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On the structure of moduli spaces of framed vector bundles on rational and ruled surfaces. (English) Zbl 0951.14028
Pragacz, Piotr (ed.) et al., Algebraic geometry: Hirzebruch 70. Proceedings of the algebraic geometry conference in honor of F. Hirzebruch’s 70th birthday, Stefan Banach International Mathematical Center, Warszawa, Poland, May 11-16, 1998. Providence, RI: American Mathematical Society. Contemp. Math. 241, 239-271 (1999).
The paper studies moduli of vector bundles on ruled surfaces in a quite particular situation: Let \(f:S\rightarrow B\) be a compact surface obtained by blowing up a finite set of distinct points on a ruled surface over the curve \(B\) and let \(N\) be a section of \(S\). The object under consideration are the pairs \((V,\phi)\) where \(V\) is a vector bundle on \(S\) and \(\phi\) an isomorphism (framing) \(V|_N \cong {\mathcal O}^n_N\), such that \(\bigwedge ^n V\cong {\mathcal O}_S\) and \(V_F\) is trivial on almost every fiber \(F\) of \(S\).
The existence of a moduli space \({\mathfrak M}={\mathfrak M}_{S,N}(n,m)\) (where \(m=c_2(V)\)) is proved, and such space turns out to be smooth.
There is a natural morphism \(\delta : {\mathfrak M}\rightarrow B^{(m)}\) (symmetric product), and the study of this morphism (mainly of its fibers) is studied in detail. It is shown that the fibers of \(\delta\) are products of spaces of framed local jumps, i.e. germs of bundles along a fiber of the ruling.
The study of this subject is also motivated by its relation with the study of moduli of instantons on connected sums of projective planes.
For the entire collection see [Zbl 0924.00033].

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14J26 Rational and ruled surfaces
14J10 Families, moduli, classification: algebraic theory