Disconnected moduli spaces of stable bundles on surfaces. (English) Zbl 1522.14059
Given a projective surface \(X\subset\mathbb{P}^n\), the moduli spaces \(M_X(2;c_1,c_2)\) of stable rank 2 bundles \(\mathcal{E}\) with Chern classes \(c_i:=c_i(\mathcal{E})\) tend to be well-behaved for large enough \(c_2\). In particular, they tend to be irreducible and reduced. On the other hand, a different behaviour can be expected for small \(c_2\).
In this paper, the authors deal with these issues constructing projective surfaces \(X\) of general type and Picard number one for which \(M_X(2;c_1,c_2)\) has a large number of components. More concretely, their examples consist on complete intersection \(X=\cap_{i=1}^{n-2}X_i\) of \(n-2\) hypersurfaces \(X_i\) in \(\mathbb{P}^n\) of degrees \(3\leq d_1\leq\dots\leq d_{n-2}\) such that \(D_1\) contains linear subspaces of codimension \(2\). It is proved that every connected component of the Fano scheme \(F(D_1)\) parameterizing such linear spaces corresponds, by means of a version of Serre correspondence, to a distinct connected component of \(M_X(2,H,(d_1-1)\prod_{i=2}^{n-2}d_i)\). Then, examples of hypersurfaces \(D_1\subset\mathbb{P}^n\) such that \(F(D_1)\) has many components are found.
In this paper, the authors deal with these issues constructing projective surfaces \(X\) of general type and Picard number one for which \(M_X(2;c_1,c_2)\) has a large number of components. More concretely, their examples consist on complete intersection \(X=\cap_{i=1}^{n-2}X_i\) of \(n-2\) hypersurfaces \(X_i\) in \(\mathbb{P}^n\) of degrees \(3\leq d_1\leq\dots\leq d_{n-2}\) such that \(D_1\) contains linear subspaces of codimension \(2\). It is proved that every connected component of the Fano scheme \(F(D_1)\) parameterizing such linear spaces corresponds, by means of a version of Serre correspondence, to a distinct connected component of \(M_X(2,H,(d_1-1)\prod_{i=2}^{n-2}d_i)\). Then, examples of hypersurfaces \(D_1\subset\mathbb{P}^n\) such that \(F(D_1)\) has many components are found.
Reviewer: Joan Pons-Llopis (Maó)
MSC:
| 14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |
| 14J29 | Surfaces of general type |
| 14D20 | Algebraic moduli problems, moduli of vector bundles |
Software:
MathOverflowReferences:
| [1] | E.Arrondo, A home‐made Hartshorne‐Serre correspondence, Rev. Mat. Complut.20 (2007), no. 2, 423-443. · Zbl 1133.14046 |
| [2] | I.Coskun, and J.Huizenga, The moduli spaces of sheaves on surfaces, pathologies, and Brill‐Noether problems, in J.Christophersen (ed.) and K.Ranestad (ed.) (eds.) Geometry of Moduli (2018), 75-105. · Zbl 1420.14098 |
| [3] | I.Coskun, and E.Riedl, Normal bundles of rational curves on complete intersections, Commun. Contemp. Math.21 (2019), no. 2. · Zbl 1436.14057 |
| [4] | I.Coskun, and M.Woolf, The stable cohomology of moduli spaces of sheaves on surfaces, J. Differential Geom., to appear. · Zbl 1505.14023 |
| [5] | S. K.Donaldson, Polynomial invariants for smooth four‐manifolds, Topology29 (1990), no. 3, 257-315. · Zbl 0715.57007 |
| [6] | L.Ein, Generalized null correlation bundles, Nagoya Math. J.111 (1988), 13-24. · Zbl 0663.14012 |
| [7] | D.Eisenbud, M.Green, and J.Harris, Cayley‐Bacharach theorems and conjectures, Bull. Amer. Math. Soc.33 (1996), no. 3, 295-324. · Zbl 0871.14024 |
| [8] | J.Fogarty, Algebraic families on an algebraic surface, Amer. J. Math.90 (1968), 511-521. · Zbl 0176.18401 |
| [9] | R.Friedman, Rank two vector bundles over regular elliptic surfaces, Invent. Math.96 (1989), 283-332. · Zbl 0671.14006 |
| [10] | R.Friedman, and J. W.Morgan, On the diffeomorphism types of certain algebraic surfaces, I. J. Differential Geom.27 (1988), 297-369. · Zbl 0669.57016 |
| [11] | D.Gieseker, On the moduli space of vector bundles on an algebraic surface, Ann. Math.106 (1977), 45-60. · Zbl 0381.14003 |
| [12] | D.Gieseker, and J.Li, Irreducibility of moduli of rank 2 bundles on algebraic surfaces, J. Differential Geom.40 (1994), 23-104. · Zbl 0827.14008 |
| [13] | L.Göttsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann.286 (1990), no. 3, 193-207. · Zbl 0679.14007 |
| [14] | D.Huybrechts, and M.Lehn, The Geometry of moduli spaces of sheaves, 2nd ed., Cambridge Mathematical Library, Cambridge Univ. Press, Cambridge, 2010. · Zbl 1206.14027 |
| [15] | A.Iarrobino, Reducibility of the families of 0‐dimensional schemes on a variety, Invent. Math.15 (1972), 72-77. · Zbl 0227.14006 |
| [16] | D.Kotschick, On manifolds diffeomorphic to \(\mathbb{C}P^2 \# \overline{\mathbb{C}P^2} \), Invent. Math.95 (1989), no. 3, 591-600. · Zbl 0691.57008 |
| [17] | M.Maruyama, Moduli of stable sheaves II, J. Math. Kyoto18 (1978), 557-614. · Zbl 0395.14006 |
| [18] | N.Mestrano, Sur le espaces de modules de fibrés vectoriels de rang deux sur des hypersurfaces de \(\mathbb{P}^3\), J. reine angew. Math.490 (1997), 65-79. · Zbl 0882.14004 |
| [19] | N.Mestrano, and C.Simpson, Obstructed bundles of rank two on a quintic surface, Int. J. Math.22 (2011), 789-836. · Zbl 1239.14003 |
| [20] | N.Mestrano, and C.Simpson, Moduli of sheaves, development of moduli theory-Kyoto 2013, Adv. Stud. Pure Math.69 (2016), no. 2016, 77-172. · Zbl 1369.14019 |
| [21] | B. G.Moishezon, Algebraic homology classes on algebraic varieties, Izv. Akad. Nauk. SSSR Ser. Mat.31 (1967), no. 2, 225-268. · Zbl 0148.41505 |
| [22] | P.Nijsse, The irreducibility of the moduli space of stable vector bundles of rank 2 on a quintic in \(\mathbb{P}^3\), arXiv:alg‐geom/9503012, 1995. |
| [23] | K. G.O’Grady, Moduli of vector bundles on projective surfaces: some basic results, Invent. Math.123 (1996), 141-207. · Zbl 0869.14005 |
| [24] | C.Okonek, and A.Van de Ven, Stable bundles and differentiable structures on certain elliptic surfaces, Invent Math.86 (1986), no. 2, 357-370. · Zbl 0613.14018 |
| [25] | J.Starr, Connectedness of moduli space, mathoverflow.net/q/323673, 2019. |
| [26] | D.Tseng, Collections of hypersurfaces containing a curve, Int. Math. Res. Not.2020, no. 13, 3927-3977. · Zbl 1454.14112 |
| [27] | R.Vakil, Murphy’s Law in algebraic geometry: badly‐behaved deformation spaces, Invent. Math.164 (2006), 569-590. · Zbl 1095.14006 |
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