The minimal Tjurina number of irreducible germs of plane curve singularities

In this paper we give a positive answer to a question of Dimca and Greuel about the quotient between the Milnor and the Tjurina numbers for any irreducible germ of plane curve singularity. This result is based on a closed formula for the minimal Tjurina number of an equisingularity class in terms of the sequence of multiplicities of the strict transform along a resolution. The key points for the proof are previous results by Genzmer, Wall and Mattei.Comment: To appear in Indiana University Mathematics Journal. Minor changes. Improvement in Corollary

Equivariant Versions of Higher Order Orbifold Euler Characteristics

There are (at least) two different approaches to define an equivariant analogue of the Euler characteristic for a space with a finite group action. The first one defines it as an element of the Burnside ring of the group. The second approach emerged from physics and includes the orbifold Euler characteristic and its higher order versions. Here we give a way to merge the two approaches together defining (in a certain setting) higher order Euler characteristics with values in the Burnside ring of a group. We give Macdonald type equations for these invariants. We also offer generalized (“motivic”) versions of these invariants and formulate Macdonald type equations for them as well

Grothendieck ring of pairs of quasi-projective varieties

We define a Grothendieck ring of pairs of complex quasi-projective varieties (that is a variety and a subvariety). We describe $\lambda$-structures and a power structure on/over this ring. We show that the conjectual symmetric power of the projective line with several orbifold points described by A.Fonarev is consistent with the symmetric power of this line with points as a pair of varieties

John Willard Milnor, 1962 Fields Medal. (Spanish: John Willard Milnor, Medalla Fields 1962)

Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasFALSEpu

On the pre-lambda-ring structure on the Grothendieck ring of stacks and the power structures over it

We describe a pre-lambda-structure on the Grothendieck ring of stacks (originally studied by Torsten Ekedahl) and the corresponding power structures over it, discuss some of their properties and give some explicit formulae for the Kapranov zeta-function for some stacks. In particular, we show that the nth symmetric power of the class of the classifying stack BGL(1) of the group GL(1) coincides, up to a power of the class L of the affine line, with the class of the classifying stack BGL(n)

A formula for the Milnor Number

We give a formula for the Milnor number of a germ (X,0) subset of (C-n+1,0) defined by f=0, f=f(d)+f(d+k)+...epsilon C {x(0),...,x(n)}, and such that Sing(D) boolean AND Z (f(d+k)) = circle divide, where D=Z (f(d)) subset of P-C(n). We prove that the topological type of (X,0) is determined by the d+k-jet of f

Monodromy conjecture for some surface singularities

In this work we give a formula for the local Denef–Loeser zeta function of a superisolated singularity of hypersurface in terms of the local Denef–Loeser zeta function of the singularities of its tangent cone. We prove the monodromy conjecture for some surfaces singularities. These results are applied to the study of rational arrangements of plane curves whose Euler–Poincaré characteristic is three