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On the geometry of the moduli spaces of semi-stable sheaves supported on plane quartics. (English) Zbl 1236.14012
Let \(M_{\mathbb{P}^2}(r,\chi)\) denote the moduli space of semistable sheaves on \(\mathbb{P}^2\) with linear Hilbert polynomial \(P(m)=rm+\chi\). The generic stable sheaves in this moduli space are line bundles on smooth plane curves of degree \(r\). It is known that the spaces \(M_{\mathbb{P}^2}(r,\chi)\) are projective and irreducible of dimension \(r^2+1\).
The authors of this paper investigate the geometry of the spaces \(M_{\mathbb{P}^2}(4,\chi)\) for \(1\leq \chi\leq4\). They decompose each moduli space into locally closed subvarieties (or strata) and describe each stratum as a good or geometric quotient of a set of morphisms of locally free sheaves.
The different strata are characterized by cohomological conditions and the resolutions of the sheaves contained in any stratum (except for one case) have been found by the second author in [“On two notions of semistability”, Pac. J. Math. 234, No. 1, 69–135 (2008; Zbl 1160.14007)] In the paper under review, one of the main difficulties is proving that the quotients are good or geometric, in particular when the group acting on the space of morphisms is not reductive. They are able to prove this result by means of a method already used by the first author in [“Varietes de modules extremales de faisceaux semi-stables sur \(\mathbb{P}^2\)”, Math. Ann. 290, No.4, 727–770 (1991; Zbl 0755.14005)].
The cohomology estimates proved in the paper show that Clifford’s theorem is true not only for the generic sheaves in the moduli spaces, but for all sheaves in \(M_{\mathbb{P}^2}(r,\chi)\), when \(0 \leq \chi < r\leq 4\). Hence, the authors conjecture that a suitable generalization of Clifford’s theorem holds for any \(r\geq1\).

MSC:
14D20 Algebraic moduli problems, moduli of vector bundles
14D22 Fine and coarse moduli spaces
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[1] Drézet J.-M.: Variétés de modules extrémales de faisceaux semi-stables sur $${\(\backslash\)mathbb{P}_2(\(\backslash\)mathbb C)}$$ . Math. Ann. 290, 727–770 (1991) · Zbl 0755.14005 · doi:10.1007/BF01459270
[2] Drézet J.-M.: Fibrés exceptionnels et suite spectrale de Beilinson généralisée sur $${\(\backslash\)mathbb{P}_2(\(\backslash\)mathbb C)}$$ . Math. Ann. 275, 25–48 (1986) · Zbl 0578.14013 · doi:10.1007/BF01458581
[3] Drézet J.-M.: Fibrés exceptionnels et variétés de modules de faisceaux semi-stables sur $${\(\backslash\)mathbb{P}_2(\(\backslash\)mathbb C)}$$ . J. Reine Angew. Math. 380, 14–58 (1987)
[4] Drézet J.-M.: Variétés de modules alternatives. Ann. Inst. Fourier 49, 57–139 (1999) · Zbl 0923.14005 · doi:10.5802/aif.1669
[5] Drézet J.-M.: Espaces abstraits de morphismes et mutations. J. Reine Angew. Math. 518, 41–93 (2000) · Zbl 0937.14030
[6] Drézet J.-M.: Faisceaux sans torsion et faisceaux quasi localement libres sur les courbes multiples primitives. Math. Nachr. 282, 919–952 (2009) · Zbl 1171.14010 · doi:10.1002/mana.200810781
[7] Drézet J.-M., Trautmann G.: Moduli spaces of decomposable morphisms of sheaves and quotients by non-reductive groups. Ann. Inst. Fourier 53, 107–192 (2003) · Zbl 1034.14023 · doi:10.5802/aif.1941
[8] Freiermuth H.-G.: On the Moduli Space $${M_P (\(\backslash\)mathbb{P}_3)}$$ of Semi-Stable Sheaves on $${\(\backslash\)mathbb{P}_3}$$ with Hilbert Polynomial P(m) = 3m + 1. Diplomarbeit, University of Kaiserslautern (2000)
[9] Freiermuth H.-G., Trautmann G.: On the moduli scheme of stable sheaves supported on cubic space curves. Am. J. Math. 126, 363–393 (2004) · Zbl 1069.14012 · doi:10.1353/ajm.2004.0013
[10] Gorodentsev A.L., Rudakov A.N.: Exceptional vector bundles on projective spaces. Duke Math. J. 54, 115–130 (1987) · Zbl 0646.14014 · doi:10.1215/S0012-7094-87-05409-3
[11] Griffiths P., Harris J.: Principles of Algebraic Geometry. Wiley, New York (1978) · Zbl 0408.14001
[12] Hartshorne R.: Introduction to Algebraic Geometry. Springer, New York (1977) · Zbl 0367.14001
[13] Hulek K.: On the classification of stable rank-r vector bundles over the projective plane, Vector bundles and differential equations Proc, Nice 1979. Prog. Math. 7, 113–144 (1980)
[14] King A.: Moduli of representations of finite dimensional algebras. Q. J. Math. Oxf. II. Ser. 45, 515–530 (1994) · Zbl 0837.16005 · doi:10.1093/qmath/45.4.515
[15] Le Potier J.: Faisceaux semi-stables de dimension 1 sur le plan projectif. Rev. Roumaine Math. Pures Appl. 38, 635–678 (1993) · Zbl 0815.14029
[16] Maican M.: Variation of GIT-Quotients. Examples Techniques and Applications to Moduli Spaces. Diplomarbeit, University of Kaiserslautern (2000)
[17] Maican M.: On two notions of semistability. Pacific J. Math. 234, 69–135 (2008) · Zbl 1160.14007 · doi:10.2140/pjm.2008.234.69
[18] Maican M.: A duality result for moduli spaces of semistable sheaves supported on projective curves. Rend. Sem. Mat. Univ. Padova 123, 55–68 (2010) · Zbl 1202.14036 · doi:10.4171/RSMUP/123-3
[19] Mumford, D., Fogarty, J., Kirwan, F.: Geometric invariant theory. 3rd enl. ed. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 Folge 34. Springer, Berlin (1993) · Zbl 0797.14004
[20] Newstead P.E.: Lectures on Introduction to Moduli Problems and Orbit Spaces, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Mathematics 51. Springer, Berlin (1978) · Zbl 0411.14003
[21] Okonek C., Schneider M., Spindler H.: Vector Bundles on Complex Projective Spaces, Progress in Mathematics 3. Birkhäuser, Boston (1980) · Zbl 0438.32016
[22] Shafarevich, I.R.: Basic algebraic geometry. Translated from the Russian by K.A. Hirsch. 2nd ed. Springer Study Edition. Springer, Berlin (1977) · Zbl 0362.14001
[23] Simpson C.T.: Moduli of representations of the fundamental group of a smooth projective variety I. Inst. Hautes Études Sci. Publ. Math. 79, 47–129 (1994) · Zbl 0891.14005 · doi:10.1007/BF02698887
[24] Świecicka J.: Good Quotients of Reductive Group Actions–Combinatorics in Projective Spaces, Algebraic Group Actions and Quotients, pp. 109–131. Hindawi Publishing Corporation, Cairo (2004) · Zbl 1102.14033
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