arXiv subjects

Baldur Sigurðsson

Publications and source records attributed to Baldur Sigurðsson.

11 recordsLinked to original sources

arXiv2026

On the Jacobian polygon and Łojasiewicz exponent of isolated complex hypersurface singularities

Given a hypersurface singularity $(X,0) \subset (\mathbb{C}^{n+1},0)$ defined by a holomorphic function $f:(\mathbb{C}^{n+1},0) \to (\mathbb{C},0)$, we introduce an alternating version of Teissier's Jacobian Newton polygon, associated with a complex isolated hypersurface singularity, and prove formulas for both these invariants in terms of an embedded resolution of $(X,0)$. The formula for the alternating version has an advantage, in that for Newton nondegenerate functions, it can be calculated in terms of volumes of faces of the Newton diagram, whereas a similar formula for the original nonalternating version includes mixed volumes. The Milnor fiber can be given a handlebody decomposition, with handles corresponding to intersection points with the polar curve in generic plane sections of the singularity. This way we obtain a Morse-Smale complex. Teissier associates with each branch of the polar curve a vanishing rate, and we show that this induces a filtration of the Morse-Smale complex. We apply this result in order to calculate the Łojasiewicz exponent in terms of the alternating Jacobian polygon, but we expect it to be of further independent interest. In the case of a Newton nondegenerate hypersurface, our result produces a formula for the Łojasiewicz exponent in terms of Newton numbers of certain subdiagrams. This statement is related to a conjecture by Brzostowski, Krasiński and Oleksik, for which we provide a counterexample. Our formula for the Łojasiewicz exponent is based on a global calculation over the Newton diagram, rather than locally specifying a subset of the facets to consider, as in this conjecture. We conjecture a similar statement, which is based on our formula and inspired by the nonnegativity of local $h$-vectors.

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arXiv2025

On the integral variation map of isolated plane curves 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. This is very much inspired by a recent work of the authors. 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 availablecomputer program written in Python.

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arXiv2025

The Milnor fiber boundary of an arrangement determines its combinatorics

The boundary of the Milnor fiber associated with a complex line arrangement is a three dimensional plumbed manifold, and it is a combinatorial invariant. We prove the reverse implication, which was conjectured Némethi and Szilárd. That is, this boundary of the Milnor fiber determines the combinatorics of the arrangement. Furthermore, we give an explicit method which constructs the poset associated with the arrangement, given a plumbing graph in normal form for the boundary.

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arXiv2025

The total spine of the Milnor fibration of a plane curve singularity

For any plane curve singularity defined by an analytic function germ $f$, we construct a spine on each Milnor fiber simultaneously, that realizes the vanishing topology. In order to do so, we study the separatrices at the origin of the vector field $-\nabla \log |f|$. Under some genericity conditions on the metric, we produce a natural partition of the set of separatrices, $S$, into a finite collection smooth strata. As a byproduct of this theory, we construct a smooth fibration which is equivalent to the Milnor fibration, and lives on a quotient of the Milnor fibration at radius $0$. The strict transform of $S$ in this space induces the aforementioned spine for each fiber of this fibration. These fibers are naturally endowed with a vector field in such a way that the spine consists of trajectories which do not escape through the boundary.

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arXiv2021

Local Newton nondegenerate Weil divisors in toric varieties

We introduce and develop the theory of Newton nondegenerate local Weil divisors $(X,0)$ in toric affine varieties. We characterize in terms of the toric combinatorics of the Newton diagram different properties of such singular germs: normality, Gorenstein property, or being an Cartier divisor in the ambient space. We discuss certain properties of their (canonical) resolution $\tilde{X}\to X $ and the corresponding canonical divisor. We provide combinatorial formulae for the delta--invariant $δ(X,0)$ and for the cohomology groups $H^i(\tilde X,\mathcal{O}_{\tilde X})$ for $i>0$. In the case $\dim(X,0)=2$, we provide the (canonical) resolution graph from the Newton diagram and we also prove that if such a Weil divisor is normal and Gorenstein, and the link is a rational homology sphere, then the geometric genus is given by the minimal path cohomology, a topological invariant of the link.

