arXiv subjects

Romain Bondil

Publications and source records attributed to Romain Bondil.

4 recordsLinked to original sources

arXiv2004

Fine polar invariants of minimal singularities of surface

We consider the polar curves $\PSO$ arising from generic projections of a germ $(S,0)$ of complex surface singularity onto $\C^2$. Taking $(S,0)$ to be a minimal singularity of normal surface (i.e. a rational singularity with reduced tangent cone), we give the $δ$-invariant of these polar curves, as well as the equisingularity-type of their generic plane projections, which are also the discriminants of generic projections of $(S,0)$. These two (equisingularity)-data for $\PSO$ are described in term, on the one side of the geometry of the tangent cone of $(S,0)$ and on the other side of the limit-trees introduced by T. de Jong and D. van Straten for the deformation theory of these minimal singularities. These trees give a combinatorial device for the description of the polar curve which makes it much clearer than in our previous Note on the subject. This previous work mainly relied on a result of M. Spivakovsky. Here we give a geometrical proof via deformations (on the tangent cone, and what we call Scott deformations) and blow-ups, although we need Spivakovsky's result at some point, extracting some other consequences of it along the way.

math.AG↗

arXiv2003

Geometry of superficial elements

In this paper, we study superficial elements of an ideal with respect to a module from a geometrical point of view, using blowing-ups. The notion of weak transform is particularly relevant to this study. We use this viewpoint to get a simple proof of a theorem by D. Kirby characterizing those superficial elements. We also indicate how the same result may be algebraically derived from a more recent theorem of Flenner and Vogel.

math.AC↗

arXiv2003

Discriminant of a generic projection of a minimal normal surface singularity

Let $(S,0)$ be a rational complex surface singularity with reduced fundamental cycle, also known as a {\em minimal} singularity. Using a fundamental result by M. Spivakovsky, we explain how to get a minimal resolution of the discriminant curve for a generic projection of $(S,0)$ onto $(\C^2,0)$ directly from the resolution graph of $(S,0)$.

math.AG↗

arXiv2003

General elements of an m-primary ideal on a normal surface singularity

In this paper, we show how to apply a theorem by Lê D.T. and the author about linear families of curves on normal surface singularities to get new results in this area. The main concept used is a specific definition of {\em general elements} of an ideal in the local ring of the surface. We make explicit the connection between this notion and the elementary notion of general element of a linear pencil, through the use of {\em reduction}. This allows us to prove the invariance of the generic Milnor number (resp. of the multiplicity of the discriminant), between two pencils generating two ideals with the same integral closure (resp. the projections associated). We also show that our theorem, applied in two special cases, on the one hand completes a theorem by Snoussi on the limits of tangent hyperplanes, and on the other hand gives an algebraic $μ$-constant theorem in linear families of planes curves.

math.AG↗