Non reduced plane curve singularities with $b_1(F)=0$ and Bobadilla's question
If the first Betti number of the Milnor fibre of a plane curve singularity is zero, then the defining function is equivalent to $x^r$.
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arXiv subjects
Dirk Siersma
Publications and source records attributed to Dirk Siersma.
Vanishing homology of projective hypersurfaces with 1-dimensional singularities
We introduce and study the vanishing homology of singular projective hypersurfaces. We prove its concentration in two levels in case of 1-dimensional singular locus $Σ$, and moreover determine the ranks of the nontrivial homology groups. These two groups depend on the monodromy at special points of $Σ$ and on the effect of the monodromy of the local system over its complement.
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On the area of constrained polygonal linkages
We study configuration spaces of linkages whose underlying graph are polygons with diagonal constrains, or more general, partial two-trees. We show that (with an appropriate definition) the oriented area is a Bott-Morse function on the configuration space. Its critical points are described and Bott-Morse indices are computed. This paper is a generalization of analogous results for polygonal linkages (obtained earlier by G. Khimshiashvili, G. Panina, and A. Zhukova).
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On the vanishing cycles of a meromorphic function on the complement of its poles
We study vanishing cycles naturally attached to a meromorphic function with isolated singularities, in both local and global settings.
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Motion planning and control of a planar polygonal linkage
For a polygonal linkage, we produce a fast navigation algorithm on its configuration space. The basic idea is to approximate the configuration space by the vertex-edge graph of its cell decomposition discovered by the first author. The algorithm has three aspects: (1) the number of navigation steps does not exceed 15 (independent of the linkage), (2) each step is a disguised flex of a quadrilateral from one triangular configuration to another, which is a well understood type of flex, and (3) each step can be performed explicitly by adding some extra bars and obtaining a mechanism with one degree of freedom.
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Milnor Fibre Homology via Deformation
In case of one-dimensional singular locus, we use deformations in order to get refined information about the Betti numbers of the Milnor fibre.
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Equilibria of point charges on convex curves
We study the equilibrium positions of three points on a convex curve under influence of the Coulomb potential. We identify these positions as orthotripods, three points on the curve having concurrent normals. This relates the equilibrium positions to the caustic (evolute) of the curve. The concurrent normals can only meet in the core of the caustic, which is contained in the interior of the caustic. Moreover, we give a geometric condition for three points in equilibrium with positive charges only. For the ellipse we show that the space of orthotripods is homeomorphic to a 2-dimensional bounded cylinder.
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Point Charges and Polygonal Linkages
We investigate the critical points of Coulomb potential of point charges placed at the vertices of a planar polygonal linkage. It is shown that, for a collection of positive charges on a pentagonal linkage, there is a unique critical point in the set of convex configurations which is the point of absolute minimum. This enables us to prove that two controlling charges are sufficient to navigate between any two convex configurations of a pentagonal linkage.
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Cross Ratios of Quadrilateral Linkages
We discuss the cross-ratio map of planar quadrilateral linkages, also in the case when one of the links is telescopic. Most of our results are valid for a planar quadrilateral linkage with generic lengths of the sides. In particular, we describe the image of cross-ratio map for quadrilateral linkage and planar robot 3-arm.
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Power diagrams and Morse Theory
We study Morse theory of the (power) distance function to a set of points in $\mathbb{R}^n$. We describe the topology of the union of the corresponding set of growing balls by a Morse poset. The Morse poset is related to the power tesselation of $\mathbb{R}^n$. We remark that the power diagrams from computer science are the spines of amoebas in algebraic geometry, or the hypersurfaces in tropical geometry. We show that there exists a discrete Morse function on the coherent triangulation, dual to the power diagram, such that its critical set equals the Morse poset of the power diagram.
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Extremal Configuration of Robot Arms in Three Dimensions
We define a volume function for a robot arms in 3-dimensional Euclidean space and give geometric conditions for its critical points. For 3-arms this volume function is an exact topological Morse function on the 3-sphere.
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Coulomb control of polygonal linkages
Equilibria of polygonal linkage with respect to Coulomb potential of point charges placed at the vertices of linkage are considered. It is proved that any convex configuration of a quadrilateral linkage is the point of global minimum of Coulomb potential for appropriate values of charges of vertices. Similar problems are treated for the equilateral pentagonal linkage. Some corollaries and applications in the spirit of control theory are also presented.
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Betti Bounds of Polynomials
We initiate a classification of complex polynomials f of degree d having the top Betti number of the general fibre close to the maximum. We find a range in which the polynomial must have isolated singularities and another range where it may have at most a line singularity of Morse transversal type, besides controlled singularities at infinity. Our method uses deformations into particular pencils with non-isolated singularities.
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Metric Properties of Conflict Sets
In this paper we show that the tangent cone of a conflict set in $R^n$ is a linear affine cone over a conflict set of smaller dimension and has dimension $n-1$. Moreover we give an example where the conflict sets is not normally embedded and not locally bi-Lipschitz equivalent to the corresponding tangent cone.
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Curvature and Gauss-Bonnet defect of global affine hypersurfaces
The total curvature of complex hypersurfaces in $\bC^{n+1}$ and its variation in families appear to depend not only on singularities but also on the behaviour in the neighbourhood of infinity. We find the asymptotic loss of total curvature towards infinity and we express the total curvature and the Gauss-Bonnet defect in terms of singularities and tangencies at infinity.
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Singularity exchange at the frontier of the space
In deformations of polynomial functions one may encounter ``singularity exchange at infinity'' when singular points disappear from the space and produce ``virtual'' singularities which have an influence on the topology of the limit polynomial. We find several rules of this exchange phenomenon, in which the total quantity of singularity turns out to be not conserved in general.
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Deformations of polynomials, boundary singularities and monodromy
We study the topology of polynomial functions by deforming them generically. We explain how the non-conservation of the total ``quantity'' of singularity in the neighbourhood of infinity is related to the variation of topology in certain families of boundary singularities along the hyperplane at infinity.
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