemma 2 the integers µs,h appearing in (2.2) are strictly positive for h = 1, while for h = 2 we have T 2 (U)(2γ) = S 3 (U)(2γ) ⊕ (U 2 )(2γ) = ⊕s OIP 3 (µs,2 ) with inf s {µs,2 } = 2γ − 3β; for h ≥ 3 we have 2γ − 3β < hγ − (h + 1)β ≤ inf s {µs,h }; the conclusion follows. (ii) Again this follows from the formula (2.3) with End E in place of E and from the lemma 3. Main Theorem I. Let 0 ≤ β < γ be two integers (γ ≥ 2), such that 2γ − 3β > 0. Let E be a bundle belonging to MIP 3 (0, 2) such that h1 End E(−2) 6= 0. The moduli space MIP 3 (0, 2γ 2 − 3β 2 ) is singular at the points corresponding to the bundles Eβ,γ . 4 Proof. By proposition 5,(i) we can define a natural map MIP 3 (0, 2) → [F ] → MIP 3 (0, 2γ 2 − 3β 2 ) [Fβ,γ ] which is clearly algebraic. Let E be as in the statement. In any neighborhood of its class [E] ∈ MIP 3 (0, 2) there is a [F ] with h1 End F (−2) = 0. From proposition 5 (ii) we obtain h1 End Eβ,γ > h1 End Fβ,γ which clearly implies that [Eβ,γ ] is a singular point of MIP 3 (0, 2γ 2 − 3β 2 ). Taking γ = 2, β = 1 we obtain: Corollary. MIP 3 (0, 5) is singular. In this particular case the singular points we have detected fall in the closure of the open set consisting of instanton bundles (see [Ra]). It would be interesting to know whether the same property is true in the general case. Let us remark that taking β = 0 we recover the result of [M]. By improving our technique we can show more generally that MIP 3 (0, kγ 2 −(k +1)β 2 ) is singular for 0 < β < γ and any integer k > 0 such that kγ − (k + 1)β > 0 [AO3]. §3 In this section we deal with singularities of the moduli spaces MIP 3 (−1, c2 ). Let 0 < β < γ be two odd integers; let f1 , . . . , f4 be homogeneous polynomials in the variables a, b, c, d without common zeroes of degree (γ − β)/2, (γ − β)/2, (γ + β)/2, (γ + β)/2 respectively. We can construct the monad 0→ C I 2 ⊗ OIP 3 (−(γ + 1)/2) Aβ/2,γ/2 → W3 Bβ/2,γ/2 → C I 2 ⊗ OIP 3 ((γ − 1)/2) → 0 (3.1) where W = OIP 3 (−(β + 1)/2) ⊕ OIP 3 ((β − 1)/2), and Aβ/2,γ/2 , Bβ/2,γ/2 are obtained from the matrices A, B in (1.1) replacing a, b, c, d by f1 , f2 , f3 , f4 respectively. The cohomology of the monad (3.1) is a rank-2 bundle Eβ/2,γ/2 whose Chern classes are c1 = −1, c2 = 2(γ/2)2 − 3(β/2)2 + 1/4. Let Eβ,γ be defined as in §2 by the monad (2.1), via the homogeneous polynomials 2 2 2 2 f1 (a , b , c , d ), . . . , f4 (a2 , b2 , c2 , d2 ). Then for t ∈ Z π ∗ Eβ/2,γ/2 (t) = Eβ,γ (2t − 1) (3.2) where π : IP 3 → IP 3 is a finite morphism of degree 8. Moreover π ∗ End Eβ/2,γ/2 (t) = End Eβ,γ (2t) 5 (3.3) In order to compute the cohomology groups of of End Eβ/2,γ/2 (t) let H 1 End E(h) = Rh (U ), where Rh is a representation of SL(U ); let V = OIP 3 (−β/2) ⊕ OIP 3 (β/2). Though L V is only a Q-bundle, by lemma 3 one easily checks that Rh (V)(hγ/2) = s OIP 3 (νs,h ) with νs,h ∈ Z. As in §2 we define sp as the dimension of the degree p summand of the artinian algebra S = C[a, I b, c, d]/(f1, f2 , f3 , f4 ); let bq (End E) = ♯ {(s, h) : νs,h = q} Then Lemma 6. P (i) h1 End Eβ/2,γ/2 (t) = p+q=t sp bq (End E) (ii) h1 End Eβ/2,γ/2 = kβ/2,γ/2 +sγ h1 End E(−2) where kβ/2,γ/2 is a constant not depending on E. Proof. (The details are left to the reader). By (3.3) the groups h1 End Eβ/2,γ/2 (t), (t ∈ Z), are uniquely determined by the groups h1 End Eβ,γ (h), (h ∈ Z); by comparison with the formula (2.3) (applied to End Eβ,γ ) one checks (i); (ii) is an immediate consequence of (i). Main Theorem II. Let 0 < β < γ be two odd integers (γ ≥ 5), such that 2γ −3β +1 > 0. Let E be a bundle belonging to MIP 3 (0, 2) such that h1 End E(−2) 6= 0. The moduli space MIP 3 (−1, 2(γ/2)2 − 3(β/2)2 + 1/4) is singular at the point corresponding to the bundle Eβ/2,γ/2 . The proof is similar to the proof of the main theorem I. The condition 2γ − 3β + 1 > 0 ensures by (3.2) the stability of Eβ/2,γ/2 , while the condition γ ≥ 5 implies sγ 6= 0 in lemma 6. Taking γ = 5, β = 3 we obtain: Corollary. MIP 3 (−1, 6) is singular. [AO1] [AO2] [AO3] [H] [Ho] REFERENCES V. Ancona, G. Ottaviani. On moduli of instanton bundles on IP 2n+1 . Pacific Journal of math. (to appear). V. Ancona, G. Ottaviani. The Horrocks bundles of rank three on IP 5 . J. reine angew. Math. 460 (1995). V. Ancona, G. Ottaviani. In preparation. R. Hartshorne. Stable vector bundles of rank 2 on IP 3 . Math. Ann. 238 (1978), 229-280. G. Horrocks. Examples of rank three vector bundles on five-dimensional projective space. J. London Math. Soc. 18 (1978), 15-27. 6 [LP] J. Le Potier. Sur l’espace des modules des fibrés de Yang et Mills. In Mathémathique et Physique, Sém. Ecole Norm. Sup. 1979-1982, 65-137, Basel-Stuttgart-Boston (1983). [M] M. Maggesi. Tesi di laurea. Universita’ di Firenze (1994). [MR] R.M. Miró-Roig. Singular moduli space of stable vector bundles on IP 3 . Pacific Journal of math. (to appear). [NT] M.S. Narasimhan, G. Trautmann. The Picard group of the compactification of MIP 3 (0, 2). J. reine angew. Math., 422 (1991), 21-44. [Ra] A.P. Rao. A family of vector bundles on IP 3 . In Lecture Notes in Math. 1266 208-231, Berlin, Heidelberg, New York: Springer 1987. Authors’ addresses: V.A.: Dipartimento di Matematica ”U. Dini” Viale Morgagni 67/A I-50134 Firenze (Italy) e-mail ancona@udini.math.unifi.it G.O.: Dipartimento di Matematica Via Vetoio, Coppito I-67010 L’Aquila (Italy) e-mail ottaviani@vxscaq.aquila.infn.it 7