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Zbl 0764.14008
Veys, W.
Congruences for numerical data of an embedded resolution.
(English)
[J] Compos. Math. 80, No.2, 151-169 (1991). ISSN 0010-437X; ISSN 1570-5846/e

The author considers the embedded resolution $h:X\to X\sb 0$ of the singularities of an hypersurface $Y$ in the affine space $X=\bbfA\sp{n+1}$. Let $Y\sb i\sp{(r)}$, $i\in I$, be the strict transforms of the irreducible components of $Y$ and $E\sb i\sp{(r)}$, $1\le i\le r$, be the irreducible components of the exceptional divisor, then $(\bigcup\sb{i\in I}Y\sb i\sp{(r)})\cup(\bigcup\sp r\sb{i=1}E\sb i\sp{(r)})$ is a normal crossings divisor on $X$. The numerical data $(N\sb i,\nu\sb i)$ are defined by: $h\sp{-1}(Y)=\sum\sb{i\in I}N\sb iY\sb i\sp{(r)}+\sum\sp r\sb{i=1}N\sb iE\sb i\sp{(r)}$ and $K\sb X=h\sp{- 1}(K\sb{X\sb 0})+\sum\sb{i\in I}(\nu\sb i-1)Y\sb i\sp{(r)}+\sum\sp r\sb{i=1}(\nu\sb i-1)E\sb i\sp{(r)}$.\par Fix one exceptional curve $E$ with numerical data $(N,\nu)$. Let $E\sb j\sp{(r)}$ be an exceptional divisor, and let $E\sb i\sp{(r)}$, $i\in T$, $T=I\cup\{1,\dots,r\}$, be the components of $h\sp{-1}(Y)$ which intersect $E$ and appear before $E\sb j\sp{(r)}$'' in the resolution process then the author gives some congruence relations between the numerical data $(N\sb i)$ and $(\nu\sb i)$, $i\in T$, which generalize the relations when $Y$ is a plane curve to any arbitrary $Y\subset A\sp{n+1}$. --- To get these relations the author looks at the succession of blowing-ups $g\sb i:X\sb{i+1}\to X\sb i$ with non-singular center $D\sb i$ such that the map $X=X\sb r\to X\sb{r-1}\to\dots\to X\sb 0$ is the embedded resolution. More precisely, he considers the strict transforms $E\sb j\sp{(i)}$ of the exceptional divisor $E=E\sb j\sp{(j)}$ of $g\sb{j-1}$, i.e. $E=g\sp{-1}\sb{j-1}(D)$ with $D=D\sb{j-1}$, and the blowing-up $E\sb j\sp{(i+1)}\to E\sb j\sp{(i)}$.\par Let $E\sb i\sp{(r)}$, $i\in T$, be the irreducible components of $h\sp{- 1}(Y)$ such that ${\cal E}\sb i=E\sb i\sp{(j)}\cap E$ are the irreducible components of $E\cap(h\sp{-1}(Y)\setminus E\sb j)$, and let $d\sb i$ be the degree of the cycle ${\cal E}\sb i\cdot F$ on the general fibre $F=\bbfP\sp{k-1}$ of $\Pi=g\sb{j-1\sb{\vert E}}$: $E\to D$, with $k=n+1- \dim D$. Then we get the congruences:\par (B1) $\sum\sb{i\in T}d\sb iN\sb i\equiv 0\pmod N$.\par (B$1'$) $\sum\sb{i\in T}d\sb i(\nu\sb i-1)+k\equiv 0\pmod\nu$.\par If $d\sb i=0$ there exists a divisor $B\sb i$ on $D$ such that ${\cal E}\sb i=\Pi\sp{-1}(B\sb i)$, and we get:\par (B2) $\sum\sb{i\in T,d\sb i\ne 0}N\sb i\Pi\sb *({\cal E}\sb i\sp k)/d\sb i\sp{k-1}+\sum\sb{i\in T,d\sb i=0}N\sb iB\sb i=0$ in $\text{Pic} D/N \text{Pic} D$.\par (B$2'$) $\sum\sb{i\in T,d\sb i\ne 0}(\nu\sb i-1)\Pi\sb *({\cal E}\sb i\sp k)/d\sb i\sp{k-1}+\sum\sb{i\in T,d\sb i=0}(\nu\sb i-1)B\sb i-kK\sb D=0$ in $\text{Pic} D/\nu \text{Pic} D$.\par If $\mu\sb t$ is the multiplicity of the generic point of $D\sb i$ on the strict transform ${\cal E}\sb t\sp{(i)}$ of ${\cal E}\sb t$ on $E\sb j\sp{(i)}$, $j\le i<r$, the author gets also the congruences:\par (A) $N\sb{i+1}\equiv\sum\sb{t\in T\cup\{1,\dots,i\}}\mu\sb tN\sb t\pmod N$\par (A$'$) $\nu\sb{i+1}\equiv\sum\sb{t\in T\cup\{1,\dots,i\}}\mu\sb t(\nu\sb t-1)+(k-1)\pmod\nu$.
[M.Vaquie (Paris)]
MSC 2000:
*14E15 Global theory of singularities
14J17 Singularities of surfaces

Keywords: embedded resolution of singularities of an hypersurface; exceptional divisor

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