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arXiv2020

Mixed tête-à-tête twists as monodromies associated with holomorphic function germs

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.

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arXiv2019

On ideal filtrations for Newton nondegenerate surface singularities

We compare three naturally occurring multi-indexed filtrations of ideals on the local ring of a Newton nondegenerate hypersurface surface singularity with rational homology sphere, which in many cases are all distinct. These are the divisorial, the order, and the image filtrations. These filtrations are indexed by the lattice associated with a toric partial resolution of the singularity, or equivalently, the free abelian group generated by the compact facets of the Newton polyhedron. We prove that there exists a top dimensional cone contained in the Lipman cone having the property that the three ideals indexed by order vectors from this cone coincide. As a corollary, if a periodic constant can be associated with the Hilbert series associated with these filtrations on the Lipman cone, then they coincide. In some cases, the Poincaré series associated with one of these filtrations has been shown to coincide with a zeta function associated with the topological type of the singularity. In the end of the article, we show that this is the case for all three filtrations considered in the case of a Newton nondegenerate suspension singularity. As a corollary, in this case, the zeta function provides a direct method of determining the Newton diagram from the link.

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arXiv2017

The boundary of the Milnor fiber of the singularity f(x,y) + zg(x,y) = 0

Let $f,g\in\mathbb{C}\{x,y\}$ be germs of functions defining plane curve singularities without common components in $(\mathbb{C}^2,0)$ and let $Φ(x,y,z) = f(x,y) + zg(x,y)$. We give an explicit algorithm producing a plumbing graph for the boundary of the Milnor fiber of $Φ$ in terms of a common resolution for $f$ and $g$.

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arXiv2016

The geometric genus and Seiberg-Witten invariant of Newton nondegenerate surface singularities

Given a normal surface singularity (X,0), its link, M is a closed differentiable three dimensional manifold which carries much analytic information. It is an interesting question to ask whether, under suitable analytic and topological conditions, the geometric genus (or other analytic invariants) can be recovered from the link. The Casson invariant conjecture predicts that p_g can be identified using the Casson invariant in the case when (X,0) is a complete intersection and M has trivial first homology with integral coefficients. The Seiberg-Witten invariant conjecture predicts that the geometric genus of a Gorenstein singularity, whose link has trivial first homology with rational coefficients, can be calculated as a normalized Seiberg-Witten invariant of the link. The first conjecture is still open, but counterexamples have been found for the second one. We prove here the Seiberg-Witten invariant conjecture for hypersurface singularities given by a function with Newton nondegenerate principal part. We provide a theory of computation sequences and of the way they bound the geometric genus. Newton nondegenerate singularities can be resolved explicitly by Oka's algorithm, and we exploit the combinatorial interplay between the resolution graph and the Newton diagram to show that in each step of the computation sequence we construct, the given bound is sharp. Our method recovers the geometric genus of (X,0) explicitly from the link, assuming that (X,0) is indeed Newton nondegenerate with a rational homology sphere link. Assuming some additional information about the Newton diagram, we recover part of the spectrum, as well as the Poincaré series associated with the Newton filtration. Finally, we show that the normalized Seiberg-Witten invariant associated with the canonical spin^c structure on the link coincides with our identification of the geometric genus.

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arXiv2014

The Milnor fiber of the singularity f(x,y) + zg(x,y) = 0

We give a description of the Milnor fiber and the monodromy of a singularity of the form f+zg = 0 where f and g define plane curves and have no common components. The description depends only on the topological type of the two plane curve germs defined by f and g. In particular, this gives a description of the boundary of the Milnor fiber. As a corollary, we give a simple formula for the monodromy zeta function and the Euler characteristic of the fiber.

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arXiv2013

The geometric genus of hypersurface singularities

Using the path lattice cohomology we provide a conceptual topological characterization of the geometric genus for certain complex normal surface singularities with rational homology sphere links, which is uniformly valid for all superisolated and Newton non--degenerate hypersurface singularities.

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