A Note on Rank Two Stable Bundles Over Surfaces
Abstract: Abstract. Let π : X → C be a fibration with integral fibers over a curve C and consider a polarization H on the surface X. Let E be a stable vector bundle of rank 2 on C. We prove that the pullback π*(E) is a H-stable bundle over X. This result allows us to relate the corresponding moduli spaces of stable bundles $${{\mathcal M}_C}(2,d)$$ and $${{\mathcal M}_{X,H}}(2,df,0)$$ through an injective morphism. We study the induced morphism at the level of B… Show more
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Cited by 2 publications
References 19 publications
Brill–Noether Theory of Stable Vector Bundles on Ruled Surfaces
Let X be a ruled surface over a nonsingular curve C of genus $$g\ge 0.$$ g ≥ 0 . Let $$M_H:=M_{X,H}(2;c_1,c_2)$$ M H : = M X , H ( 2 ; c 1 , c 2 ) be the moduli space of H-stable rank 2 vector bundles E on X with fixed Chern classes $$c_i:=c_i(E)$$ c i : = c i ( E ) for $$i=1,2.$$ i = 1 , 2 . The main goal of this paper is to contribute to a better understanding of the geometry of the moduli space $$M_H$$ M H in terms of its Brill–Noether locus $$W_H^k(2;c_1,c_2),$$ W H k ( 2 ; c 1 , c 2 ) , whose points correspond to stable vector bundles in $$M_H$$ M H having at least k independent sections. We deal with the non-emptiness of this Brill–Noether locus, getting in most of the cases sharp bounds for the values of k such that $$W_H^k(2;c_1,c_2)$$ W H k ( 2 ; c 1 , c 2 ) is non-empty.
Brill–Noether Theory of Stable Vector Bundles on Ruled Surfaces
Let X be a ruled surface over a nonsingular curve C of genus $$g\ge 0.$$ g ≥ 0 . Let $$M_H:=M_{X,H}(2;c_1,c_2)$$ M H : = M X , H ( 2 ; c 1 , c 2 ) be the moduli space of H-stable rank 2 vector bundles E on X with fixed Chern classes $$c_i:=c_i(E)$$ c i : = c i ( E ) for $$i=1,2.$$ i = 1 , 2 . The main goal of this paper is to contribute to a better understanding of the geometry of the moduli space $$M_H$$ M H in terms of its Brill–Noether locus $$W_H^k(2;c_1,c_2),$$ W H k ( 2 ; c 1 , c 2 ) , whose points correspond to stable vector bundles in $$M_H$$ M H having at least k independent sections. We deal with the non-emptiness of this Brill–Noether locus, getting in most of the cases sharp bounds for the values of k such that $$W_H^k(2;c_1,c_2)$$ W H k ( 2 ; c 1 , c 2 ) is non-empty.
Spin(8,C)-Higgs pairs over a compact Riemann surface
Let X X be a compact Riemann surface of genus g ≥ 2 g\ge 2 , G G be a semisimple complex Lie group and ρ : G → GL ( V ) \rho :G\to {\rm{GL}}\left(V) be a complex representation of G G . Given a principal G G -bundle E E over X X , a vector bundle E ( V ) E\left(V) whose typical fiber is a copy of V V is induced. A ( G , ρ ) \left(G,\rho ) -Higgs pair is a pair ( E , φ ) \left(E,\varphi ) , where E E is a principal G G -bundle over X X and φ \varphi is a holomorphic global section of E ( V ) ⊗ L E\left(V)\otimes L , L L being a fixed line bundle over X X . In this work, Higgs pairs of this type are considered for G = Spin ( 8 , C ) G={\rm{Spin}}\left(8,{\mathbb{C}}) and the three irreducible eight-dimensional complex representations which Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) admits. In particular, the reduced notions of stability, semistability, and polystability for these specific Higgs pairs are given, and it is proved that the corresponding moduli spaces are isomorphic, and a precise expression for the stable and not simple Higgs pairs associated with one of the three announced representations of Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) is described.
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