Search results for "singularity."
showing 10 items of 346 documents
Singularities of lightlike hypersurfaces in Minkowski four-space
2006
We classify singularities of lightlike hypersurfaces in Minkowski 4-space via the contact invariants for the corresponding spacelike surfaces and lightcones.
On stability of logarithmic tangent sheaves. Symmetric and generic determinants
2021
We prove stability of logarithmic tangent sheaves of singular hypersurfaces D of the projective space with constraints on the dimension and degree of the singularities of D. As main application, we prove that determinants and symmetric determinants have stable logarithmic tangent sheaves and we describe an open dense piece of the associated moduli space.
Cluster tilting for one-dimensional hypersurface singularities
2008
In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with the representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological methods, using higher almost split sequences and results from birational geometry. We obtain a large class of 2-CY tilted algebras which are finite dimensional symmetric and satisfy $\tau^2=\id$. In particular, we compute 2-CY tilted algebras for simple and minimally elliptic curve singularities.
On a Theorem of Greuel and Steenbrink
2017
A famous theorem of Greuel and Steenbrink states that the first Betti number of the Milnor fibre of a smoothing of a normal surface singularity vanishes. In this paper we prove a general theorem on the first Betti number of a smoothing that implies an analogous result for weakly normal singularities.
Stable Images and Discriminants
2020
We show that the discriminant/image of a stable perturbation of a germ of finite \(\mathcal {A}\)-codimension is a hypersurface with the homotopy type of a wedge of spheres in middle dimension, provided the target dimension does not exceed the source dimension by more than one. The number of spheres in the wedge is called the discriminant Milnor number/image Milnor number. We prove a lemma showing how to calculate this number, and show that when the target dimension does not exceed the source dimension, the discriminant Milnor number and the \(\mathcal {A}\)-codimension obey the “Milnor–Tjurina relation” familiar in the case of isolated hypersurface singularities. This relation remains conj…
Flat lightlike hypersurfaces in Lorentz–Minkowski 4-space
2009
Abstract The lightlike hypersurfaces in Lorentz–Minkowski space are of special interest in Relativity Theory. In particular, the singularities of these hypersurfaces provide good models for the study of different horizon types. We introduce the notion of flatness for these hypersurfaces and study their singularities. The classification result asserts that a generic classification of flat lightlike hypersurfaces is quite different from that of generic lightlike hypersurfaces.
Families of ICIS with constant total Milnor number
2021
We show that a family of isolated complete intersection singularities (ICIS) with constant total Milnor number has no coalescence of singularities. This extends a well-known result of Gabriélov, Lazzeri and Lê for hypersurfaces. We use A’Campo’s theorem to see that the Lefschetz number of the generic monodromy of the ICIS is zero when the ICIS is singular. We give a pair applications for families of functions on ICIS which extend also some known results for functions on a smooth variety.
Logarithmic Vector Fields and the Severi Strata in the Discriminant
2017
The discriminant, D, in the base of a miniversal deformation of an irreducible plane curve singularity, is partitioned according to the genus of the (singular) fibre, or, equivalently, by the sum of the delta invariants of the singular points of the fibre. The members of the partition are known as the Severi strata. The smallest is the δ-constant stratum, D(δ), where the genus of the fibre is 0. It is well known, by work of Givental’ and Varchenko, to be Lagrangian with respect to the symplectic form Ω obtained by pulling back the intersection form on the cohomology of the fibre via the period mapping. We show that the remaining Severi strata are also co-isotropic with respect to Ω, and mor…
Invariants of unipotent groups
1987
I’ll give a survey on the known results on finite generation of invariants for nonreductive groups, and some conjectures. You know that Hilbert’s 14th problem is solved for the invariants of reductive groups; see [12] for a survey. So the general case reduces to the case of unipotent groups. But in this case there are only a few results, some negative and some positive. I assume that k is an infinite field, say the complex numbers, but in most instances an arbitrary ring would do it.
A Property on Singularities of NURBS Curves
2002
We prove that if a.n open Non Uniform Rational B-Spline curve of order k has a singular point, then it belongs to both curves of order k - 1 defined in the k - 2 step of the de Boor algorithm. Moreover, both curves are tangent at the singular point.