Fine polar invariants of minimal singularities of surface

In two parts, we present a bigness criterion for the cotangent bundle of resolutions of orbifold surfaces of general type. As a corollary, we obtain the \textit{canonical model singularities} (CMS) criterion that can be applied to determine when a birational class of surfaces has smooth representatives with big cotangent bundle and compare it with other known criteria. We then apply this criterion to the problem of finding the minimal degrees $d$ for which the deformation equ...

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 survey on local additive invariants of real and complex definable singular germs we systematically present classical or more recent invariants of different nature as emerging from a tame degeneracy principle. For this goal, we associate to a given singular germ a specific deformation family whose geometry degenerates in such a way that it eventually gives rise to a list of invariants attached to this germ. Complex analytic invariants, real curvature invariants and mot...

In this work, we study families of singular surfaces in $\mathbb{C}^3$ parametrized by $\mathcal{A}$-finitely determined map germs. We consider the topological triviality and Whitney equisingularity of an unfolding $F$ of a finitely determined map germ $f:(\mathbb{C}^2,0)\rightarrow(\mathbb{C}^3,0)$. We investigate the following conjecture: topological triviality implies Whitney equisingularity of the unfolding $F$? We provide a complete answer to this conjecture, given count...

In this short note we prove two theorems, the first one is a sharpening of a result of Lange and Sernesi: the discriminant curve W of a general Abelian surface $A$ endowed with an irreducible polarization $D$ of type $(1,3)$ is an irreducible curve of degree $18$ whose singularities are exactly $36$ nodes and $72$ cusps. Moreover, we analyze the degeneration of the discriminant curve $W$ and its singularities as $A$ tends to the product of two equal elliptic curves. The sec...

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.

The purpose of this article is to study the deformations of smooth surfaces $X$ of general type whose canonical map is a finite, degree 2 morphism onto a minimal rational surface or onto $\mathbf F_1$, embedded in projective space by a very ample complete linear series. Among other things, we prove that any deformation of the canonical morphism of such surfaces $X$ is again a morphism of degree 2. A priori, this is not at all obvious, for the invariants $(p_g(X),c_1^2(X))$ of...

Let $(f,g): (S,s) \to (\mathbb{C}^2, 0)$ be a finite morphism from a germ of normal complex analytic surface to the germ of $\mathbb{C}^2$ at the origin. We show that the affine algebraic curve in $\mathbb{C}^2$ defined by the initial Newton polynomial of a defining series of the discriminant germ of $(f,g)$ depends up to toric automorphisms only on the germs of curves defined by $f$ and $g$. This result generalizes a theorem of Gryszka, Gwo\'zdziewicz and Parusi\'nski, which...

We present an algorithm which, given a deformation with trivial section of a reduced plane curve singularity, computes equations for the equisingularity stratum (that is, the mu-constant stratum in characteristic 0) in the parameter space of the deformation. The algorithm works for any, not necessarily reduced, parameter space and for algebroid curve singularities C defined over an algebraically closed field of characteristic 0 (or of characteristic p>ord(C)). It provides at ...

In this paper we develop the theory of equisingular deformations of plane curve singularities in arbitrary characteristic. We study equisingular deformations of the parametrization and of the equation and show that the base space of its semiuniveral deformation is smooth in both cases. Our approach through deformations of the parametrization is elementary and we show that equisingular deformations of the parametrization form a linear subfunctor of all deformations of the para...