The Bruce-Roberts Numbers of 1-Forms on an ICIS
We relate the Bruce-Roberts numbers of a 1-form with respect to an ICIS to other invariants as the GSV-index, Tjurina and Milnor numbers.
math.AG↗
arXiv subjects
Bruna Oréfice-Okamoto
Publications and source records attributed to Bruna Oréfice-Okamoto.
The Bruce-Roberts Numbers of a Function on an ICIS
We give formulas for the Bruce-Roberts number $μ_{BR}(f,X)$ and its relative version $μ_{BR}^{-}(f,X)$ of a function $f$ with respect to an ICIS $(X,0)$. We show that $μ_{BR}^{-}(f,X)=μ(f^{-1}(0)\cap X,0)+μ(X,0)-τ(X,0)$, where $μ$ and $τ$ are the Milnor and Tjurina numbers, respectively, of the ICIS. The formula for $μ_{BR}(f,X)$ is more complicated and also involves $μ(f)$ and some lengths in terms of the ideals $I_X$ and $Jf$. We also consider the logarithmic characteristic variety, $LC(X)$, and its relative version, $LC(X)^{-}$. We show that $LC(X)^{-}$ is Cohen-Macaulay and that $LC(X)$ is Cohen-Macaulay at any point not in $X\times\{0\}$. We generalize previous results presented by the authors when $(X,0)$ has codimension one and by Bruce and Roberts when it is weighted homogeneous of any codimension.
math.AG↗
Families of ICIS with constant total Milnor number
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 Gabrielov, 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.
math.AG↗
Equisingularity of families of functions on isolated determinantal singularities
We study the equisingularity of a family of function germs $\{f_t\colon(X_t,0)\to (\mathbb{C},0)\}$, where $(X_t,0)$ are $d$-dimensional isolated determinantal singularities. We define the $(d-1)$th polar multiplicity of the fibers $X_t\cap f_t^{-1}(0)$ and we show how the constancy of the polar multiplicities is related to the constancy of the Milnor number of $f_t$ and the Whitney equisingularity of the family.
math.AG↗
The Bruce-Roberts number of a function on a hypersurface with isolated singularity
Let $(X,0)$ be an isolated hypersurface singularity defined by $ϕ\colon(\mathbb C^n,0)\to(\mathbb C,0)$ and $f\colon(\mathbb C^n,0)\to\mathbb C$ such that the Bruce-Roberts number $μ_{BR}(f,X)$ is finite. We first prove that $μ_{BR}(f,X)=μ(f)+μ(ϕ,f)+μ(X,0)-τ(X,0)$, where $μ$ and $τ$ are the Milnor and Tjurina numbers respectively of a function or an isolated complete intersection singularity. Second, we show that the logarithmic characteristic variety $LC(X,0)$ is Cohen-Macaulay. Both theorems generalize the results of a previous paper by some of the authors, in which the hypersurface $(X,0)$ was assumed to be weighted homogeneous.
math.AG↗