Let M be the moduli space of stable bundles of rank 2 and with fixed
determinant \mathcal{L} of degree d on a smooth projective curve C of genus g>=
2. When g=3 and d is even, we prove, for any point [W]\in M, there is a minimal
rational curve passing through [W], which is not a Hecke curve. This
complements a theorem of Xiaotao Sun

Min, Liu

English

arXiv

Min Liu

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 354 (2016) 1013–1017Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comAlgebraic geometryRemarks on minimal rational curves on moduli spaces of stable bundles ✩Remarques sur les courbes rationnelles minimales sur les espaces des modules de faisceaux stablesLiu MinSchool of Mathematics and Statistics, Qingdao University, Qingdao 266071, PR Chinaa r t i c l e i n f o a b s t r a c tArticle history:Received 19 July 2016Accepted after revision 31 August 2016Available online 9 September 2016Presented by Claire VoisinLet C be a smooth projective curve of genus g ≥ 2 over an algebraically closed field of characteristic zero, and M be the moduli space of stable bundles of rank 2 and with fixed determinant L of degree d on the curve C . When g = 3 and d is even, we prove that, for any point [W ] ∈ M , there is a minimal rational curve passing through [W ], which is not a Hecke curve. This complements a theorem of Xiaotao Sun.© 2016 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éSoient C une courbe projective lisse de genre g ≥ 2 et M l’espace des modules de faisceaux stables de rang 2 et de déterminant fixe L de degré d sur C . Nous prouvons que, lorsque g = 3 et d est pair, il existe, pour tout point [W ] ∈ M , une courbe rationnelle minimale passant par [W ], qui n’est pas une courbe de Hecke. Cela complète un théorème de Xiaotao Sun.© 2016 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionThroughout this paper, we assume that C is a smooth projective curve of genus g ≥ 2 over an algebraically closed field of characteristic zero. Let M := SUC (r, L) be the moduli space of stable vector bundles of rank r ≥ 2 and with the fixed determinant L of degree d, which is a smooth quasi-projective Fano variety with P ic(M) = Z ·  and −K M = 2(r, d), where  is an ample divisor ([9,1]). By a rational curve of M , we mean a nontrivial proper morphism φ : P1 → M and its degree is defined to be deg φ∗(−K M) (with respect to the ample anti-canonical line bundle −K M ).In [10], Xiaotao Sun has determined all rational curves of minimal degree passing through generic points of M except in the case where g = 3, r = 2, and d is even.✩ Supported by the National Natural Science Foundation of China (Grant No. 11401330).E-mail address: liumin@amss.ac.cn.http://dx.doi.org/10.1016/j.crma.2016.08.0071631-073X/© 2016 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1014 M. Liu / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 1013–1017Theorem 1.1. (Theorem 1 of [10]) If g ≥ 3, then any rational curve φ : P1 → M passing through the generic point has degree at least 2 r. It has degree 2 r if and only if it is a Hecke curve unless g = 3, r = 2, and d is even.This implies that all the rational curves of (−K M )-degree smaller than 2 r, called small rational curves, must lie in a proper closed subset [3,4]. In this note, we remark that the condition in Sun’s Theorem is necessary:Theorem 1.2. If g = 2, r = 2 and d is odd, then, for any [W ] ∈ M, there exists a rational curve passing through it, which has degree 2.If g = 3, r = 2 and d is even, then, for any point [W ] ∈ M, there exists a rational curve of degree 4 passing through it, which is not a Hecke curve.Recall that, by Lemma 2.1 of [10], any rational curve φ : P1 → M is defined by a vector bundle E on f : X = C ×P1 → C .If E is semi-stable on generic fiber Xξ = f −1(ξ) (tensoring a pullback of line bundle on P1, we can assume the restriction of E to a generic fiber is of the form O ⊕rXξ ), according to the arguments of section 2 in [10], there is a finite set S ⊂ C of points and a vector bundle V on C such that E just suits in the exact sequence0 → f ∗V → E →⊕p∈SQp → 0where Qp is a vector bundle on Xp = {p} × P1. The curves defined by such E were said to be of Hecke type in [8,11](since a Hecke curve by definition is defined by a vector bundle E suited in 0 → f ∗V → E → OXp (−1) → 0). If E is not semi-stable on the generic fiber Xξ (curves defined by such E were said of split type in [11]) and the curve has minimal degree 2(r, d), then E must suit in0 → f ∗V 1 ⊗ π∗OP1(1) → E → f ∗V 2 → 0where π : X → P1 is the projection and V 1, V 2 are stable vector bundles on C of rank r1, r2, and degrees d1, d2 satisfy r1d − rd1 = (r, d). Note that rational curves of degree 2(r, d) have degree 1 with respect to  because of −K M = 2(r, d), which will be called lines in M .The rational curves we constructed in Theorem 1.2 are of split type (thus they are not Hecke curves). We have in fact a more general result. Let M = SUC (2, L) be the moduli space of rank-two stable bundles with fixed determinant L on a smooth projective curve C of genus g ≥ 3. Let Ms ⊂ M be the locus of stable bundles [W ] ∈ M with the Segre invariant s(W ) = s (refer to Section 3 for the definition of Segre invariant). Then we have the following theorem.Theorem 1.3. When d is even, for any [W ] ∈ M2 , there is a rational curve of split type passing through it, which has degree 4. If d is odd, for any [W ] ∈ M1 , there is a rational curve of split type passing through it, which has degree 2.When g = 3 and d is even, we have M2 = M (see Lemma 3.1). Thus Theorem 1.2 is a corollary of Theorem 1.3.2. Rational curves of split typeLet C be a smooth projective curve with genus g ≥ 2 over an algebraically closed field of characteristic zero, W be a stable bundle of rank r and of degree d with determinant L over C . Assume that there is a stable subbundle V 1 of W such thatr1d − d1r = (r,d), (1)where r1 = rank V 1, d1 = deg V 1 and d = deg W . Let V 2 := V /V 1 be the quotient bundle, then W fits a non-trivial extension0 → V 1 → W → V 2 → 0. (2)It is known that there is a family of vector bundles {Ep}p∈P on C parametrized by P = P Ext1(V 2, V 1) so that for each p ∈ P , Ep is isomorphic to the bundle obtained as the extension of V 2 by V 1 given by p (see Lemma 2.3 of [9]). Let l be a line in P = P Ext1(V 2, V 1) passing through the point p0, where p0 is the point in P given by (2). If it happens that Ep is stable for each p ∈ l, then{Ep}p∈lwill define a rational curve of degree 2(r, d) (with respect to −K M ) passing through [W ] ∈ SUC (r, L) ([10,4]). Such a rational curve in SUC (r, L) will be called a rational curve of split type.It is known that an extension 0 → E → W → F → 0, where E , W , F are vector bundles on C , gives rise to an element δ(W ) ∈ H1(C, Hom(F , E)), which is the image of the identity homomorphism in H0(C, Hom(F , F )) by the connecting ho-momorphism H0(C, Hom(F , F )) → H1(C, Hom(F , E)). This gives a one:one correspondence between the set of equivalent classes of extensions of F by E and H1(C, Hom(F , E)) (refer to section 2 in [9]).M. Liu / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 1013–1017 1015Lemma 2.1. Let d be an even number, and 0 → L1 → W → L2 → 0 be any non-trivial extension of L2 by L1 , where L1 (resp. L2) is a line bundle of degree d2 − 1 (resp. d2 + 1). Then(i) W is semi-stable;(ii) W is non-stable if and only if the element δ(W ) ∈ H1(C, L−12 ⊗ L1) corresponding to W is in the kernel of the mapH1(C, L−12 ⊗ L1) −→ H1(C, L−12 ⊗ L1 ⊗ Lx),for some x ∈ C , where Lx = OC (x) is the line bundle defined by x. In this case, W is S-equivalent to L2 ⊗ L−1x ⊕ L1 ⊗ Lx (refer to section 2 of [7] for the definition of S-equivalent).Proof. (i) See Lemma 2.2 in [4] and [5].