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The Bruce-Roberts Numbers of a Function on an ICIS
Algebraic Geometry · Submitted 21 Mar 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We give formulas for the Bruce-Roberts number $\mu_{BR}(f,X)$ and its relative version $\mu_{BR}^{-}(f,X)$ of a function $f$ with respect to an ICIS $(X,0)$.
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We give formulas for the Bruce-Roberts number $\mu_{BR}(f,X)$ and its relative version $\mu_{BR}^{-}(f,X)$ of a function $f$ with respect to an ICIS $(X,0)$. We show that $\mu_{BR}^{-}(f,X)=\mu(f^{-1}(0)\cap X,0)+\mu(X,0)-\tau(X,0)$, where $\mu$ and $\tau$ are the Milnor and Tjurina numbers, respectively, of the ICIS. The formula for $\mu_{BR}(f,X)$ is more complicated and also involves $\mu(f)$ 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We give formulas for the Bruce-Roberts number $\mu_{BR}(f,X)$ and its relative version $\mu_{BR}^{-}(f,X)$ of a function $f$ with respect to an ICIS $(X,0)$. | contribution 0.80 | 0.96 | 0.50 | asserted 1.00 |
| s1 | We show that $\mu_{BR}^{-}(f,X)=\mu(f^{-1}(0)\cap X,0)+\mu(X,0)-\tau(X,0)$, where $\mu$ and $\tau$ are the Milnor and Tjurina numbers, respectively, of the ICIS. | result 0.99 | 0.94 | 0.43 | asserted 1.00 |
| s2 | The formula for $\mu_{BR}(f,X)$ is more complicated and also involves $\mu(f)$ and some lengths in terms of the ideals $I_X$ and $Jf$. | result 0.84 | 0.92 | 0.14 | asserted 1.00 |
| s3 | We also consider the logarithmic characteristic variety, $LC(X)$, and its relative version, $LC(X)^{-}$. | contribution 0.99 | 0.76 | 0.04 | asserted 0.69 |
| s4 | We show that $LC(X)^{-}$ is Cohen-Macaulay and that $LC(X)$ is Cohen-Macaulay at any point not in $X\times\{0\}$. | result 0.99 | 0.94 | 0.12 | asserted 1.00 |
| s5 | 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. | contribution 0.80 | 0.70 | 0.12 | asserted 1.00 |
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Original abstract
We give formulas for the Bruce-Roberts number $\mu_{BR}(f,X)$ and its relative version $\mu_{BR}^{-}(f,X)$ of a function $f$ with respect to an ICIS $(X,0)$. We show that $\mu_{BR}^{-}(f,X)=\mu(f^{-1}(0)\cap X,0)+\mu(X,0)-\tau(X,0)$, where $\mu$ and $\tau$ are the Milnor and Tjurina numbers, respectively, of the ICIS. The formula for $\mu_{BR}(f,X)$ is more complicated and also involves $\mu(f)$ 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.
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How the literature uses it
2 arXiv papers cite this in 9 sentences
Papers citing it · 6 sentences without a confident label
Cites it as support 1 paper
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Since $f_2$ has isolated singularity at the origin, $X∩{f_2=0}$ is an ICIS and $𝒱={W∖{f_2=0}, W∩{f_2=0}∖{0}, {0}, W∈𝒲}$ is a good stratification of $X$ induced by $f_2.$ In [ref] , the authors proved that the relative logarithmic characteristic variety $LC(X∩{f_2=0})^-$ of $X∩{f_2=0}$ is Cohen-Macaulay, since $X∩{f_2=0}$ is an ICIS.
Relative Bruce-Roberts number and Chern obstruction · The particular case of an ICIS · 0.97
Cites it as background 2 papers
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We indicate the following recent papers for these formulas, [ref] , when $(X,0)$ is a weighted homogeneous hypersurface with isolated singularity, [ref] and [ref] for any Isolated Hypersurface Singularity (IHS), and [ref] for any ICIS (see also [ref] ).
Relative Bruce-Roberts number and Chern obstruction · Introduction · 0.88
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[ref] , [ref] , [ref] , [ref] , [ref] , [ref] - [ref] , [ref] ) it has been called the Bruce-Roberts Milnor number of $(f,X)$ (or of $f$ with respect to $X$).
Bruce-Roberts Numbers and Quasihomogeneous Functions on Analytic Varieties · Introduction-Main Results · 0.99
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