We prove that every reduced real plane curve singularity admits a real morsification. This settles a question of A'Campo and Gusein-Zade, later stated as conjectures by Leviant–Shustin and by Fomin–Pylyavskyy–Shustin–Thurston. In particular we overcome the main obstruction that remained posed by conjugate pairs of nonreal branches. Our new main ingredient is a construction that produces the divide from a nodal smoothing of two normalization disks. This is what we call the trace map. For real branches, it recovers Gusein-Zade's construction using Chebyshev polynomials. For pairs of complex conjugate branches with distinct tangents, the construction gives an explicit formula for the divide in terms of the Puiseux data. The general method consists in a delicate combination of the trace map with A'Campo's translations and contractions to produce divides and real morsifications for all reduced real plane curve singularities.

The integral variation map and algebraic monodromy of isolated plane curve singularities are important homological invariants of the singularity which are still far from being completely understood. This work provides effective ways of computing them with respect to an explicit geometric basis of the homology. For any given topological type of plane curve singularity, we construct an analytic model of it, along with a vector field on our version of its A'Campo space. This vector field is tangent to the Milnor fibers at radius zero and the union of the stable manifolds of their singularities yields a spine of each fiber, which can be described explicitly. Our first main contribution is the algorithmic computation of the algebraic monodromy and integral variation map as matrices with explicit bases for any Milnor fiber in the Milnor fibration, not merely congruence classes. For our second contribution, we introduce gyrographs which are graphs equipped with angular data and rational weights. We prove that the invariant spine naturally carries a gyrograph structure were the weights are given by the Hironaka numbers, and that this structure recovers the geometric monodromy as a homotopy class, as well as the integral variation map. This provides a combinatorial framework for computation by hand. Our methods are further implemented in a publicly available computer program written in Python.

The variation operator associated with an isolated hypersurface singularity is a classical topological invariant that relates relative and absolute homologies of the Milnor fiber via a non trivial isomorphism. Here we work with a topological version of this operator that deals with proper arcs and closed curves instead of homology cycles. Building on the classical framework of geometric vanishing cycles, we introduce the concept of vanishing arcsets as their counterpart using this geometric variation operator. We characterize which properly embedded arcs are sent to geometric vanishing cycles by the geometric variation operator in terms of intersections numbers of the arcs and their images by the geometric monodromy. Furthermore, we prove that for any distinguished collection of vanishing cycles arising from an A'Campo's divide, there exists a topological exceptional collection of arcsets whose variation images match this collection.

Generic relative immersions of compact one-manifolds in the closed unit disk, i.e. divides, provide a powerful combinatorial framework, and allow a topological construction of fibered classical links, for which the monodromy diffeomorphism is explicitly given as a product of Dehn twists. Complex isolated plane curve singularities provide a classical fibered link, the Milnor fibration, with its Milnor monodromy, monodromy group, and vanishing cycles. This survey puts together much of the work done on divides and their role in the topology of isolated plane curve singularities. We review two complementary approaches for constructing divides: one via embedded resolution techniques and controlled real deformations, and another via Chebyshev polynomials, which yield explicit real morsifications. A combinatorial description of the Milnor fiber is developed, leading to an explicit factorization of the geometric monodromy as a product of right-handed Dehn twists. We further explore the structure of reduction curves that arise from the Nielsen description of quasi-finite mapping classes and from iterated cabling operations on divides. The interplay between the geometric and integral homological monodromies is analyzed, with special attention to symmetries induced by complex conjugation and strong invertibility phenomena. In particular, the integral homological monodromy for isolated plane curve singularities can be computed effectively. In contrast, for complex hypersurface singularities in higher dimensions no method of computation of the integral homology monodromy is known. Connections with mapping class groups, contact and symplectic geometry, and Lefschetz fibrations are also discussed.

We study the separatrices at the origin of the vector field -\nabla \log |f| where f is any plane curve singularity. Under some genericity conditions on the metric we produce a natural partition of the set of separatrices S into segments and disks. As a byproduct of this theory we construct a smooth fibration equivalent to the Milnor fibration that lives on a quotient of the Milnor fibration at radius 0, and, furthermore we see how the strict transform of S in this space induces a spine for each Milnor fiber of this fibration.

We study the nilpotent part of certain pseudoperiodic automorphisms of surfaces appearing in singularity theory. We associate a quadratic form \tilde{Q} defined on the first (relative to the boundary) homology group of the Milnor fiber F of any germ analytic curve on a normal surface. Using the twist formula and techniques from mapping class group theory, we prove that the form \tilde{Q} obtained after killing \ker N is definite positive, and that its restriction to the absolute homology group of F is even whenever the Nielsen-Thurston graph of the monodromy automorphism is a tree. The form \tilde{Q} is computable in terms of the Nielsen-Thurston or the dual graph of the semistable reduction, as illustrated with several examples. Numerical invariants associated to \tilde{Q} are able to distinguish plane curve singularities with different topological types but same spectral pairs or Seifert form. Finally, we discuss a generic linear germ defined on a superisolated surface with not smooth ambient space.

Let f be an isolated plane curve singularity with Milnor fiber of genus at least 5. For all such f, we give (a) an intrinsic description of the geometric monodromy group that does not invoke the notion of the versal deformation space, and (b) an easy criterion to decide if a given simple closed curve in the Milnor fiber is a vanishing cycle or not. With the lone exception of singularities of type A_n and D_n, we find that both are determined completely by a canonical framing of the Milnor fiber induced by the Hamiltonian vector field associated to f. As a corollary we answer a question of Sullivan concerning the injectivity of monodromy groups for all singularities having Milnor fiber of genus at least 7.

We generalize a classical result concerning smooth germs of surfaces, by proving that monodromies on links of isolated complex surface singularities associated with reduced holomorphic map germs admit a positive factorization. As a consequence of this and a topological characterization of these monodromies by Anne Pichon, we conclude that a pseudoperiodic homeomorphism on an oriented surface with boundary with positive fractional Dehn twist coefficients and screw numbers, admits a positive factorization. We use the main theorem to give a sufficiency criterion for certain pseudoperiodic homeomorphisms with negative screw numbers to admit a positive factorization.

Tête-à-tête graphs were introduced by N. A'Campo in 2010 with the goal of modeling the monodromy of isolated plane curves. Mixed tête-à-tête graphs provide a generalization which define mixed tête-à-tête twists, which are pseudo-periodic automorphisms on surfaces. We characterize the mixed tête-à-tête twists as those pseudo-periodic automorphisms that have a power which is a product of right-handed Dehn twists around disjoint simple closed curves, including all boundary components. It follows that the class of tête-à-tête twists coincides with that of monodromies associated with reduced function germs on isolated complex surface singularities.

We study monodromies of plane curve singularities and pseudo-periodic homeomorphisms of oriented surfaces with boundary, following an original idea of the first author: tête-à-tête graphs and twists. We completely characterize mapping classes that can be represented by tête-à-tête twists, and generalize the notion to be able to represent any class of the mapping class group relative to the boundary which is boundary-free periodic. This improves previous work on the subject by C. Graf. Furthermore, we introduce the class of mixed tête-à-tête graphs and twists, and prove that mixed tête-à-tête twists contain monodromies of irreducible plane curve singularities. In a sequel paper, the fourth author and B. Sigurdsson have extended this to the reducible case.