Abstract
Let \({(X,{\mathcal{O}}_X(1))}\) be a polarized smooth projective variety over the complex numbers. Fix \({{\mathcal{D}} \in {\mathrm{coh}}(X)}\) and a nonnegative rational polynomial \({\delta}\). Using GIT we contruct a coarse moduli space for \({\delta}\)-semistable pairs \({({\mathcal{E}},\varphi)}\) consisting of a coherent sheaf \({{\mathcal{E}}}\) and a homomorphism \({\varphi \colon {\mathcal{D}} \rightarrow {\mathcal{E}}}\). We prove a chamber structure result and establish a connection to the moduli space of coherent systems constructed by Le Potier in (Faisceaux semi-stable et systèmes cohérents. Cambridge University Press, Cambridge, 1995; Systèmes cohérents et structures de niveau. Astérisque 214, 1993).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Li J., Tian G.: Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties. J. Am. Math. Soc. 11, 119–174 (1998)
Pandharipande, R., Thomas, R.P.: Curve counting via stable pairs in the derived category. arXiv:0707.2348v3
Le Potier, J.: Systèmes cohérents et structures de niveau. Astérisque, vol. 214, p 604. Société mathématique de France, Montrouge (1993)
Hartshorne R.: Algebraic Geometry, GTM 52. Springer, New York (1977)
Huybrechts D., Lehn M.: The Geometry of Moduli Spaces of Sheaves, Aspects of Mathematics, vol. 31. Friedr Vieweg & Sohn, Braunschweig (1997)
Huybrechts D., Lehn M.: Framed modules and their moduli. Int. J. Math. 6, 297–324 (1995)
Simpson C.T.: Moduli of representations of the fundamental group of a smooth projective variety I. Inst. Hautes études Sci. Publ. Math. IHES 79, 47–129 (1994)
Schmitt, A.H.W.: Geometric Invariant Theory and Decorated Principal Bundles, Zurich Lectures in Advanced Mathematics. European Mathematical Society Publishing House, Zurich (2008)
Gómez, T.L., Sols, I.: Stable tensors and moduli space of orthogonal sheaves. arXiv:math/0103150
Grothendieck, A.: Techniques de construction et théoremes d’existence en géometrie algébrique IV: Les schémas de Hilbert, Séminaire Bourbaki, 1960/61, no. 221
Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, 3rd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) 34. Springer, Berlin (1994)
Le Potier, J.: Faisceaux semi-stable et systèmes cohérents, Vector Bundles in Algebraic Geometry, London Math. Soc. Lecture Note Ser., vol. 208, pp. 179–239. Cambridge University Press, Cambridge (1995)
Schmitt A.H.W.: Moduli problems of sheaves associated with oriented trees. Algebras Represent. Theory 6, 1–32 (2003)