(ii) Let L′ be a line bundle of degree d2 . Then, since H0(C, Hom(L′, L1)) = 0, it is easy to see that H0(C, Hom(L′, W )) = 0if and only if L′ is of the form L2 ⊗ L−1x for some x ∈ C and the natural map L2 ⊗ L−1x → L2 can be lifted into a map L2 ⊗ L−1x → W .Consider the commutative diagram of vector bundles0 → Hom(L2, L1) −−−−→ Hom(L2, W ) −−−−→ Hom(L2, L2) → 0⏐⏐⏐⏐⏐⏐0 → Hom(L2 ⊗ L−1x , L1) −−−−→ Hom(L2 ⊗ L−1x , W ) −−−−→ Hom(L2 ⊗ L−1x , L2) → 0,where the horizontal sequences are exact and the vertical maps are induced by the natural map L2 ⊗ L−1x → L2. From this, we deduce the commutative diagram0 → H0(C, Hom(L2, W )) −−−−→ H0(C, Hom(L2, L2)) −−−−→ H1(C, Hom(L2, L1)) → ·· ·⏐⏐⏐⏐⏐⏐0 → H0(C, Hom(L2 ⊗ L−1x , W )) −−−−→ H0(C, Hom(L2 ⊗ L−1x , L2)) −−−−→ H1(C, Hom(L2 ⊗ L−1x , L1)) → ·· ·which implies the lemma. Remark 2.2. Lemma 2.1 (ii) asserts that the non-stable bundles in PH1(L−12 ⊗ L1) correspond precisely to the image of C in PH1(L−12 ⊗ L1) under the map given by the linear system KC ⊗ L−11 ⊗ L2. Which implies that the dimension of the subset of non-stable bundles in PH1(L−12 ⊗ L1) is at most 1.3. Proof of Theorem 1.3Let C be a smooth irreducible curve over an algebraically closed field of characteristic zero, W a vector bundle of rank 2over C , setm(W ) := max{deg(L)|L ⊂ W is a sub line bundle of W }. (3)A sub line bundle L of W of maximal degree m(W ) is called a maximal sub line bundle. The Segre invariant is defined bys(W ) := deg(W ) − 2m(W ). (4)Note that s(W ) ≡ deg(W ) (mod 2) and that W is stable (resp. semi-stable) if and only if s(W ) ≥ 1 (resp. s(W ) ≥ 0). Nagata proved in [6] thats(W ) ≤ g.It is easy to see thatLemma 3.1. If g = 3, then, for any stable bundle W over C of rank 2 and with even degree d, we have s(W ) = 2.In general, the function s : M −→ Z defined by [W ] −→ s(W ) is lower semicontinuous and gives a stratification of Minto locally closed subsets Ms according to the value of s. Then, by Proposition 3.1 in [2], we haveProposition 3.2. ([2]) Suppose that 1 ≤ s ≤ g − 2 and s ≡ d (mod 2). Then Ms is an irreducible algebraic variety of dimension 2g + s − 2.The proof of Theorem 1.3 follows the following two propositions.1016 M. Liu / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 1013–1017Proposition 3.3. Suppose that g ≥ 3, r = 2, d is even and M2 is non-empty. Then, for any [W ] ∈ M2 , there is a rational curve of split type passing through it, which has degree 4.Proof. For any [W ] ∈ M2, there is a sub line bundle L1 of W with deg L1 = d2 − 1, where d = degL. Let L2 := W /L1 be the quotient bundle, which has degree d2 + 1. It is easy to see that1 × d − (d2− 1) × 2 = 2 = (2,d).Let i : L1 → W be the natural injection, then0 −−−−→ L1 i−−−−→ W −−−−→ L2 −−−−→ 0is a non-trivial extension (otherwise, we have W ∼= L1 ⊕ L2, which contradicts the stability of W ).It is known that there is a family of vector bundles E on C parametrized by P (L1,L2) = P Ext1(L2, L1) so that for each p ∈ P (L1,L2) , the Ep is isomorphic to the bundle obtained as the extension of L2 by L1 given by p (see Lemma 2.3 of [9]). More precisely, there is a universal extension0 → f ∗L1 ⊗ π∗OP(L1,L2) (1) → E → f ∗L2 → 0 (5)on C × P (L1,L2) , where f : C × P (L1,L2) → C and π : C × P (L1,L2) → P (L1,L2) are projections. Then E is a family of semi-stable bundles of rank 2 and with fixed determinant det(L1) ⊗ det(L2) ∼= L (Lemma 2.1). Thus, the universal extension (5) defines a morphism(L1,L2) : P (L1,L2) −→ UC (2,L), (6)where UC (2, L) denotes the moduli space of semi-stable bundles of rank 2 and with fixed determinant L, which is a projective compactification of