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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 \(r\geq1\).

14D20 Algebraic moduli problems, moduli of vector bundles
14D22 Fine and coarse moduli spaces
Full Text: DOI
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