We consider the "representation type" of the classification problem of vector bundles on a projective curve. We prove that this problem is always either finite, or tame, or wild and we completely describe those curves which are of finite, resp. tame, vector bundle type. We also give a complete list of indecomposable vector bundles for the finite and tame cases.
Let P2r be the projective plane blown up at r generic points. Denote by E0; E1; : : : ; Er the strict transform of a generic straight line on P2 and the exceptional divisors of the blown-up points on P2r respectively. We consider the variety Virr of all irreducible curves C with k nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness, irreducibility and non-emptiness. Moreover, we extend our conditions for the smoothness and the irreducibility on families of reducible curves. For r ^ 9 we give the complete answer concerning the existence of nodal curves in Virr.
We study families V of curves in P2(C) of degree d having exactly r singular points of given topological or analytic types. We derive new sufficient conditions for V to be T-smooth (smooth of the expected dimension), respectively to be irreducible. For T-smoothness these conditions involve new invariants of curve singularities and are conjectured to be asymptotically proper, i.e., optimal up to a constant factor. To obtain the results, we study the Castelnuovo function, prove the irreducibility of the Hilbert scheme of zero-dimensional schemes associated to a cluster of infinitely near points of the singularities and deduce new vanishing theorems for ideal sheaves of zero-dimensional schemes in P2. Moreover, we give a series of examples of cuspidal curves where the family V is reducible, but where ss1(P2nC) coincides (and is abelian) for all C 2 V .
After the notion of Gröbner bases and an algorithm for constructing them was introduced by Buchberger [Bu1, Bu2] algebraic geometers have used Gröbner bases as the main computational tool for many years, either to prove a theorem or to disprove a conjecture or just to experiment with examples in order to obtain a feeling about the structure of an algebraic variety. Nontrivial problems coming either from logic, mathematics or applications usually lead to nontrivial Gröbner basis computations, which is the reason why several improvements have been provided by many people and have been implemented in general purpose systems like Axiom, Maple, Mathematica, Reduce, etc., and systems specialized for use in algebraic geometry and commutative algebra like CoCoA, Macaulay and Singular. The present paper starts with an introduction to some concepts of algebraic geometry which should be understood by people with (almost) no knowledge in this field. In the second chapter we introduce standard bases (generalization of Gr"obner bases to non-well-orderings), which are needed for applications to local algebraic geometry (singularity theory), and a method for computing syzygies and free resolutions. The last chapter describes a new algorithm for computing the normalization of a reduced affine ring and gives an elementary introduction to singularity theory. Then we describe algorithms, using standard bases, to compute infinitesimal deformations and obstructions, which are basic for the deformation theory of isolated singularities. It is impossible to list all papers where Gr"obner bases have been used in local and global algebraic geometry, and even more impossible to give an overview about these contributions. We have, therefore, included only references to papers mentioned in this tutorial paper. The interested reader will find many more in the other contributions of this volume and in the literature cited there.