M .It is easy to see that P (L1,L2) is a projective space of dimension g ≥ 3. By Lemma 2.1 and Remark 2.2, there is a line l in P (L1,L2) passing throughq = [0 −−−−→ L1 i−−−−→ W −−−−→ L2 −−−−→ 0]such that Ep is stable for each p ∈ l. Thus, (L1,L2)(l) ⊂ M = SUC (2, L) and(L1,L2)|l : l → M = SUC (2,L) (7)is a rational curve of split type passing through the point [W ] ∈ M . Proposition 3.4. Suppose g ≥ 2, r = 2, d is odd and M1 is non-empty. Then, for any [W ] ∈ M1 , there is a rational curve of split type passing through it, which has degree 2.Proof. Let [W ] be a point in M1, then we have s(W ) = 1 and there is a sub line bundle L1 of W with deg L1 = d−12 , where d = degL. Let L2 := W /L1, which is a line bundle of degree d+12 . It is easy to see that1 × d − d − 12× 2 = 1 = (2,d).Let ι : L1 → W be the natural injection, then0 −−−−→ L1 ι−−−−→ W −−−−→ L2 −−−−→ 0is a non-trivial extension because W is a stable bundle.It is known that there is a family of vector bundles {Ep} on C parametrized by P (L1,L2) = P Ext1(L2, L1) such that for each p ∈ P (L1,L2) , Ep is isomorphic to the bundle obtained as the extension of L2 by L1 given by p (Lemma 2.3 of [9]). By Lemma 3.1 of [10], {Ep} is a family of stable bundles of rank 2 and with fixed determinant det(L1) ⊗ det(L2) ∼= L, which defines a morphism(L1,L2) : P (L1,L2) −→ SUC (2,L) = M. (8)Let l be a line in P (L1,L2) passing throughq = [0 −−−−→ L1 ι−−−−→ W −−−−→ L2 −−−−→ 0],then(L1,L2)|l : l −→ M = SUC (2,L) (9)is a rational curve of split type passing through the point [W ] ∈ M , which has degree 2. When g = 2, the same as Lemma 3.1, we have:M. Liu / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 1013–1017 1017Lemma 3.5. If g = 2, r = 2 and d is odd, for any [W ] ∈ M, s(W ) = 1.By Lemma 3.5 and Proposition 3.4, we have:Proposition 3.6. If g = 2, r = 2 and d is odd, then, for any [W ] ∈ M, there exists a rational curve of split type passing through it, which has degree 2.AcknowledgementsThe author is grateful to her supervisor Prof. Xiaotao Sun for his helpful suggestions in the preparation of this paper and to Prof. Meng Chen and Prof. Kejian Xu for their help.References[1] J.-M. Drezet, M.S. Narasimhan, Groupe de Picard des variétés de modules de fibrés semistables sur les courbes algébriques, Invent. Math. 97 (1989) 53–94.[2] H. Lang, M.S. Narasimhan, Maximal subbundles of rank two vector bundles on curves, Math. Ann. 266 (1983) 55–72.[3] M. Liu, Small rational curves on the moduli space of stable bundles, Int. J. Math. 23 (8) (2012) 1250085.[4] N. Mok, X. Sun, Remarks on lines and minimal rational curves, Sci. China Ser. A, Math. 52 (4) (2009) 617–630.[5] N. Mok, X. Sun, Erratum to: Remarks on lines and minimal rational curves, Sci. China Ser. A, Math. 5 (9) (2014) 1992.[6] M. Nagata, On self intersection number of vector bundles of rank 2 on a Riemann surface, Nagoya Math. J. 37 (1970) 191–196.[7] M.S. Narasimhan, S. Ramanan, Moduli of vector bundles on a compact Riemann surface, Ann. of Math. (2) 89 (1969) 14–51.[8] M.S. Narasimhan, S. Ramanan, Geometry of Hecke cycles I, in: C.P. Ramanujam – A Tribute, Tata Institute, Springer Verlag, 1978, pp. 291–345.[9] S. Ramanan, The moduli spaces of vector bundles over an algebraic curve, Math. Ann. 200 (1973) 69–84.[10] X. Sun, Minimal rational curves on the moduli spaces of stable bundles, Math. Ann. 331 (2005) 925–937.[11] X. Sun, Elliptic curves in moduli spaces of stable bundles, in: Special Issue: In Memory of Eckart Viehweg 1761–1783, Pure Appl. Math. Q. 7 (4) (2011).

Remarks on minimal rational curves on moduli spaces of stable bundles

Liu, Min

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Remarks on minimal rational curves on moduli spaces of stable bundles

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