Moduli spaces of decomposable morphisms of sheaves and quotients by non-reductive groups
Annales de l'Institut Fourier, Volume 53 (2003) no. 1, pp. 107-192.

We extend the methods of geometric invariant theory to actions of non–reductive groups in the case of homomorphisms between decomposable sheaves whose automorphism groups are non–reductive. Given a linearization of the natural action of the group Aut (E)× Aut (F) on Hom(E,F), a homomorphism is called stable if its orbit with respect to the unipotent radical is contained in the stable locus with respect to the natural reductive subgroup of the automorphism group. We encounter effective numerical conditions for a linearization such that the corresponding open set of semi- stable homomorphisms admits a good and projective quotient in the sense of geometric invariant theory, and that this quotient is in addition a geometric quotient on the set of stable homomorphisms.

Nous étendons les méthodes utilisées en géométrie invariante à l’étude de l’action de G= Aut (E)× Aut (F) sur Hom (E,F),E,F étant des faisceaux cohérents décomposables, dont les groupes d’automorphismes ne sont pas nécessairement réductifs. Étant donné une linéarisation de cette action, un homomorphisme est dit stable si son orbite relativement au radical unipotent de G est contenue dans le lieu stable relativement à l’action du sous-groupe réductif naturel de G. Nous donnons des conditions numériques effectives portant sur la linéarisation pour que l’ouvert correspondant des points semi-stables admette un bon quotient au sens de la géométrie invariante, qui soit projectif, et pour que ce quotient restreint à l’ouvert des homomorphismes stables soit un quotient géométrique.

DOI: 10.5802/aif.1941
Classification: 14L30,  14D20
Keywords: algebraic quotients, good quotients, non-reductive groups, moduli spaces
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Drézet, Jean-Marc; Trautmann, Günther. Moduli spaces of decomposable morphisms of sheaves and quotients by non-reductive groups. Annales de l'Institut Fourier, Volume 53 (2003) no. 1, pp. 107-192. doi : 10.5802/aif.1941. https://aif.centre-mersenne.org/articles/10.5802/aif.1941/

[1] A. Bialynicki-Birula; J. Świȩcika A recipe for finding open subsets of vector spaces with a good quotient, Colloq. Math., Tome 77 (1998) no. 1, pp. 97-114 | MR | Zbl

[2] A. Bialynicki-Birula; J. Świȩcika Open subsets of projective space with a good quotient by an action of a reductive group, Transf. Groups, Tome 1 (1996) no. 3, pp. 153-185 | DOI | MR | Zbl

[3] J. Dixmier Quelques aspects de la théorie des invariants, Gazette, Soc. Math. de France, 1989 | MR | Zbl

[4] J. Dixmier; M. Raynaud Sur le quotient d'une variété algébrique par un groupe algébrique, Advances in Math., Suppl. Studies, Tome vol. 7A (1981), pp. 327-344 | MR | Zbl

[5] H. Dolgachev; Yi Hu Variation of Geometric Invariant Theory Quotients (e-print, alg-geom/9402008)

[6] J.-M. Drézet Fibrés exceptionnels et variétés de modules de faisceaux semi-stables sur $\P^2(\C)$, J. reine angew. Math., Tome 380 (1987), pp. 14-58 | DOI | MR | Zbl

[7] J.-M. Drézet Cohomologie des variétés de modules de hauteur nulle, Math. Ann., Tome 281 (1988), pp. 43-85 | DOI | MR | Zbl

[8] J.-M. Drézet Variétés de modules extrémales de faisceaux semi-stables sur $\P^2(\C)$, Math. Ann., Tome 290 (1991), pp. 727-770 | DOI | MR | Zbl

[9] J.-M. Drézet Exceptional bundles and moduli spaces of stables sheaves on $\P_n$, Vector Bundles in Algebraic Geometry, Proceedings Durham 1993 (London Math. Soc. Lecture Note) Tome 208 (1995) | Zbl

[10] J.-M. Drézet Quotients algébriques par des groupes non réductifs et variétés de modules de complexes, Intern. J. Math., Tome 9 (1998) no. 7, pp. 769-819 | DOI | MR | Zbl

[11] J.-M. Drézet Variétés de modules alternatives, Ann. Inst. Fourier, Tome 49 (1999) no. 1, pp. 57-139 | DOI | Numdam | MR | Zbl

[12] J.-M. Drézet Espaces abstraits de morphismes et mutations, J. reine angew. Math., Tome 518 (2000), pp. 41-93 | DOI | MR | Zbl

