- Lineare Algebra I & II (2000)
- Inhalte der Grundvorlesungen Lineare Algebra I und II im Winter- und Sommersemester 1999/2000: Gruppen, Ringe, Körper, Vektorräume, lineare Abbildungen, Determinanten, lineare Gleichungssysteme, Polynomring, Eigenwerte, Jordansche Normalform, endlich-dimensionale Hilberträume, Hauptachsentransformation, multilineare Algebra, Dualraum, Tensorprodukt, äußeres Produkt, Einführung in Singular.
- Description of SINGULAR: A Computer Algebra System for Singularity Theory, Algebraic Geometry and Commutative Algebra (1999)
- On the classification of vector bundles on projective curves (1999)
- 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.
- Geometry of families of nodal curves on the blown-up projective plane (1999)
- 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.
- Castelnuvo Function, Zero-dimensional Schemes and Singular Plane Curves (1999)
- 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 .