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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 +0200Geometry of families of nodal curves on the blown-up projective plane
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/779
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.Gert-Martin Greuel; Christop Lossen; Eugenii Shustinpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/779Mon, 03 Apr 2000 00:00:00 +0200New asymptotics in the geometry of equisingular families of curves
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/781
Gert-Martin Greuel; Christop Lossen; Eugenii Shustinpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/781Mon, 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 +0200Geometry of Equisingular Families of Curves
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/785
Singular algebraic curves, their existence, deformation, families (from the local and global point of view) attract continuous attention of algebraic geometers since the last century. The aim of our paper is to give an account of results, new trends and bibliography related to the geometry of equisingular families of algebraic curves on smooth algebraic surfaces over an algebraically closed field of characteristic zero. This theory is founded in basic works of Plücker, Severi, Segre, Zariski, and has tight links and finds important applications in singularity theory, topology of complex algebraic curves and surfaces, and in real algebraic geometry.Gert-Martin Greuel; Eugenii Shustinpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/785Mon, 03 Apr 2000 00:00:00 +0200