On Harder-Narasimhan stratification over quot schemes. (English) Zbl 0587.14008

Hernández, Rafael

J. Reine Angew. Math. 371, 115-124 (1986).

Let C be a complete smooth curve of genus g over an algebraically closed field of characteristic zero and let \(Q(n,r,d)^ 0\) be the open subscheme parametrizing quotient bundles \(0_ c^ n\twoheadrightarrow E\) of rank r and degree d with \(h^ 1(E)=0\). The aim of the paper is to prove the following theorem: The Harder-Narasimhan strata [S. S. Shatz, Compos. Math. 35, 163-187 (1977; Zbl 0371.14010)] in \(Q(n,r,d)^ 0\) given by the universal quotient bundle are irreducible and smooth.
This generalizes partially a previous work of J. L. Verdier for the case \(g=0\) [in Group theoretical methods in physics, Proc. XIth internat. Colloq., Istanbul 1982; Lect. Notes Phys. 180, 136-141 (1983; Zbl 0528.58008)] and A. Bruguières [”Le schéma des morphismes d’une courbe elliptique dans une grassmannienne” (These, Paris 1984)]. The smoothness part of the statement is also a consequence of a theorem of A. Bruguières [in Module des fibres stables sur les courbes algébriques, Notes Éc. Norm. Super., Paris 1983, Prog. Math. 54, 81- 104 (1984; Zbl 0577.14012)].

MSC:

14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
14M17	Homogeneous spaces and generalizations
57N80	Stratifications in topological manifolds

Keywords:

complete smooth curve; Harder-Narasimhan strata; universal quotient bundle

Citations:

Zbl 0371.14010; Zbl 0528.58008; Zbl 0577.14012