29 research outputs found
Plane curves of minimal degree with prescribed singularities
We prove that there exists a>0 such that for any integer d>2 and any
topological types S_1,...,S_n of plane curve singularities, satisfying
, there exists a reduced irreducible plane
curve of degree d with exactly n singular points of types S_1,...,S_n,
respectively. This estimate is optimal with respect to the exponent of d. In
particular, we prove that for any topological type S there exists an
irreducible polynomial of degree having a singular
point of type S.Comment: 33 pages, LaTeX 2e, corrected some typos, simplified proofs of Lemmas
3.1, 4.
Equianalytic and equisingular families of curves on surfaces
We consider flat families of reduced curves on a smooth surface S such that
each member C has the same number of singularities of fixed singularity types
and the corresponding (locally closed) subscheme H of the Hilbert scheme of S.
We are mainly concerned with analytic resp. topological singularity types and
give a sufficient condition for the smoothness of H (at C). Our results for
S=P^2 seem to be quite sharp for families of cuves of small degree d.Comment: LaTeX v 2.0