Deformation and Smoothing of Singularities

On the normal sheaf of Gorenstein curves

“…Furthermore, [LS67, Lemma 3.1.2] assures that the 𝒪 𝐶 -module T 1 can be taken as the cokernel of the map 𝒯 𝐶/P 𝑟 → 𝒩 𝐶/P 𝑟 . Given a singular point 𝑃 ∈ 𝐶, 𝜏 𝑃 := dim 𝑘 (T 1 𝑃 ) stands for the Tjurina number at 𝑃 ∈ 𝐶, see [Gre20] for the connection between the Tjurina algebra and the cotangent complex of an analytic germ. As an immediate consequence of Lemma 4.1.1, we get deg(T 1 ) = ∑︁ 𝑃 ∈Sing(𝐶) 𝜏 𝑃 =: 𝜏, that is the so-called Tjurina number of 𝐶.…”

“…The notion of a deformation preserving the singularity degree appears in the literature when we (formally) deform an analytic germ of a singularity. In a survey, [Gre20] by Greuel, the author shows that the singularity degree of an analytic germ is preserved if, and only if, it is contractible, cf. [Gre20,Corollary 2.46].…”

“…In a survey, [Gre20] by Greuel, the author shows that the singularity degree of an analytic germ is preserved if, and only if, it is contractible, cf. [Gre20,Corollary 2.46]. We also should mention a result due to Teissier, cf.…”

On the quotient of Milnor and Tjurina numbers for two‐dimensional isolated hypersurface singularities

“…In this way, the difference $$\mu -\tau$$ or alternatively the quotient $$\mu /\tau$$ must be considered as a measure of the non‐quasihomogeneity of the singularity and not as a measure of the exactness of the Poincaré complex. Remark We refer to [14, Sections 7.2.4, 7.2.6] for a survey by Greuel on Milnor versus Tjurina number, not only for complete intersections, but also for reduced curve singularities.…”