# Positivity of braid monodromy of curve singularities

I recall the notion of braid monodromy. Let $C \subset \mathbb{C}^2$ be an algebraic curve, and choose a projection $\pi: \mathbb{C}^2 \to \mathbb{C}$ such that the restriction $\pi: C \to \mathbb{C}$ is finite of degree $n$. Let $B \subset \mathbb{C}$ be the branch divisor, i.e., the locus where the preimage consists of fewer than $n$ points. Let $\mathrm{Conf}_n(\mathbb{C})$ denote the space parameterizing subsets of $\mathbb{C}$ of cardinality $n$. Then there is a map

$\mathbb{C} \setminus B \to \mathrm{Conf}_n(\mathbb{C})$ given by $z \mapsto [\pi|_C^{-1}(z) \subset \pi^{-1}(z)]$.

Picking a basepoint $p \in \mathbb{C}$ and lifting loops gives a map:

$BM: \pi_1(\mathbb{C} \setminus B, p) \to \pi_1(\mathrm{Conf}_n(\mathbb{C}), [\pi|_C^{-1}(p) \subset \pi^{-1}(p)] )$

which is called the braid monodromy since $\pi_1(\mathrm{Conf}_n(\mathbb{C}))$ is the braid group on $n$ strands.

Suppose now that $C$ is smooth and moreover $\pi|_C$ has only simple ramification; i.e., the preimage of every point $B$ has cardinality $n-1$. Consider a disc $\mathbb{D} \subset \mathbb{C}$ with no ramification points on the boundary, and the braid $BM(\partial \mathbb{D})$. This is quasipositive'' -- i.e., it can be decomposed into a product of braids in which two of the points are exchanged by a counterclockwise half-twist. To see this, just factor $\partial \mathbb{D}$ into a sequence of loops each containing exactly one ramification point. Since a small deformation of any $C$ will be of this form, in fact the braid monodromy is always quasipositive.

However, in the knot theory literature, there is a rather stronger notion of positivity. The braid group on $n$ strands is generated by the $n-1$ counterclockwise half-twists $\tau_1,\ldots,\tau_{n-1}$ which exchange adjacent strands, and their inverses. (In the description of the braid group as $\pi_1(\mathrm{Conf}_n(\mathbb{C}))$, numbering the strands is done by fixing an ordering on the set of points corresponding to the basepoint.) A braid is said to be positive'' if it can be written as a product of the $\tau_i$.

Let $C_0$ be a (reduced) plane curve with singularity at the origin, and fix a projection $\pi: \mathbb{C}^2 \to \mathbb{C}$ as above. Consider a sufficiently small disc $\mathbb{D} \subset \mathbb{C}$ encircling the image of the singularity at the origin; in particular it should not contain the images of any other singularities or ramification points. It is known that $BM(\partial \mathbb{D})$ is not just quasipositive, but in fact positive. I would like a factorization into positive half-twists to be given in the following manner: first deform $C_0$ to some smooth $C$ such that $\pi|_C$ has simple ramification. Then factor $\partial \mathbb{D}$ into loops which each encircle one of these ramification points. Can this be done?

• Wasn't braid monodromy introduced a century earlier by Wirthinger? – Ryan Budney May 19 '11 at 20:01
• I have no idea. People always seem to say Moishezon, but if you have a reference I'll be happy to be corrected. – Vivek Shende May 19 '11 at 20:53
• This is the earliest occurance I know of: V. Puiseux, Recherches sur les fonctions algebriques, Journal des Mathematiques Pures et Appliquees (I) 15 (1850), 365--480. – Ryan Budney May 19 '11 at 22:03
• I got the reference out of M. Epple's survey in "History of Topology". It appears Wirthinger was the person who really ran with this idea, applying it to a larger class of algebraic varieties, and bringing in more connections with knot theory. – Ryan Budney May 19 '11 at 22:04