Computing log canonical thresholds and equisingular deformations of plane curves

Nontechnical description:

A plane curve is defined as the locus of points in the (x,y)-plane satisfying a polynomial equation f(x,y)=0. Most people first encounter plane curves in high school analytic geometry. Circles, parabolas, ellipses, and hyperbolas are all examples, e.g., the polynomial equation xy-1=0 defines a hyperbola. A degenerate case is the union of the coordinate axes xy=0, which has a singularity where the axes cross.

The problem of classifying plane curves goes back to antiquity. Mathematicians use invariants to distinguish different curves. An example of such an invariant is the eccentricity of a conic section, which can be used to differentiate ellipses, parabolas, and hyperbolas. Our main focus is to describe how invariants of plane curve singularities can be computed effectively by a computer. This makes it possible to extract these invariants for large numbers of examples, and allows us to analyze their dependence on the coefficients of the defining polynomials.