Consider a projective variety $X \subset \mathbb{ P}^n$ (over an algebraically closed field of characteristic zero), together with a (reduced) simple normal crossings divisor $E \subset \mathbb{ P}^n$, where the degrees of both $X$ and $E$ are at most $d$. We show there is a pair $(n',d')$ which can be explicitly computed in terms of $(n,d)$, such that $(X,E)$ has a log resolution of singularities $(X',E')$, where $(X',E')$ can be embedded in $\mathbb{ P}^{ n'}$ and both $X'$ and $E'$ have degrees at most $d'$ in $\mathbb{ P}^{ n'}$.
Categories: math.AG cs.IT math.CV math.IT