Quasi-unipotency is a well defined notion at any point of the discriminant. If we have a proper family $f : X \to S$ of varieties with a smooth total space and a smooth base, and if $p \in D \subset S$ is a point of the discriminant, then we say that the local monodromy of the family near $p$ is quasi-unipotent if we can find a small analytic neighborhood $p \in U \subset S$ of $p$ in $S$, so that if $o \in U - D$ is a base point, then the monodromy representation '$mon$mon : \pi_{1}(U-D,o) \to GL(H^{i}(X_{o},\mathbb{C})$' has an image whose Zariski closure$G$is a unipotent quasi-unipotent linear algebraic group (that is, the quotient of$G$by its unipotent radical is a finite-group). In general it is rare for the local monodromy to be quasi-unipotent. If$p$happens to be a very singular point of the discriminant, then the local monodromy tends to be big and is often as big as it can be, and not quasi-unipotent at all. However, if$p$is at worst a normal crossing singularity of$D$, then the local monodromy is quasi-unipotent. 1 Quasi-unipotency is a well defined notion at any point of the discriminant. If we have a proper family$f : X \to S$of varieties with a smooth total space and a smooth base, and if$p \in D \subset S$is a point of the discriminant, then we say that the local monodromy of the family near$p$is quasi-unipotent if we can find a small analytic neighborhood$p \in U \subset S$of$p$in$S$, so that if$o \in U - D$is a base point, then the monodromy representation '$mon : \pi_{1}(U-D,o) \to GL(H^{i}(X_{o},\mathbb{C})$' has an image whose Zariski closure$G$is a unipotent linear algebraic group (that is, the quotient of$G$by its unipotent radical is a finite-group). In general it is rare for the local monodromy to be quasi-unipotent. If$p$happens to be a very singular point of the discriminant, then the local monodromy tends to be big and is often as big as it can be, and not quasi-unipotent at all. However, if$p$is at worst a normal crossing singularity of$D\$, then the local monodromy is quasi-unipotent.