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Abstract

Let $k$ an algebraically closed field, ${\rm char}\, k=0$. We study multiplicity-$r$ structures on varieties for $r\in \mathbb N$, $r\ge 2$. Let $Z$ be a reduced irreducible nonsingular $(N-2)$-dimensional variety such that $rZ=X\cap F$, where $X$ is a normal $(N-1)$-fold of degree $n$, $F$ is a smooth $(N-1)$-fold of degree $m$ in $\mathbb P^{N}$, such that $r\in \mathbb N$, $r\ge 2$, $Z\cap \text {Sing} (X)\not =\emptyset $. There are effective divisors $V$ and $D_1$ on $Z$ such that $O_{Z}(V-(r-1)D_1)\simeq {\omega _{Z}}^{r}(-rm-n+(N+1)r)$, where $\omega _{Z}$ is the canonical sheaf of $Z$. Let $Z \subset \mathbb P^{N}$ be a reduced irreducible subvariety of codimension 2. Let $Y$ be an irreducible hypersurface in $\mathbb P^{N}$, $Z \subset Y$. Let ${\omega ^{o}}_{Z}$ be the dualizing sheaf of $Z$. Then, there exists a hypersurface $X$ in $\mathbb P^{N}$ such that $Z=Y\cap X$ is a scheme-theoretical complete intersection if and only if

$\bullet $ ${\omega ^{o}}_{Z}\simeq \omega _{\mathbb P^{N}}\otimes {\wedge }^2{\mathcal N }_{Z}|_{\mathbb P^{N}}$.

$\bullet $ $\text {deg}\,Y$ divides $\text {deg}\,Z $.

$\bullet $ ${\omega ^{o}}_{Z}\simeq O_{Z} (\text {deg}\,Y+(\frac {\text {deg}\,Z}{\text {deg}\,Y})-N-1)$.