[13] J.-M. Drézet; J. Le Potier Fibrés stables et fibrés exceptionnels sur $\P^2(\C)$, Ann. École Norm. Sup., Tome 18 (1985), pp. 193-244 | Numdam | MR | Zbl

[14] G. Ellingsrud; R. Piene; S.A. Str{\O}mme On the variety of nets of quadrics defining twisted cubic curves, Space Curves (Lect. Notes in Math.) Tome 1266 (1987) | Zbl

[15] G. Ellingsrud; S.A. Str{\O}mme On the Chow ring of a geometric quotient, Ann. of Math., Tome 130 (1989), pp. 159-187 | DOI | MR | Zbl

[16] A. Fauntleroy Geometric invariant theory for general algebraic groups, Comp. Math., Tome 55 (1985), pp. 63-87 | Numdam | MR | Zbl

[17] A. Fauntleroy Invariant theory for linear algebraic groups II, Comp. Math., Tome 68 (1983), pp. 23-29 | Numdam | MR | Zbl

[18] H.G. Freiermuth On the moduli space $M_p(\P_3)$ of semi-stable sheaves on $\P_3$ with Hilbert polynomial $P(m)=3m+1$ (2000) (Diplomarbeit, Kaiserslautern)

[19] G.-M. Greuel; G. Pfister Geometric quotients of unipotent group actions, Proc. Lond. Math. Soc., Tome 67 (1993), pp. 75-105 | DOI | MR | Zbl

[20] B.V. Karpov Semi-stable sheaves on a two-dimensional quadric and Kronecker modules, Math. Izvestiya AMS Transl., Tome 40 (1993), pp. 33-66 | MR | Zbl

[21] A. King Moduli of representations of finite dimensional algebras, Quart. J. Math. Oxford, Tome 45 (1994), pp. 515-530 | DOI | MR | Zbl

[22] J. KollÁr Quotient spaces modulo algebraic groups, Ann. of Math., Tome 145 (1997), pp. 33-79 | DOI | MR | Zbl

[23] J. Le Potier Systèmes cohérents et structures de niveau, Astérisque, Tome 214, Soc. Math. France, 1993 | MR | Zbl

[24] J. Le Potier Faisceaux semi-stables de dimension 1 sur le plan projectif, Rev. Roumaine Math. Pures Appl., Tome 318 (1993), pp. 635-678 | MR | Zbl

[25] R.M. Miró-Roig Some moduli spaces for rank 2 stable reflexive sheaves on $\P^3$, Trans. Amer. Math. Soc., Tome 299 (1987), pp. 699-717 | MR | Zbl

[26] R.M. Miró-Roig; G. Trautmann The moduli scheme $M(0,2,4)$ over $\P^3$, Math. Z., Tome 216 (1994), pp. 283-315 | DOI | MR | Zbl

[27] D. Mumford; J. Fogarty Geometric invariant theory, Ergeb. Math. Grenzgeb., Springer, Berlin-Heidelberg-New York, 1982 | MR | Zbl

[28] M. Nagata On the 14th problem of Hilbert, Proc. Intern. Cong. Math. 1958, Edinburgh (1960), pp. 459-462 | Zbl

[29] P.E. Newstead Introduction to moduli problems and orbit spaces, TIFR Lect. Notes in Math., Tome 51, Springer, Berlin-Heidelberg-New York, 1978 | MR | Zbl

[30] C. Okonek Moduli extremer reflexiver Garben auf $\P^n$, J. reine angew. Math., Tome 338 (1983), pp. 183-194 | DOI | MR | Zbl

[31] V.L. Popov; E.G. Vinberg Invariant theory, Algebraic Geometry, IV: Linear algebraic groups, invariant theory (Encycl. Math. Sci.) Tome vol. 55 (1994), pp. 123-278 | Zbl

[32] M. Reid What is a flip (1992) (Preprint)

[33] A. Schofield General representations of quivers, Proc. Lond. Math. Soc., Tome 65 (1992), pp. 46-64 | DOI | MR | Zbl

[34] C.S. Seshadri Mumford's conjecture for $\GL(2)$ and applications, Proc. Int. Colloq. on Algebraic Geometry, Tome vol. 347 (1968) | Zbl

[35] M. Thaddeus Geometric invariant theory and flips, J. Amer. Math. Soc., Tome 9 (1996), pp. 691-723 | DOI | MR | Zbl

[36] E.N. Tj{\O}tta Rational curves on the space of determinantal nets of conics (1998) (e-print, math.AG/9802037)

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