A goal of this paper is a characterization of singularities according to a new invariant, Mather discrepancy. We also show some evidences convincing us that Mather discrepancy is a reasonable invariant in a view point of birational geometry.

Nous nous donnons, dans l’anneau des germes de fonctions holomorphes à l’origine de ${ℂ}^2$, une fonction $f$ définissant une singularité isolée et nous nous intéressons à l’équation $u{f}_x^{\prime }+v{f}_y^{\prime }=wf$, lorsque la fonction $w$ est donnée. Nous introduisons les multiplicités d’intersection relatives de $w$ et $f_y^{\prime }$ le long des branches de $f$ et nous étudions les solutions à l’aide de ces valuations. Grâce aux résultats ainsi démontrés, nous construisons explicitement une équation fonctionnelle vérifiée par $f$.

We investigate different concepts of modular deformations of germs of isolated singularities (infinitesimal, Artinian, formal). An obstruction calculus based on the graded Lie algebra structure of the tangent cohomology for modular dcformations is introduced. As the main result the characterisation of the maximal infinitesimally modular subgerm of the miniversal family as flattening stratum of the relative Tjurina module is extended from ICIS to space curve singularities.

Let A be a finitely generated associative algebra over an algebraically closed field. We characterize the finite-dimensional A-modules whose orbit closures are local hypersurfaces. The result is reduced to an analogous characterization for orbit closures of quiver representations obtained in Section 3.

We construct the generic component of the moduli space of the germs of Legendrian curves with generic plane projection topologically equivalent to a curve $y^n=x^m$.

Let $Y$ be an $(m+1)$-dimensional irreducible smooth complex projective variety embedded in a projective space. Let $Z$ be a closed subscheme of $Y$, and $\delta$ be a positive integer such that ${ℐ}_{Z,Y}\left(\delta \right)$ is generated by global sections. Fix an integer $d\ge \delta +1$, and assume the general divisor $X\in |H^0(Y,{ℐ}_{Z,Y}\left(d\right))|$ is smooth. Denote by $H^m{(X;\mathbb{Q})}_{\perp Z}^{\mathrm{van}}$ the quotient of $H^m(X;\mathbb{Q})$ by the cohomology of $Y$ and also by the cycle classes of the irreducible components of dimension $m$ of $Z$. In the present paper we prove that the monodromy representation on $H^m{(X;\mathbb{Q})}_{\perp Z}^{\mathrm{van}}$ for the family of smooth...

To a given analytic function germ $f:({ℝ}^d,0)\to (ℝ,0)$, we associate zeta functions $Z_{f,+}$, $Z_{f,-}\in \mathbb{Z}\left[\left[T\right]\right]$, defined analogously to the motivic zeta functions of Denef and Loeser. We show that our zeta functions are rational and that they are invariants of the blow-analytic equivalence in the sense of Kuo. Then we use them together with the Fukui invariant to classify the blow-analytic equivalence classes of Brieskorn polynomials of two variables. Except special series of singularities our method classifies as well the blow-analytic...