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Abstract

Let $k$ be an algebraically closed field, ${\rm char}\, k=0$. Let $C$ be an irreducible nonsingular curve such that $rC=S\cap F$, $r\in \Bbb N$, where $S$ and $F$ are two surfaces and all the singularities of $F$ are of the form $z^3=x^{3s}-y^{3s}$, $s\in \Bbb N $. We prove that $C$ can never pass through such kind of singularities of a surface, unless $r=3a$, $a\in \Bbb N$. We study multiplicity-$r$ structures on varieties $r\in \Bbb N$. Let $Z$ be a reduced irreducible nonsingular $(n-1)$-dimensional variety such that $rZ=X\cap F$, where $X$ is a normal $n$-fold, $F$ is a $(N-1)$-fold in $\Bbb P^{N}$, such that $Z\cap \operatorname{Sing} (X)\ne \emptyset$. We study the singularities of $X$ through which $Z$ passes.