Determining plane curve singularities from its polars

This paper addresses a very classical topic that goes back at least to Pl\"ucker: how to understand a plane curve singularity using its polar curves. Here, we explicitly construct the singular points of a plane curve singularity directly from the weighted cluster of base points of its polars. In particular, we determine the equisingularity class (or topological equivalence class) of a germ of plane curve from the equisingularity class of generic polars and combinatorial data about the non-singular points shared by them.Comment: 22 pages. Final version, to appear in Advances in Mat

Experiència pràctica de comunicació de Matemàtiques a la ciutadania i a secundària

Peer Reviewe

Constancy regions of mixed multiplier ideals in two-dimensional local rings with rational singularities

The aim of this paper is to study mixed multiplier ideals associated with a tuple of ideals in a two-dimensional local ring with a rational singularity. We are interested in the partition of the real positive orthant given by the regions where the mixed multiplier ideals are constant. In particular we reveal which information encoded in a mixed multiplier ideal determines its corresponding jumping wall and we provide an algorithm to compute all the constancy regions, and their corresponding mixed multiplier ideals, in any desired range.Peer ReviewedPostprint (author's final draft

Developable surfaces with prescribed boundary

It is proved that a generic simple, closed, piecewise regular curve in space can be the boundary of only finitely many developable surfaces with nonvanishing mean curvature. The relevance of this result in the context of the dynamics of developable surfaces is discussed.Comment: Submitted to the conference 'Women in Geometry and Topology', held on September 25-27, 2019, at the Centre de Recerca Matem\`atica, Bellaterra, Spai

The ultrametric space of plane branches

We study properties of the space of irreducible germs of plane curves (branches), seen as an ultrametric space. We provide various geometrical methods to measure the distance between two branches and to compare distances between branches, in terms of topological invariants of the singularity which comprises some of the branches. We show that, in spite of being very close to the notion of intersection multiplicity between two germs, this notion of distance behaves very differently. For instance, any value in [0, 1] ¿ Q is attained as the distance between a fixed branch and some other branch, in contrast with the fact that the semigroup of the fixed branch has gaps. We also present results that lead to interpret this distance as a sort of geometric distance between the topological equivalence or equisingularity classes of branches.This research has been partially supported by the Spanish Committee for Science and Technology (CAICYT) through the project M2009-14163-C02-02, and the Catalan Research Commission through the project 2009 SGR 1284. Ignasi Abío and Víctor González-Alonso completed this work supported each of them by a postgraduate scholarship of the FPU program from the Spanish Ministerio de Educación y Ciencia.Peer Reviewe

Multiplicity and Poincaré series for mixed multiplier ideals

Postprint (published version

Depth from the visual motion of a planar target induced by zooming

Robot egomotion can be estimated from an acquired video stream up to the scale of the scene. To remove this uncertainty (and obtain true egomotion), a distance within the scene needs to be known. If no a priori knowledge on the scene is assumed, the usual solution is to derive “in some way” the initial distance from the camera to a target object. This paper proposes a new, very simple way to obtain such a distance, when a zooming camera is available and there is a planar target in the scene. Similarly to “two-grid calibration” algorithms, no estimation of the camera parameters is required, and no assumption on the optical axis stability between the different focal lengths is needed. Quite the reverse, the non stability of the optical axis between the different focal lengths is the key ingredient that enables to derive our depth estimate, by applying a result in projective geometry. Experiments carried out on a mobile robot platform show the promise of the approach.Peer Reviewe