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Mon, 03 Apr 2000 00:00:00 +0200Mon, 03 Apr 2000 00:00:00 +0200Plane curves of minimal degree with prescribed singularities
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/757
We prove that there exists a positive \(\alpha\) such thatfor any integer \(\mbox{$d\ge 3$}\) and any topological types \(\mbox{$S_1,\dots,S_n$}\) of plane curve singularities, satisfying \(\mbox{$\mu(S_1)+\dots+\mu(S_n)\le\alpha d^2$}\), there exists a reduced irreducible plane curve of degree \(d\) with exactly \(n\) singular points of types \(\mbox{$S_1,\dots,S_n$}\), respectively. This estimate is optimal with respect to theexponent of \(d\). In particular, we prove that for any topological type \(S\) there exists an irreducible polynomial of degree \(\mbox{$d\le 14\sqrt{\mu(S)}$}\) having a singular point of type \(S\).Gert-Martin Greuel; Christoph Lossen; Eugenii Shustinarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/757Mon, 03 Apr 2000 00:00:00 +0200Castelnuvo Function, Zero-dimensional Schemes and Singular Plane Curves
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/782
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 .Gert-Martin Greuel; Christoph Lossen; Eugenii Shustinpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/782Mon, 03 Apr 2000 00:00:00 +0200Equianalytic and equisingular families of curves on surfaces
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/808
Gert-Martin Greuel; Christoph Lossenpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/808Mon, 03 Apr 2000 00:00:00 +0200