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.