957 research outputs found
Geometry of obstructed equisingular families of projective hypersurfaces
We study geometric properties of certain obstructed equisingular families of
projective hypersurfaces with emphasis on smoothness, reducibility, being
reduced, and having expected dimension.
In the case of minimal obstructness, we give a detailed description of such
families corresponding to quasihomogeneous singularities.
Next we study the behavior of these properties with respect to stable
equivalence of singularities. We show that under certain conditions,
stabilization of singularities ensures the existence of a reduced component of
expected dimension. For minimally obstructed families the whole family becomes
irreducible.
As an application we show that if the equisingular family of a projective
hypersurface H has a reduced component of expected dimension then the
deformation of H induced by the linear system |H| is complete with respect to
one-parameter deformations.Comment: 30 pages. v2: more detailed explanations. v3: minor corrections,
version to appear in the Journal of Pure and Applied Algebr
On the enumeration of complex plane curves with two singular points
We study equi-singular strata of plane curves with two singular points of
prescribed types. The method of the previous work [Kerner06] is generalized to
this case. In particular we consider the enumerative problem for plane curves
with two singular points of linear singularity types. First the problem for two
ordinary multiple points of fixed multiplicities is solved. Then the
enumeration for arbitrary linear types is reduced to the case of ordinary
multiple points and to the understanding of "merging" of singular points. Many
examples and numerical answers are given.Comment: 24 pages, the Mathematica file with explicit calculations is
attached. Some typos removed. To appear in the International Mathematics
Research Notice
Minimal generating sets of non-modular invariant rings of finite groups
It is a classical problem to compute a minimal set of invariant polynomial
generating the invariant ring of a finite group as an algebra. We present here
an algorithm for the computation of minimal generating sets in the non-modular
case. Apart from very few explicit computations of Groebner bases, the
algorithm only involves very basic operations, and is thus rather fast.
As a test bed for comparative benchmarks, we use transitive permutation
groups on 7 and 8 variables. In most examples, our algorithm implemented in
Singular works much faster than the one used in Magma, namely by factors
between 50 and 1000. We also compute some further examples on more than 8
variables, including a minimal generating set for the natural action of the
cyclic group of order 11 in characteristic 0 and of order 15 in characteristic
2.
We also apply our algorithm to the computation of irreducible secondary
invariants.Comment: 14 pages v3: Timings updated. One example adde
Some remarks on the planar Kouchnirenko's Theorem
We consider different notions of non-degeneracy, as introduced by
Kouchnirenko (NND), Wall (INND) and Beelen-Pellikaan (WNND) for plane curve
singularities and introduce the new notion of weighted
homogeneous Newton non-degeneracy (WHNND). It is known that the Milnor number
resp. the delta-invariant can be computed by explicit formulas
resp. from the Newton diagram of if is NND resp.
WNND. It was however unknown whether the equalities resp.
can be characterized by a certain non-degeneracy condition on
and, if so, by which one. We show that resp.
is equivalent to INND resp. WHNND and give some applications and interesting
examples related to the existence of "wild vanishing cycles". Although the
results are new in any characteristic, the main difficulties arise in positive
characteristic.Comment: 23 pages, 2 figures. Final versio
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