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Vector bundles of type \(T_ 3\) over a curve. (English) Zbl 0823.14018

The author introduces the following definition: a semistable vector bundle \(E\) over a curve is of type \(T_ r\) if \(\text{End} (E)\) has dimension \(r\) and there exists a stable vector bundle \(F\) with \(\mu (E)= \mu (F)\) together with a filtration \(0= E_ 0 \subset E_ 1 \subset \ldots \subset E_ r= E\) such that \(E_ i/ E_{i-1} \cong F\) for all \(i= 1,\ldots, r\). Vector bundles of type \(T_ 1\) are exactly the stable ones, and it is well-known that they have (always fixing the rank and the degree) a moduli space that is a quasiprojective variety of dimension \(n^ 2 (g-1)+1\) (\(n\) being the rank). The idea for defining these new types of bundles is that they have nice moduli spaces, while just the set of isomorphism classes of strictly semistable bundles do not have a good structure.
More precisely, the author shows that the set of (isomorphism classes of) vector bundles of type \(T_ 2\) has a structure of projective bundle over the moduli space of stable vector bundles (of suitable rank and degree). In the same way, vector bundles of type \(T_ 3\) with algebra of endomorphisms isomorphic to \(\mathbb{C} [t, s]/ (t,s)^ 2\) have a moduli space which consists of two Grassmann bundles over the moduli space of stable vector bundles. Finally, the author constructs a moduli space for vector bundles of type \(T_ 3\) with algebra of endomorphisms isomorphic to \(\mathbb{C} [t]/ (t^ 3)\). In all cases she computes the dimension and shows when there exists a Poincaré bundle.
Reviewer: E.Arrondo (Madrid)

MSC:

14H60 Vector bundles on curves and their moduli
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14H10 Families, moduli of curves (algebraic)
14D20 Algebraic moduli problems, moduli of vector bundles
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