Local invariants of minimal generic curves on rational surfaces

We prove that if (C,0) is a reduced curve germ on a rational surface singularity (X,0) then its delta invariant can be recovered by a concrete expression associated with the embedded topological type of the pair (X,C). Furthermore, we also identify it with another (a priori) embedded analytic invariant, which is motivated by the theory of adjoint ideals. Finally, we connect our formulae with the local correction term at singular points of the global Riemann--Roch formula, val...

In this article we study abstract and embedded invariants of reduced curve germs via topological techniques. One of the most important numerical analytic invariants of an abstract curve is its delta invariant. Our primary goal is to develop delta invariant formulae for curves embedded in rational singularities in terms of embedded data. The topological machinery not only produces formulae, but it also creates deep connections with the theory of (analytical and topological) mu...

A viable and still unproved conjecture states that, if $X$ is a smooth algebraic surface and $C$ is a smooth algebraic curve in $X$, then $C$ realizes the smallest possible genus amongst all smoothly embedded $2$-manifolds in its homology class. A proof is announced here for this conjecture, for a large class of surfaces $X$, under the assumption that the normal bundle of $C$ has positive degree.

We consider families of schemes over arbitrary fields resp. analytic varieties with finitely many (not necessarily reduced) isolated non-normal singularities, in particular families of generically reduced curves. We define a modified delta invariant for isolated non-normal singularities of any dimension that takes care of embedded points and prove that it behaves upper semicontinuous in flat families parametrized by an arbitrary principal ideal domain. Moreover, if the fibers...

We complete the classification of local stability thresholds for smooth del Pezzo surfaces of degree 2. In particular, we show that this number is irrational if and only if the tangent plane at the point intersecting the surface is the union of a line and a smooth cubic curve meeting transversally at the point.

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 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...

In this article, we compute $\delta$-invariant of Du Val del Pezzo surfaces of degree $\ge 4$.

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.

This paper deals with a complete invariant $R_X$ for cyclic quotient surface singularities. This invariant appears in the Riemann Roch and Numerical Adjunction Formulas for normal surface singularities. Our goal is to give an explicit formula for $R_X$ based on the numerical information of $X$, that is, $d$ and $q$ as in $X=X(d;1,q)$. In the process, the space of curvettes and generic curves is explicitly described. We also define and describe other invariants of curves in $X...