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On the moduli spaces of semi-stable plane sheaves of dimension one and multiplicity five. (English) Zbl 1273.14027

The moduli space of semistable sheaves \(M_{\mathbb P^2}(5, \chi)\) of multiplicity 5 and Euler characteristic \(\chi\), with one-dimensional support on the complex projective plane \(\mathbb P^2\) is considered. Spaces \(M_{\mathbb P^2}(r,\chi)\) for \(r\leq 3\) were studied in detail in [J. Le Potier, Rev. Roum. Math. Pures Appl. 38, No. 7–8, 635–678 (1993; Zbl 0815.14029)]. Spaces \(M_{\mathbb P^2}(4,\chi)\) were understood in [J.-M. Drézet and M. Maican, Geom. Dedicata 152, 17–49 (2011; Zbl 1236.14012)]. The present article continues the program designed in the cited article of J.-M.Drézet and M. Maican in the case of sheaves supported on quintics. Due to results from the cited article of J. Le Potier, the varieties \(M_{\mathbb P^2}(5, \chi)\) are projective, irreducible and locally factorial, have equal dimension and are smooth at points corresponding to stable sheaves. The author finds their decompositions into union of locally closed subvarieties (strata). Each stratum is characterized by means of locally free presentations of sheaves plus cohomological conditions and also by means of quotients of appropriate spaces by some (mostly non-reductive) linear algebraic groups. In some cases the description of sheaves in the strata is given also in terms of extensions. Results from [M. Maican, Pac. J. Math. 234, No. 1, 69–135 (2008, Zbl 1160.14007)] are widely used.
Due to the isomorphism \(M_{\mathbb P^2}(r, \chi) \cong M_{\mathbb P^2} (r, \chi+r)\) the author considers cases \(0\geq \chi \geq 4\). By duality all the study is subdivided into tree parts devoted to \(M_{\mathbb P^2}(5,3)\cong M_{\mathbb P^2}(5,2)\), \(M_{\mathbb P^2}(5,4)\cong M_{\mathbb P^2}(5,1)\) and \(M_{\mathbb P^2}(5,0)\) correspondently.

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
14D22 Fine and coarse moduli spaces
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14H60 Vector bundles on curves and their moduli
14J10 Families, moduli, classification: algebraic theory
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References:

[1] J.-M. Drézet, Fibrés exceptionnels et variétés de modules de faisceaux semi-stables sur \(¶_2(\C)\) , J. Reine Angew. Math. 380 (1987), 14-58. · Zbl 0613.14013 · doi:10.1515/crll.1987.380.14
[2] J.-M. Drézet, Variétés de modules alternatives , Ann. Inst. Fourier 49 (1999), 57-139.
[3] J.-M. Drézet, Espaces abstraits de morphismes et mutations , J. Reine Angew. Math. 518 (2000), 41-93. · Zbl 0937.14030 · doi:10.1515/crll.2000.007
[4] J.-M. Drézet and M. Maican, On the geometry of the moduli spaces of semi-stable sheaves supported on plane quartics , Geom. Dedicata 152 (2011), 17-49. · Zbl 1236.14012 · doi:10.1007/s10711-010-9544-1
[5] J.-M. Drézet and G. Trautmann, Moduli spaces of decomposable morphisms of sheaves and quotients by non-reductive groups , Ann. Inst. Fourier 53 (2003), 107-192. · Zbl 1034.14023 · doi:10.5802/aif.1941
[6] D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves , Aspects of Mathematics, vol. E31, Vieweg, Braunschweig, 1997. · Zbl 0872.14002
[7] A. D. King, Moduli of representations of finite dimensional algebras , Q. J. Math. 45 (1994), 515-530. · Zbl 0837.16005 · doi:10.1093/qmath/45.4.515
[8] J. Le Potier, Faisceaux semi-stables de dimension 1 sur le plan projectif , Rev. Roumaine Math. Pures Appl. 38 (1993), 635-678. · Zbl 0815.14029
[9] M. Maican, On two notions of semistability , Pacific J. Math. 234 (2008), 69-135. · Zbl 1160.14007 · doi:10.2140/pjm.2008.234.69
[10] M. Maican, A duality result for moduli spaces of semistable sheaves supported on projective curves , Rend. Semin. Mat. Univ. Padova 123 (2010), 55-68. · Zbl 1202.14036 · doi:10.4171/RSMUP/123-3
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