Tame and Wild Projective Curves and Classification of Vector Bundles

Abstract

We propose a new method of classifying vector bundles on projective curves, especially singular ones, according to their "representation type." In particular, we prove that the classification problem of vector bundles, respectively of torsion-free sheaves, on projective curves is always finite, tame, or wild. We completely classify curves which are of finite, respectively tame, vector bundle type by their dual graph. Moreover, our methods yield a geometric description of all indecomposable vector bundles and torsion-free sheaves on finite and tame curves.  2001 Elsevier Science

Simple vector bundles on plane degenerations of an elliptic curve

“…An algebraic variety X is called vector-bundle-wild or vector-bundle-tame if the category VB X of vector bundles on X is wild or respectively tame (see [DG01]). …”

“…It was originally proposed in [DG01]; see also [BDG01] and [BBDG] for some further elaborations. The idea of this method can be explained as follows.…”

“…In the case of a cycle of N projective lines (Kodaira cycles I N ), the obtained matrix problem turns out to be representation-tame; see [Bon92] and [CB89]. As a result, it allows us to obtain a complete classification of indecomposable torsion-free sheaves on these genus one curves; see [DG01] and [BBDG].…”

“…They are called Gelfand problems or representations of bunches of chains (see [GP68,Bon92]). For an application of Gelfand problems to the classification of torsion-free sheaves on cycles of projective lines, we refer to [DG01] (see also [BBDG]). …”

EXAMPLES OF SMOOTH SURFACES IN ℙ3 WHICH ARE ULRICH-WILD

“…Inspired by an analogous classification for k-algebras of finite type, T. Drozd and G. M. Greuel proposed in [14] a classification of aCM varieties as…”