Delta invariant of curves on rational surfaces I. The analytic approach

The article investigates the behaviour of the characteristic zero resolution invariant when transcribed suitably to the case of surfaces in positive characteristic. By Moh's jumping phenomenon -- or the occurrence of kangaroo singularities -- one knows that the invariant may increase, thus destroying any induction. We describe in the paper how one can modify in the purely inseparable surface case the invariant by adding a subtle correction term so as to prohibit its occasio...

We describe the topology of singular real algebraic curves in a smooth surface. We enumerate and bound in terms of the degree the number of topological types of singular algebraic curves in the real projective plane.

We estimate $\delta$-invariants of some singular del Pezzo surfaces with quotient singularities, which we studied ten years ago. As a result, we show that each of these surfaces admits an orbifold K\"ahler--Einstein metric.

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 $\delta$-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 ...

In this paper, we will give a precise formula to compute delta invariants of projective bundles and projective cones of Fano type.

We compute the $\delta$-invariant of a curve singularity parameterized by generic sparse polynomials. We apply this to describe topological types of generic singularities of sparse resultants and ``algebraic knot diagrams'' (i.e. generic algebraic spatial curve projections). Our approach is based on some new results on zero loci of Schur polynomials, on transversality properties of maps defined by sparse polynomials, and on a new refinement of the notion of tropicalization ...

We study deformations of rational curves and their singularities in positive characteristic. We use this to prove that if a smooth and proper surface in positive characteristic $p$ is dominated by a family of rational curves such that one member has all $\delta$-invariants (resp. Jacobian numbers) strictly less than $(p-1)/2$ (resp. $p$), then the surface has negative Kodaira dimension. We also prove similar, but weaker results hold for higher dimensional varieties. Moreover,...

In this paper we observe that the {\L}ojasiewicz exponent $\mathcal{L}_0(X)$ of an ADE-type singularity $X$ can be computed by means of invariants of certain ideals in the local ring ${\mathcal O}_{X,0}$. After extending the notion of {\L}ojasiewicz exponent to rational singularities of higher multiplicities we make a similar observation for RTP-type singularities.

In this paper we solve the problem of analytic classification of plane curves singularities with two branches by presenting their normal forms. This is accomplished by means of a new analytic invariant that relates vectors in the tangent space to the orbits under analytic equivalence in a given equisingularity class to K\"ahler differentials on the curve.

In this paper we extend the concept of Milnor fiber and Milnor number of a curve singularity allowing the ambient space to be a quotient surface singularity. A generalization of the local {\delta}-invariant is defined and described in terms of a Q-resolution of the curve singularity. In particular, when applied to the classical case (the ambient space is a smooth surface) one obtains a formula for the classical {\delta}-invariant in terms of a Q-resolution, which simplifies c...