Moduli spaces of vector bundles on a real nodal curve
- PDF / 326,696 Bytes
- 12 Pages / 439.37 x 666.142 pts Page_size
- 23 Downloads / 187 Views
Moduli spaces of vector bundles on a real nodal curve Usha N. Bhosle1 Received: 22 March 2019 / Accepted: 10 February 2020 © The Managing Editors 2020
Abstract Let Y be a geometrically irreducible nodal projective algebraic curve of arithmetic genus g ≥ 2 defined over R. Let YC = Y ×R C. Fix an R-valued point ξ of Picd (Y ). For integers r ≥ 2 and d, let M(r , ξ ) [respectively U (r , ξ )] be the moduli stack of vector bundles (respectively the moduli space of stable vector bundles) of rank r and determinant ξ on Y . We determine the Picard group of M(r , ξ ). We compute the Picard group and Brauer group of U (r , ξ ). Keywords Real nodal curve · Moduli spaces · Picard group · Brauer group Mathematics Subject Classification 14H60
1 Introduction Let YC be an irreducible reduced complex curve of arithmetic genus g ≥ 2, with ordinary nodes as singularities. Fix a line bundle ξC of degree d on YC . Let SLYC (r , d) denote the moduli stack parametrising vector bundles F on YC of rank r , degree d together with an isomorphism δY : r (F) → ξC . We show that the Picard group of SLYC (r , d) is isomorphic to the group of integers and it is generated by the determinant line bundle L(det)YC . In case YC is non-singular, this was done in Beauville and Laszlo (1994). The formalism of infinite Grassmannians used in Beauville and Laszlo (1994) is not available in the case of nodal curves. In nodal case, we define generalised quasiparabolic S L(r , C)-bundles (QPBs in short) on the normalisation X C of YC and show q pb that there is a canonical isomorphism from the stack SL X C (r , d) of QPBs on X C to the stack SLYC (r , d) . We use the formalism of infinite Grassmannians to compute q pb the Picard group of SL X C (r , d). We then study Picard group of the moduli stack of bundles over a real nodal curve. Let Y be a geometrically irreducible nodal projective algebraic curve of arithmetic
B 1
Usha N. Bhosle [email protected] Indian Statistical Institute, Bangalore, India
123
Beitr Algebra Geom
genus g ≥ 2 defined over R. Let p : X −→ Y be the normalisation of Y . Let X C = X ×R C (respectively YC ) be the irreducible projective complex algebraic curve obtained from X (respectively Y ) by base change to C. Let ξ be an R-valued point of the Picard variety Picd (Y ). Then the complexification ξC corresponds to a line bundle ξC over YC which has a real or quaternionic structure. This line bundle descends to Y if and only if it has a real structure. For fixed integers r ≥ 2 and d, let M(r , ξ ) be the moduli stack of vector bundles of rank r and determinant ξ on Y and M gs (r , ξ ) ⊂ M(r , ξ ) its open substack of geometrically stable vector bundles (i.e. vector bundles F for which FC = F ⊗R C over YC is stable). Let U (r , ξ ) denote their moduli space. Let L(det) denote the determinant of cohomology line bundle over M(r , ξ ). Denote by L(ξ ) the real point of Pic M(r , ξ ) such that the fibre of the line bundle L(ξ )C over M(r , ξ )C over the point corresponding to a vector bundle E is Hom (ξC , det E). The line bundle L(
Data Loading...
